{"id":270,"date":"2020-08-21T19:43:24","date_gmt":"2020-08-21T23:43:24","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/visualize-fractions\/"},"modified":"2025-09-12T16:41:39","modified_gmt":"2025-09-12T20:41:39","slug":"visualize-fractions","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/visualize-fractions\/","title":{"raw":"2.1 Visualize Fractions","rendered":"2.1 Visualize Fractions"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Find equivalent fractions<\/li>\r\n \t<li>Simplify fractions<\/li>\r\n \t<li>Multiply fractions<\/li>\r\n \t<li>Divide fractions<\/li>\r\n \t<li>Simplify expressions written with a fraction bar<\/li>\r\n \t<li>Translate phrases to expressions with fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1>Find Equivalent Fractions<\/h1>\r\n<strong data-effect=\"bold\">Fractions<\/strong> are a way to represent parts of a whole. The fraction \\(\\frac{1}{3}\\) means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_001\">(Figure 1)<\/a>. The fraction \\(\\frac{2}{3}\\) represents two of three equal parts. In the fraction \\(\\frac{2}{3}\\), the 2 is called the numerator and the 3 is called the denominator.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"230\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_ElemAlg_Figure_01_05_001_img_new.jpg\" alt=\"Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled \u201cone third\u201d in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.\" width=\"230\" height=\"106\" data-media-type=\"image\/jpeg\" \/> Figure 1.[\/caption]\r\n\r\nThe circle on the left has been divided into 3 equal parts. Each part is \\(\\frac{1}{3}\\) of the 3 equal parts. In the circle on the right, \\(\\frac{2}{3}\\) of the circle is shaded (2 of the 3 equal parts).\r\n<div id=\"fs-id1170653797209\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Fraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170653894126\">A <strong data-effect=\"bold\">fraction<\/strong> is written \\(\\frac{a}{b}\\), where \\(b\\ne 0\\) and<\/p>\r\n\r\n<ul id=\"fs-id1166424908857\" data-bullet-style=\"bullet\">\r\n \t<li><em data-effect=\"italics\">a<\/em> is the <strong data-effect=\"bold\">numerator<\/strong> and <em data-effect=\"italics\">b<\/em> is the <strong data-effect=\"bold\">denominator<\/strong>.<\/li>\r\n<\/ul>\r\nA fraction represents parts of a whole. The denominator <em data-effect=\"italics\">b<\/em> is the number of equal parts the whole has been divided into, and the numerator <em data-effect=\"italics\">a<\/em> indicates how many parts are included.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653931062\">If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate \\(\\frac{6}{6}\\) pieces, or, in other words, one whole pie.<\/p>\r\n<span id=\"fs-id1170654015188\" data-type=\"media\" data-alt=\"A circle is shown and is divided into six section. All sections are shaded.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_002_new.jpg\" alt=\"A circle is shown and is divided into six section. All sections are shaded.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170653905316\">So \\(\\frac{6}{6}=1\\). This leads us to the property of one that tells us that any number, except zero, divided by itself is 1<\/p>\r\n\r\n<div id=\"fs-id1170654116040\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Property of One<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n\\(\\begin{array}{cccc}\\frac{a}{a}=1\\hfill &amp; &amp; &amp; \\left(a\\ne 0\\right)\\hfill \\end{array}\\)\r\n\r\nAny number, except zero, divided by itself is one.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654211143\">If a pie was cut in \\(6\\) pieces and we ate all 6, we ate \\(\\frac{6}{6}\\) pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate \\(\\frac{8}{8}\\) pieces, or one whole pie. We ate the same amount\u2014one whole pie.<\/p>\r\n<p id=\"fs-id1170653885603\">The fractions \\(\\frac{6}{6}\\) and \\(\\frac{8}{8}\\) have the same value, 1, and so they are called equivalent fractions. <strong data-effect=\"bold\">Equivalent fractions<\/strong> are fractions that have the same value.<\/p>\r\nLet\u2019s think of pizzas this time. <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_003\">(Figure 2)<\/a> shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that \\(\\frac{1}{2}\\) is equivalent to \\(\\frac{4}{8}\\). In other words, they are <span class=\"no-emphasis\" data-type=\"term\">equivalent fractions<\/span>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"258\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_003_img_new.jpg\" alt=\"A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.\" width=\"258\" height=\"165\" data-media-type=\"image\/jpeg\" \/> Figure 2.[\/caption]\r\n\r\nSince the same amount is of each pizza is shaded, we see that \\(\\frac{1}{2}\\) is equivalent to \\(\\frac{4}{8}\\). They are equivalent fractions.\r\n<div id=\"fs-id1170653736114\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Equivalent Fractions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong data-effect=\"bold\">Equivalent fractions<\/strong> are fractions that have the same value.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nHow can we use mathematics to change \\(\\frac{1}{2}\\) into \\(\\frac{4}{8}?\\) How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into \\(8\\) pieces instead of just 2. Mathematically, what we\u2019ve described could be written like this as \\(\\frac{1\\cdot 4}{2\\cdot 4}=\\frac{4}{8}\\). See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_004_new\">(Figure 3)<\/a>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"119\"]<img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_004_new.jpg\" alt=\"A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.\" width=\"119\" height=\"119\" data-media-type=\"image\/jpeg\" \/> Figure 3.[\/caption]\r\n\r\nCutting each half of the pizza into \\(4\\) pieces, gives us pizza cut into 8 pieces: \\(\\frac{1\\cdot4}{2\\cdot4}=\\frac{4}{8}\\).\r\n<div id=\"CNX_ElemAlg_Figure_01_05_004_new\" class=\"bc-figure figure\"><\/div>\r\n<p id=\"fs-id1170654116758\">This model leads to the following property:<\/p>\r\n\r\n<div id=\"fs-id1170654068263\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Equivalent Fractions Property<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654016289\">If \\(a,b,c\\) are numbers where \\(b\\ne 0,c\\ne 0\\), then<\/p>\r\n\\(\\frac{a}{b}=\\frac{a\\cdot c}{b\\cdot c}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653751355\">If we had cut the pizza differently, we could get<\/p>\r\n<span id=\"fs-id1170653800497\" data-type=\"media\" data-alt=\"An image shows three rows of fractions. In the first row are the fractions \u201c1, times 2, divided by 2, times 2, equals two fourths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half, equals two fourths. The second row reads \u201c1, times 3, divided by 2 times 3, equals three sixths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, three sixths\u201d. The third row reads \u201c1 times 10, divided by 2 times 10, ten twentieths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, ten twentieths\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_005_img_new.jpg\" alt=\"An image shows three rows of fractions. In the first row are the fractions \u201c1, times 2, divided by 2, times 2, equals two fourths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half, equals two fourths. The second row reads \u201c1, times 3, divided by 2 times 3, equals three sixths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, three sixths\u201d. The third row reads \u201c1 times 10, divided by 2 times 10, ten twentieths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, ten twentieths\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654114644\">So, we say \\(\\frac{1}{2},\\frac{2}{4},\\frac{3}{6},\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{20}\\) are equivalent fractions.<\/p>\r\n\r\n<div id=\"fs-id1166424997376\" class=\"manipulative-math\" data-type=\"note\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654115770\" data-type=\"problem\">\r\n<p id=\"fs-id1170653910320\">Find three fractions equivalent to \\(\\frac{2}{5}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654069037\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1170654125509\">To find a fraction equivalent to \\(\\frac{2}{5}\\), we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let\u2019s multiply them by 2, 3, and then 5.<\/p>\r\n<span id=\"fs-id1170654014939\" data-type=\"media\" data-alt=\"A row of fractions reads \u201c2 times 2, divided by 5 times 2, equals four tenths\u201d. Next to this is \u201c2, times 3, divided by 5 times 3, equals six fifteenths\u201d. Next to this is \u201c2 times 5, divided by 5 times 5, equals ten twenty-fifths\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_006_img_new.jpg\" alt=\"A row of fractions reads \u201c2 times 2, divided by 5 times 2, equals four tenths\u201d. Next to this is \u201c2, times 3, divided by 5 times 3, equals six fifteenths\u201d. Next to this is \u201c2 times 5, divided by 5 times 5, equals ten twenty-fifths\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\nSo, \\(\\frac{4}{10},\\frac{6}{15},\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{25}\\) are equivalent to \\(\\frac{2}{5}\\).\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653906562\" data-type=\"problem\">\r\n<p id=\"fs-id1170653832505\">Find three fractions equivalent to \\(\\frac{3}{5}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653881636\" data-type=\"solution\"><details><summary>Show answer<\/summary>\\(\\frac{6}{10},\\frac{9}{15},\\frac{12}{20};\\) answers may vary\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654015429\" data-type=\"problem\">\r\n<p id=\"fs-id1170654013584\">Find three fractions equivalent to \\(\\frac{4}{5}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654061051\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653905408\">\\(\\frac{8}{10},\\frac{12}{15},\\frac{16}{20};\\) answers may vary<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Simplify Fractions<\/h1>\r\n<p id=\"fs-id1170654116001\">A fraction is considered <strong data-effect=\"bold\">simplified<\/strong> if there are no common factors, other than 1, in its numerator and denominator.<\/p>\r\n<p id=\"fs-id1170654027609\">For example,<\/p>\r\n\r\n<ul id=\"fs-id1166422535686\" data-bullet-style=\"bullet\">\r\n \t<li>\\(\\frac{2}{3}\\) is simplified because there are no common factors of 2 and 3.<\/li>\r\n \t<li>\\(\\frac{10}{15}\\) is not simplified because \\(5\\) is a common factor of 10 and 15.<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1170654047581\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Simplified Fraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA fraction is considered simplified if there are no common factors in its numerator and denominator.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654203066\">The phrase <em data-effect=\"italics\">reduce a fraction<\/em> means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.<\/p>\r\n<p id=\"fs-id1170654016508\">In <a class=\"autogenerated-content\" href=\"#fs-id1170654115770\">Example 1<\/a>, we used the equivalent fractions property to find equivalent fractions. Now we\u2019ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.<\/p>\r\n\r\n<div id=\"fs-id1170653882056\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Equivalent Fractions Property<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170653904200\">If \\(a,b,c\\) are numbers where \\(b\\ne 0,c\\ne 0\\),<\/p>\r\n\\(\\text{then}\\phantom{\\rule{1em}{0ex}}\\frac{a}{b}=\\frac{a \\cdot c}{b \\cdot c}\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\frac{a \\cdot c}{b \\cdot c}=\\frac{a}{b}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653886029\" data-type=\"problem\">\r\n<p id=\"fs-id1170654075477\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{32}{56}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653897505\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-id1169750893704\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction \u201cnegative thirty two fifty-sixths\u201d. The second row reads \u201cRewrite the numerator and denominator showing the common factors.\u201d Next to this in the right column reads \u201cnegative four times 8, divided by 7 times 8\u201d. The third row reads \u201cSimplify using the equivalent fractions properly\u201d. Next to this in the right column, it reads and \u201cnegative four sevenths\u201d.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{32}{56}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator showing the common factors.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169750893744\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_007_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify using the equivalent fractions property.<\/td>\r\n<td data-align=\"center\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{7}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the fraction \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{7}\\) is simplified because there are no more common factors.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1166424799823\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653897688\" data-type=\"problem\">\r\n<p id=\"fs-id1170654061444\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{42}{54}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654148628\" data-type=\"solution\"><details><summary>Show answer<\/summary>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{9}\\)\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653741575\" data-type=\"problem\">\r\n<p id=\"fs-id1170653898228\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{45}{81}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653808872\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653886874\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{9}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654060626\">Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into <span class=\"no-emphasis\" data-type=\"term\">prime number<\/span>s. Then divide out the common factors using the equivalent fractions property.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div data-type=\"title\">How to Simplify a Fraction<\/div>\r\n<div id=\"fs-id1170653749775\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654190237\" data-type=\"problem\">\r\n<p id=\"fs-id1170654147077\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{210}{385}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654117064\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<span id=\"fs-id1170653880713\" data-type=\"media\" data-alt=\"A table is shown with three columns and three rows. The first row of the left column reads \u201cStep 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree\u201d. Next to this in the middle column, it reads \u201crewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads \u201cnegative 210 divided by 385.\u201d Under this, is the equation \u201ctwo times three times five times seven.\u201d The five and 7 are blue and red respectively.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008a_new.jpg\" alt=\"A table is shown with three columns and three rows. The first row of the left column reads \u201cStep 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree\u201d. Next to this in the middle column, it reads \u201crewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads \u201cnegative 210 divided by 385.\u201d Under this, is the equation \u201ctwo times three times five times seven.\u201d The five and 7 are blue and red respectively.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655026820\" data-type=\"media\" data-alt=\"The next row down reads \u201cStep 2. Simplify using the equivalent fractions property by dividing out common factors.\u201d Next to this in the middle column, it reads, \u201cMark the common factors 5 and 7.\u201d Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation \u201cnegative two times 3 divided by 11.\u201d\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008b_new.jpg\" alt=\"The next row down reads \u201cStep 2. Simplify using the equivalent fractions property by dividing out common factors.\u201d Next to this in the middle column, it reads, \u201cMark the common factors 5 and 7.\u201d Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation \u201cnegative two times 3 divided by 11.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655164362\" data-type=\"media\" data-alt=\"The next row reads, \u201cStep 3. Multiply the remaining factors, if necessary.\u201d Next to this in the right column is negative six elevenths.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008c_new.jpg\" alt=\"The next row reads, \u201cStep 3. Multiply the remaining factors, if necessary.\u201d Next to this in the right column is negative six elevenths.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654221756\" data-type=\"problem\">\r\n<p id=\"fs-id1170653892448\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{69}{120}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654125391\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653742729\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{23}{40}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654010172\" data-type=\"problem\">\r\n<p id=\"fs-id1170653906336\">Simplify: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{120}{192}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654073679\" data-type=\"solution\"><details><summary>Show answer<\/summary>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{8}\\)\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653768282\">We now summarize the steps you should follow to simplify fractions.<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Simplify a Fraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1166424912916\" class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the numerator and denominator to show the common factors.\r\nIf needed, factor the numerator and denominator into prime numbers first.<\/li>\r\n \t<li>Simplify using the equivalent fractions property by dividing out common factors.<\/li>\r\n \t<li>Multiply any remaining factors, if needed.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170653905375\" class=\"howto\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653758697\" data-type=\"problem\">\r\n<p id=\"fs-id1170654031464\">Simplify: \\(\\frac{5x}{5y}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654007558\" data-type=\"solution\">\r\n<div data-type=\"title\"><strong>\r\nSolution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-id1169753205881\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction 5x divided by 5y\u201d. The second row reads \u201cRewrite showing the common factors, then divide out the factors.\u201d Next to this in the right column is a fraction equation \u201c5 times \u201cx\u201d, over five times \u201cy\u201d. The fives are crossed out in both the numerator and the denominator. The third row reads \u201cSimplify\u201d. Next to this in the right column, it reads \u201cx\u201d over \u201cy\u201d.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(\\frac{5x}{5y}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing the common factors, then divide out the common factors.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169753269750\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_009_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td data-align=\"center\">\\(\\frac{x}{y}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653888423\" data-type=\"problem\">\r\n<p id=\"fs-id1170653912078\">Simplify: \\(\\frac{7x}{7y}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653902694\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653879192\">\\(\\frac{x}{y}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653888423\" data-type=\"problem\">\r\n<p id=\"fs-id1170653912078\">Simplify: \\(\\frac{7x}{7y}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653902694\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653879192\">\\(\\frac{x}{y}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Multiply Fractions<\/h1>\r\n<p id=\"fs-id1170653834744\">Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.<\/p>\r\n<p id=\"fs-id1170653909812\">We\u2019ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let\u2019s start with \\(\\frac{3}{4}\\).<\/p>\r\n<span id=\"fs-id1170653895041\" data-type=\"media\" data-alt=\"A rectangle made up of four squares in a row. The first three squares are shaded.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_010_img_new.jpg\" alt=\"A rectangle made up of four squares in a row. The first three squares are shaded.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654080788\">Now we\u2019ll take \\(\\frac{1}{2}\\) of \\(\\frac{3}{4}\\).<\/p>\r\n<span id=\"fs-id1170653889891\" data-type=\"media\" data-alt=\"A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_011_img_new.jpg\" alt=\"A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170653869811\">Notice that now, the whole is divided into 8 equal parts. So \\(\\frac{1}{2}\\cdot\\frac{3}{4}=\\frac{3}{8}\\).<\/p>\r\nTo multiply fractions, we multiply the numerators and multiply the denominators.\r\n<div id=\"fs-id1170653801922\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Fraction Multiplication<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654115792\">If \\(a,b,c\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\) are numbers where \\(b\\ne 0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\ne 0\\), then<\/p>\r\n\\(\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{ac}{bd}\\)\r\n\r\nTo multiply fractions, multiply the numerators and multiply the denominators.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654066406\">When <span class=\"no-emphasis\" data-type=\"term\">multiplying fractions<\/span>, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the <span class=\"no-emphasis\" data-type=\"term\">product<\/span> as the first step. In <a class=\"autogenerated-content\" href=\"#fs-id1170654025836\">Example 5<\/a>, we will multiply negative and a positive, so the product will be negative.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654025836\" data-type=\"problem\">\r\n<p id=\"fs-id1170653750622\">Multiply: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{11}{12}\\cdot \\frac{5}{7}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654012761\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1170654030468\">The first step is to find the sign of the product. Since the signs are the different, the product is negative.<\/p>\r\n\r\n<table id=\"eip-201\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{11}{12}\\cdot\\frac{5}{7}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product; multiply.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{11\\cdot5}{12\\cdot7}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any common factors in the numerator and the demoninator? No.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{55}{84}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653750640\" data-type=\"problem\">\r\n<p id=\"fs-id1170653887277\">Multiply: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{28}\\cdot\\frac{8}{15}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654008561\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654202807\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{21}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654061006\" data-type=\"problem\">\r\n<p id=\"fs-id1170654061283\">Multiply: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{20}\\cdot\\frac{5}{12}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653900368\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654285392\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{16}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654026520\">When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, <em data-effect=\"italics\">a<\/em>, can be written as \\(\\frac{a}{1}\\). So, for example, \\(3=\\frac{3}{1}\\).<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653870152\" data-type=\"problem\">\r\n<p id=\"fs-id1170653907777\">Multiply: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{5}\\cdot \\left(-20x\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654080945\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1166420388500\">Determine the sign of the product. The signs are the same, so the product is positive.<\/p>\r\n\r\n<table id=\"eip-id1169753124719\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction twelve fifths times negative twenty \u201cx\u201d in parentheses. The second row reads \u201cWrite negative twenty \u201cx\u201d as a fraction.\u201d Next to this in the right column is a fraction equation 12 times 20 times x over five times 1. The third row reads \u201cShow the common factors and then remove them\u201d. Next to this in the right column, it shows negative twelve times 4 times 5 over 5 times 1. The fives are crossed out; one in the numerator and one in the denominator. The last row reads, \u201cMultiply the remaining factors; simplify. To the right of this is negative 48 \u201cx\u201d.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{5}\\cdot \\left(-20x\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write \\(20x\\) as a fraction.<\/td>\r\n<td data-align=\"center\">\\(\\frac{12}{5}\\cdot \\left(\\frac{20x}{1}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite 20 to show the common factor 5 and divide it out.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169747225310\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_012_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td data-align=\"center\">\\(48x\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654113920\" data-type=\"problem\">\r\n<p id=\"fs-id1170654148643\">Multiply: \\(\\frac{11}{3}\\cdot \\left(-9a\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653761249\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653743318\">\\(-33a\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654165872\" data-type=\"problem\">\r\n<p id=\"fs-id1170652633105\">Multiply: \\(\\frac{13}{7}\\cdot \\left(-14b\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654185150\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654129788\">\\(-2b\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Divide Fractions<\/h1>\r\n<p id=\"fs-id1170653750930\">Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.<\/p>\r\n<p id=\"fs-id1170654185172\">The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of \\(\\frac{2}{3}\\) is \\(\\frac{3}{2}\\).<\/p>\r\nNotice that \\(\\frac{2}{3}\\cdot \\frac{3}{2}=1\\). A number and its reciprocal multiply to 1.\r\n<p id=\"fs-id1170653770746\">To get a <span class=\"no-emphasis\" data-type=\"term\">product<\/span> of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.<\/p>\r\n<p id=\"fs-id1170654128489\">The reciprocal of \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{7}\\) is \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{10}\\), since \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{7}\\cdot \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{10}\\right)=1\\).<\/p>\r\n\r\n<div id=\"fs-id1170654286462\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Reciprocal<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <strong data-effect=\"bold\">reciprocal<\/strong> of \\(\\frac{a}{b}\\) is \\(\\frac{b}{a}\\).\r\n\r\nA number and its reciprocal multiply to one \\(\\frac{a}{b}\\cdot \\frac{b}{a}=1\\).\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654235623\">To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/p>\r\n\r\n<div id=\"fs-id1170652619282\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Fraction Division<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654189237\">If \\(a,b,c\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\) are numbers where \\(b\\ne 0,c\\ne 0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\ne 0\\), then<\/p>\r\n\\(\\frac{a}{b}\\div \\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653745038\">We need to say \\(b\\ne 0,c\\ne 0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\ne 0\\) to be sure we don\u2019t divide by zero!<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654125521\" data-type=\"problem\">\r\n<p id=\"fs-id1170653786596\">Divide: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\div \\frac{n}{5}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654239446\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-381\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\div \\frac{n}{5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>To divide, multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\cdot \\frac{5}{n}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{3n}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654068819\" data-type=\"problem\">\r\n<p id=\"fs-id1170653756083\">Divide: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{5}\\div \\frac{p}{7}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653924388\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653879363\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{21}{5p}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654114180\" data-type=\"problem\">\r\n<p id=\"fs-id1170654116769\">Divide: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{8}\\div \\frac{q}{3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654284311\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654239718\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{15}{8q}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654150092\" data-type=\"problem\">\r\n<p id=\"fs-id1170654022376\">Find the quotient: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{8}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{14}{27}\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654010810\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-id1169747240983\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation seven eighteenths divided by fourteen twenty sevenths. The second row reads \u201cMultiply the first fraction by the reciprocal of the second.\u201d Next to this in the right column is a fraction equation seven eighteenths times twenty seven fourteenths. The third row reads \u201cMultiply\u201d. Next to this in the right column, is a fraction equation seven times twenty seven over eighteen times fourteen. The next row reads \u201crewrite showing common factors. Next to this in the right column is seven times nine times three over nine times two times seven times two. The seven and the nines are crossed out in the numerator and denominator. The next row reads \u201crewrite showing common factors.\u201d Next to this is \u201c3 over 2 times 2.\u201d The last row reads \u201csimplify.\u201d Next to this in the right column is the fraction three fourths.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{18}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{14}{27}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>To divide, multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td data-align=\"center\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{18}\\cdot -\\phantom{\\rule{0.2em}{0ex}}\\frac{27}{14}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product, and then multiply..<\/td>\r\n<td data-align=\"center\">\\(\\frac{7\\cdot 27}{18\\cdot 14}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing common factors.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169750795691\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_013_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3}{2\\cdot 2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3}{4}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653800619\" data-type=\"problem\">\r\n<p id=\"fs-id1170654114238\">Find the quotient: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{27}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{35}{36}\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653898026\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653898028\">\\(\\frac{4}{15}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653790948\" data-type=\"problem\">\r\n<p id=\"fs-id1170653991340\">Find the quotient: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{14}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{15}{28}\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653767637\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654166274\">\\(\\frac{2}{3}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653930534\">There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.<\/p>\r\n\r\n<ul id=\"fs-id1166424848300\" data-bullet-style=\"bullet\">\r\n \t<li>\u201cTo multiply fractions, multiply the numerators and multiply the denominators.\u201d<\/li>\r\n \t<li>\u201cTo divide fractions, multiply the first fraction by the reciprocal of the second.\u201d<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170653795085\">Another way is to keep two examples in mind:<\/p>\r\n<span id=\"fs-id1170654007613\" data-type=\"media\" data-alt=\"This is an image with two columns. The first column reads \u201cOne fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the centre of each one representing one half. The halves are labeled \u201cone half\u201d. Under this is the equation \u201c2 times 1 fourth\u201d. Under this is another equation \u201ctwo over 1 times 1 fourth.\u201d Under this is the fraction two fourths and under this is the fraction one half. The next column reads \u201cthere are eight quarters in two dollars.\u201d Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation \u201ctwo over one divided by one fourth.\u201d Under this is two over one times four over one. Under this is the answer \u201c8\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_016_img_new.jpg\" alt=\"This is an image with two columns. The first column reads \u201cOne fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the centre of each one representing one half. The halves are labeled \u201cone half\u201d. Under this is the equation \u201c2 times 1 fourth\u201d. Under this is another equation \u201ctwo over 1 times 1 fourth.\u201d Under this is the fraction two fourths and under this is the fraction one half. The next column reads \u201cthere are eight quarters in two dollars.\u201d Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation \u201ctwo over one divided by one fourth.\u201d Under this is two over one times four over one. Under this is the answer \u201c8\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654286468\">The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a <strong data-effect=\"bold\">complex fraction<\/strong>.<\/p>\r\n\r\n<div id=\"fs-id1170653887308\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Complex Fraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA complex fraction is a fraction in which the numerator or the denominator contains a fraction.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653914822\">Some examples of complex fractions are:<\/p>\r\n\\(\\frac{\\frac{6}{7}}{3}\\phantom{\\rule{1em}{0ex}}\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\phantom{\\rule{1em}{0ex}}\\frac{\\frac{x}{2}}{\\frac{5}{6}}\\)\r\n<p id=\"fs-id1170652622719\">To simplify a complex fraction, we remember that the fraction bar means <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. For example, the complex fraction \\(\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\) means \\(\\frac{3}{4}\\div \\frac{5}{8}\\).<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654076126\" data-type=\"problem\">\r\n<p id=\"fs-id1170653897982\">Simplify: \\(\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653896387\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-id1169752928919\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation three fourths divided by five eighths. The second row on the left reads, \u201cRewrite as division.\u201d Next to this in the right column is a fraction equation three fourths divided by, using the division symbol, five eighths. The third row reads \u201cMultiply the first fraction by the reciprocal of the second\u201d. Next to this in the right column, is the fraction equation three fourths times five eighths. The next row down in the left column it has \u201cMultiply.\u201d Next to this in the right column, is the fraction equation three times eight divided by four times six. The next row reads \u201cLook for common factors. Next to this in the right column is the fraction equations three times four times two divided by four times six. The fours are crossed out in the numerator and denominator. The next row reads \u201cdivide out common factors and simplify.\u201d Next to this shows the fraction six fifths.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as division.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3}{4}\\div \\frac{5}{8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3}{4}\\cdot \\frac{8}{5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3\\cdot 8}{4\\cdot 5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169753189361\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_014_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide out common factors and simplify.<\/td>\r\n<td data-align=\"center\">\\(\\frac{6}{5}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654034116\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170654239534\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654122792\" data-type=\"problem\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654122792\" data-type=\"problem\">\r\n<p id=\"fs-id1170654122795\">Simplify: \\(\\frac{\\frac{2}{3}}{\\frac{5}{6}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654080854\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653901056\">\\(\\frac{4}{5}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653902067\" data-type=\"problem\">\r\n<p id=\"fs-id1170654128462\">Simplify: \\(\\frac{\\frac{3}{7}}{\\frac{6}{11}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654061219\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170652622093\">\\(\\frac{11}{14}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652618856\" data-type=\"problem\">\r\n<p id=\"fs-id1170654007229\">Simplify: \\(\\frac{\\frac{x}{2}}{\\frac{xy}{6}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170652631581\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-id1169754096098\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation \u201cx\u201d over 2 divided by \u201cxy\u201d over six. The second row on the left reads, \u201cRewrite as division.\u201d Next to this in the right column is a fraction equation \u201cx over 2 divided by, using the division symbol, \u201cxy\u201d over 6. The third row reads \u201cMultiply the first fraction by the reciprocal of the second\u201d. Next to this in the right column, is the fraction equation \u201cx\u201d over 2 times six over \u201cxy\u201d. The next row down in the left column it has \u201cMultiply.\u201d Next to this in the right column, is the fraction equation \u201cx\u201d times six divided by two times \u201cxy\u201d. The next row reads \u201cLook for common factors\u201d. Next to this in the right column is the fraction equation \u201cx\u201d times three times two divided by two times x times y\u201d The \u201cx\u201d\u2019s and 2\u2019s are crossed out in the numerator and denominator. The next row reads \u201cdivide out common factors and simplify.\u201d Next to this shows the fraction 3 over y.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\\(\\frac{\\frac{x}{2}}{\\frac{xy}{6}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as division.<\/td>\r\n<td data-align=\"center\">\\(\\frac{x}{2}\\div\\frac{xy}{6}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td data-align=\"center\">\\(\\frac{x}{2}\\cdot \\frac{6}{xy}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td data-align=\"center\">\\(\\frac{x\\cdot 6}{2\\cdot xy}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td data-align=\"center\"><span id=\"eip-id1169748377665\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_015_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide out common factors and simplify.<\/td>\r\n<td data-align=\"center\">\\(\\frac{3}{y}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170653902065\" data-type=\"exercise\">\r\n<div id=\"fs-id1170653902067\" data-type=\"problem\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 10.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654238021\" data-type=\"problem\">\r\n<p id=\"fs-id1170654238023\">Simplify: \\(\\frac{\\frac{a}{8}}{\\frac{ab}{6}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170652622839\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170652622841\">\\(\\frac{3}{4b}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654150057\" data-type=\"problem\">\r\n<p id=\"fs-id1170654150060\">Simplify: \\(\\dfrac{\\dfrac{p}{2}}{\\dfrac{pq}{8}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654234015\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654234017\">\\(\\frac{4}{q}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Simplify Expressions with a Fraction Bar<\/h1>\r\n<p id=\"fs-id1170653878324\">The line that separates the numerator from the denominator in a fraction is called a fraction bar. A <span class=\"no-emphasis\" data-type=\"term\">fraction bar<\/span> acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.<\/p>\r\n<p id=\"fs-id1170654032774\">To simplify the expression \\(\\frac{5-3}{7+1}\\), we first simplify the numerator and the denominator separately. Then we divide.<\/p>\r\n\r\n<div id=\"fs-id1170653894422\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\frac{5-3}{7+1}\\)<\/div>\r\n<div id=\"fs-id1170654068842\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\frac{2}{8}\\)<\/div>\r\n<div id=\"fs-id1170654150959\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\frac{1}{4}\\)<\/div>\r\n<div data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Simplify an Expression with a Fraction Bar<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1166424867009\" class=\"stepwise\" type=\"1\">\r\n \t<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\r\n \t<li>Simplify the fraction.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654286084\" class=\"howto\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654032705\" data-type=\"exercise\">\r\n<div id=\"fs-id1170653910175\" data-type=\"problem\">\r\n<p id=\"fs-id1170653910178\">Simplify: \\(\\frac{4-2\\left(3\\right)}{{2}^{2}+2}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653752520\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<\/div>\r\n<\/div>\r\n<table id=\"eip-149\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(\\frac{4-2\\left(3\\right)}{{2}^{2}+2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the order of operations to simpliy the numerator and the denominator.<\/td>\r\n<td>\\(\\frac{4-6}{4+2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerator and the denominator.<\/td>\r\n<td>\\(\\frac{-2}{6}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify. A negative divided by a positive is negative.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 11.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654283295\" data-type=\"problem\">\r\n<p id=\"fs-id1170654283297\">Simplify: \\(\\frac{6-3\\left(5\\right)}{{3}^{2}+3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654061370\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654061372\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653797687\" data-type=\"problem\">\r\n<p id=\"fs-id1170654154932\">Simplify: \\(\\frac{4-4\\left(6\\right)}{{3}^{2}+3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653990815\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653990817\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654237820\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170654283293\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654283295\" data-type=\"problem\">Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654185949\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccccc}\\frac{-1}{3}=-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\hfill &amp; &amp; &amp; &amp; &amp; \\frac{\\text{negative}}{\\text{positive}}=\\text{negative}\\hfill \\\\ \\frac{1}{-3}=-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\hfill &amp; &amp; &amp; &amp; &amp; \\frac{\\text{positive}}{\\text{negative}}=\\text{negative}\\hfill \\end{array}\\)<\/div>\r\n<div id=\"fs-id1170653878972\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Placement of Negative Sign in a Fraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654286245\">For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<\/p>\r\n\\(\\frac{-a}{b}=\\frac{a}{-b}=\\phantom{\\rule{0.2em}{0ex}}-\\frac{a}{b}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654066922\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652633027\" data-type=\"problem\">\r\n<p id=\"fs-id1170652633029\">Simplify: \\(\\frac{4\\left(-3\\right)+6\\left(-2\\right)}{-3\\left(2\\right)-2}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654286156\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-599\" summary=\"-\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(\\begin{array}{cccc}&amp; \\frac{4\\left(-3\\right)+6\\left(-2\\right)}{-3\\left(2\\right)-2}\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(\\frac{-12+\\left(-12\\right)}{-6-2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(\\frac{-24}{-8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>\\(3\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 12.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654235874\" data-type=\"problem\">\r\n<p id=\"fs-id1170653885349\">Simplify: \\(\\frac{8\\left(-2\\right)+4\\left(-3\\right)}{-5\\left(2\\right)+3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654038199\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654283278\">4<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654073995\" data-type=\"problem\">\r\n<p id=\"fs-id1170654073997\">Simplify: \\(\\frac{7\\left(-1\\right)+9\\left(-3\\right)}{-5\\left(3\\right)-2}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654060112\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654060114\">2<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Translate Phrases to Expressions with Fractions<\/h1>\r\n<p id=\"fs-id1170654164343\">Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.<\/p>\r\n<p id=\"fs-id1170653890410\">The English words <span class=\"no-emphasis\" data-type=\"term\">quotient<\/span> and <span class=\"no-emphasis\" data-type=\"term\">ratio<\/span> are often used to describe fractions. Remember that \u201cquotient\u201d means <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. The quotient of \\(a\\) and \\(b\\) is the result we get from dividing \\(a\\) by \\(b\\), or \\(\\frac{a}{b}\\).<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654281372\" data-type=\"problem\">\r\n<p id=\"fs-id1170654281374\">Translate the English phrase into an algebraic expression: the quotient of the difference of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>, and <em data-effect=\"italics\">p<\/em>.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654154539\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1170654235420\">We are looking for the <em data-effect=\"italics\">quotient<\/em> <em data-effect=\"italics\">of<\/em> the difference of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>, <em data-effect=\"italics\">and<\/em> <em data-effect=\"italics\">p<\/em>. This means we want to divide the difference of \\(m\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}n\\phantom{\\rule{0.2em}{0ex}}\\text{by}\\phantom{\\rule{0.2em}{0ex}}p\\).<\/p>\r\n\r\n<div id=\"fs-id1166425083474\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\frac{m-n}{p}\\)<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 13.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1170652623005\">Translate the English phrase into an algebraic expression: the quotient of the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">cd<\/em>.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654292142\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654292144\">\\(\\frac{a-b}{cd}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 13.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653794596\" data-type=\"problem\">\r\n<p id=\"fs-id1170654150368\">Translate the English phrase into an algebraic expression: the quotient of the sum of \\(p\\) and \\(q\\), and \\(r\\)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653904815\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653904817\">\\(\\frac{p+q}{r}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"fs-id1170654149151\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Equivalent Fractions Property:<\/strong> If \\(a,b,c\\) are numbers where \\(b\\ne 0,c\\ne 0\\), then\r\n\\(\\frac{a}{b}=\\frac{a \\cdot c}{b \\cdot c}\\) and \\(\\frac{a \\cdot c}{b \\cdot c}=\\frac{a}{b}\\).<\/li>\r\n \t<li><strong data-effect=\"bold\">Fraction Division:<\/strong> If \\(a,b,c\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\) are numbers where \\(b\\ne 0,c\\ne 0,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\ne 0\\), then \\(\\frac{a}{b}\\div \\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}\\). To divide fractions, multiply the first fraction by the reciprocal of the second.<\/li>\r\n \t<li><strong data-effect=\"bold\">Fraction Multiplication:<\/strong> If \\(a,b,c\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\) are numbers where \\(b\\ne 0,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d\\ne 0\\), then \\(\\frac{a}{b} \\cdot \\frac{c}{d}=\\frac{ac}{bd}\\). To multiply fractions, multiply the numerators and multiply the denominators.<\/li>\r\n \t<li><strong data-effect=\"bold\">Placement of Negative Sign in a Fraction:<\/strong> For any positive numbers \\(a\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}b\\), \\(\\frac{-a}{b}=\\frac{a}{-b}=-\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{b}\\).<\/li>\r\n \t<li><strong data-effect=\"bold\">Property of One:<\/strong>\\(\\frac{a}{a}=1;\\) Any number, except zero, divided by itself is one.<\/li>\r\n \t<li><strong data-effect=\"bold\">Simplify a Fraction<\/strong>\r\n<ol class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.<\/li>\r\n \t<li>Simplify using the equivalent fractions property by dividing out common factors.<\/li>\r\n \t<li>Multiply any remaining factors.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Simplify an Expression with a Fraction Bar<\/strong>\r\n<ol id=\"fs-id1170654237200\" class=\"stepwise\" type=\"1\">\r\n \t<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\r\n \t<li>Simplify the fraction.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<h1>Glossary<\/h1>\r\n<div class=\"textbox shaded\">\r\n<dl id=\"fs-id1166424866332\">\r\n \t<dt>complex fraction<\/dt>\r\n \t<dd id=\"fs-id1166424866337\">A complex fraction is a fraction in which the numerator or the denominator contains a fraction.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166424866347\">\r\n \t<dt>denominator<\/dt>\r\n \t<dd id=\"fs-id1166425223004\">The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166425223009\">\r\n \t<dt>equivalent fractions<\/dt>\r\n \t<dd id=\"fs-id1166425223015\">Equivalent fractions are fractions that have the same value.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166425223019\">\r\n \t<dt>fraction<\/dt>\r\n \t<dd id=\"fs-id1166422580993\">A fraction is written \\(\\frac{a}{b}\\), where \\(b\\ne 0\\) \\(a\\) is the numerator and \\(b\\) is the denominator. A fraction represents parts of a whole. The denominator \\(b\\) is the number of equal parts the whole has been divided into, and the numerator \\(a\\) indicates how many parts are included.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166424896994\">\r\n \t<dt>numerator<\/dt>\r\n \t<dd id=\"fs-id1166424897000\">The numerator is the value on the top part of the fraction that indicates how many parts of the whole are included.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166424844292\">\r\n \t<dt>reciprocal<\/dt>\r\n \t<dd id=\"fs-id1166424844298\">The reciprocal of \\(\\frac{a}{b}\\) is \\(\\frac{b}{a}\\). A number and its reciprocal multiply to one: \\(\\frac{a}{b}\\cdot \\frac{b}{a}=1\\).<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166422601976\">\r\n \t<dt>simplified fraction<\/dt>\r\n \t<dd id=\"fs-id1166422601981\">A fraction is considered simplified if there are no common factors in its numerator and denominator.<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h1>Practice Makes Perfect<\/h1>\r\n<h2>Find Equivalent Fractions<\/h2>\r\n<p id=\"fs-id1166424799887\">In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">1. \\(\\frac{3}{8}\\)<\/td>\r\n<td style=\"width: 50%;\">2. \\(\\frac{5}{8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">3. \\(\\frac{5}{9}\\)<\/td>\r\n<td style=\"width: 50%;\">4. \\(\\frac{1}{8}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Simplify Fractions<\/h2>\r\n<p id=\"fs-id1166422693579\">In the following exercises, simplify.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 90px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">5. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{40}{88}\\)<\/td>\r\n<td style=\"width: 50%; height: 15px;\">6. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{63}{99}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">7. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{108}{63}\\)<\/td>\r\n<td style=\"width: 50%; height: 15px;\">8. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{104}{48}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">9. \\(\\frac{120}{252}\\)<\/td>\r\n<td style=\"width: 50%; height: 15px;\">10. \\(\\frac{182}{294}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">11. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3x}{12y}\\)<\/td>\r\n<td style=\"width: 50%; height: 15px;\">12. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4x}{32y}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">13. \\(\\frac{14{x}^{2}}{21y}\\)<\/td>\r\n<td style=\"width: 50%; height: 15px;\">14. \\(\\frac{24a}{32{b}^{2}}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Multiply Fractions<\/h2>\r\n<p id=\"fs-id1166424880356\">In the following exercises, multiply.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">15. \\(\\frac{3}{4}\\cdot \\frac{9}{10}\\)<\/td>\r\n<td style=\"width: 50%;\">16. \\(\\frac{4}{5}\\cdot \\frac{2}{7}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">17. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\cdot\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">18. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}\\cdot\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{9}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">19. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{9}\\cdot \\frac{3}{10}\\)<\/td>\r\n<td style=\"width: 50%;\">20. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}\\cdot \\frac{4}{15}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">21. \\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{14}{15}\\right) \\cdot \\left(\\frac{9}{20}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">22. \\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{10}\\right) \\cdot \\left(\\frac{25}{33}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">23. \\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{63}{84}\\right)\\cdot \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{44}{90}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">24. \\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{63}{60}\\right) \\cdot \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{40}{88}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">25. \\(4\\cdot \\frac{5}{11}\\)<\/td>\r\n<td style=\"width: 50%;\">26. \\(5\\cdot \\frac{8}{3}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">27. \\(\\frac{3}{7}\\cdot 21n\\)<\/td>\r\n<td style=\"width: 50%;\">28. \\(\\frac{5}{6}\\cdot 30m\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">29. \\(-8\\cdot \\left(\\frac{17}{4}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">30. \\(\\left(-1\\right)\\cdot\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{6}{7}\\right)\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Divide Fractions<\/h2>\r\n<p id=\"fs-id1166422785479\">In the following exercises, divide.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">31. \\(\\frac{3}{4}\\div \\frac{2}{3}\\)<\/td>\r\n<td style=\"width: 50%;\">32. \\(\\frac{4}{5}\\div \\frac{3}{4}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">33. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{9}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{4}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">34. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{6}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{6}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">35. \\(\\frac{3}{4}\\div \\frac{x}{11}\\)<\/td>\r\n<td style=\"width: 50%;\">36. \\(\\frac{2}{5}\\div \\frac{y}{9}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">37. \\(\\frac{5}{18}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{15}{24}\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">38. \\(\\frac{7}{18}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{14}{27}\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">39. \\(\\frac{8u}{15}\\div \\frac{12v}{25}\\)<\/td>\r\n<td style=\"width: 50%;\">40. \\(\\frac{12r}{25}\\div \\frac{18s}{35}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">41. \\(-5\\div \\frac{1}{2}\\)<\/td>\r\n<td style=\"width: 50%;\">42. \\(-3\\div \\frac{1}{4}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">43. \\(\\frac{3}{4}\\div \\left(-12\\right)\\)<\/td>\r\n<td style=\"width: 50%;\">44. \\(-15\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{3}\\right)\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170654215028\">In the following exercises, simplify.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">45. \\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{21}}{\\frac{12}{35}}\\)<\/td>\r\n<td style=\"width: 50%;\">46. \\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{16}}{\\frac{33}{40}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">47. \\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}}{2}\\)<\/td>\r\n<td style=\"width: 50%;\">48. \\(\\frac{5}{\\frac{3}{10}}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">49. \\(\\frac{\\frac{m}{3}}{\\frac{n}{2}}\\)<\/td>\r\n<td style=\"width: 50%;\">50. \\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}}{-\\phantom{\\rule{0.2em}{0ex}}\\frac{y}{12}}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Simplify Expressions Written with a Fraction Bar<\/h2>\r\n<p id=\"fs-id1166424900706\">In the following exercises, simplify.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">51. \\(\\frac{22+3}{10}\\)<\/td>\r\n<td style=\"width: 50%;\">52. \\(\\frac{19-4}{6}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">53. \\(\\frac{48}{24-15}\\)<\/td>\r\n<td style=\"width: 50%;\">54. \\(\\frac{46}{4+4}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">55. \\(\\frac{-6+6}{8+4}\\)<\/td>\r\n<td style=\"width: 50%;\">56. \\(\\frac{-6+3}{17-8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">57. \\(\\frac{4\\cdot3}{6\\cdot6}\\)<\/td>\r\n<td style=\"width: 50%;\">58. \\(\\frac{6\\cdot6}{9\\cdot2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">59. \\(\\frac{{4}^{2}-1}{25}\\)<\/td>\r\n<td style=\"width: 50%;\">60. \\(\\frac{{7}^{2}+1}{60}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">61. \\(\\frac{8\\cdot3+2\\cdot9}{14+3}\\)<\/td>\r\n<td style=\"width: 50%;\">62. \\(\\frac{9\\cdot6-4\\cdot7}{22+3}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">63. \\(\\frac{5\\cdot6-3\\cdot4}{4\\cdot5-2\\cdot3}\\)<\/td>\r\n<td style=\"width: 50%;\">64. \\(\\frac{8\\cdot9-7\\cdot6}{5\\cdot6-9\\cdot2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">65. \\(\\frac{{5}^{2}-{3}^{2}}{3-5}\\)<\/td>\r\n<td style=\"width: 50%;\">66. \\(\\frac{{6}^{2}-{4}^{2}}{4-6}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">67. \\(\\frac{7\\cdot4-2\\left(8-5\\right)}{9\\cdot3-3\\cdot5}\\)<\/td>\r\n<td style=\"width: 50%;\">68. \\(\\frac{9\\cdot7-3\\left(12-8\\right)}{8\\cdot7-6\\cdot6}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">69. \\(\\frac{9\\left(8-2\\right)-3\\left(15-7\\right)}{6\\left(7-1\\right)-3\\left(17-9\\right)}\\)<\/td>\r\n<td style=\"width: 50%;\">70. \\(\\frac{8\\left(9-2\\right)-4\\left(14-9\\right)}{7\\left(8-3\\right)-3\\left(16-9\\right)}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Translate Phrases to Expressions with Fractions<\/h2>\r\n<p id=\"fs-id1166425080667\">In the following exercises, translate each English phrase into an algebraic expression.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">71. the quotient of <em data-effect=\"italics\">r<\/em> and the sum of <em data-effect=\"italics\">s<\/em> and 10<\/td>\r\n<td style=\"width: 50%;\">72. the quotient of <em data-effect=\"italics\">A<\/em> and the difference of 3 and <em data-effect=\"italics\">B<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">73. the quotient of the difference of \\(x\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}y,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}-3\\)<\/td>\r\n<td style=\"width: 50%;\">74. the quotient of the sum of \\(m\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}n,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}4q\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Everyday Math<\/h2>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">75. <strong data-effect=\"bold\">Baking.<\/strong> A recipe for chocolate chip cookies calls for \\(\\frac{3}{4}\\) cup brown sugar. Imelda wants to double the recipe. a) How much brown sugar will Imelda need? Show your calculation. b) Measuring cups usually come in sets of \\(\\frac{1}{4},\\frac{1}{3},\\frac{1}{2},\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}1\\) cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.<\/td>\r\n<td style=\"width: 50%;\">76. <strong data-effect=\"bold\">Baking.<\/strong> Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs \\(\\frac{2}{3}\\) cup of condensed milk. a) How much condensed milk will Nina need? Show your calculation. b) Measuring cups usually come in sets of \\(\\frac{1}{4},\\frac{1}{3},\\frac{1}{2},\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}1\\) cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for \\(4\\) pans of fudge.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">77. <strong data-effect=\"bold\">Portions<\/strong> Don purchased a bulk package of candy that weighs \\(5\\) pounds. He wants to sell the candy in little bags that hold \\(\\frac{1}{4}\\) pound. How many little bags of candy can he fill from the bulk package?<\/td>\r\n<td style=\"width: 50%;\">78. <strong data-effect=\"bold\">Portions<\/strong> Kristen has \\(\\frac{3}{4}\\) yards of ribbon that she wants to cut into \\(6\\) equal parts to make hair ribbons for her daughter\u2019s 6 dolls. How long will each doll\u2019s hair ribbon be?<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Writing Exercises<\/h2>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn\u2019t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael\u2019s reasoning.<\/td>\r\n<td style=\"width: 50%;\">80. Give an example from everyday life that demonstrates how \\(\\frac{1}{2}\\cdot \\frac{2}{3}\\phantom{\\rule{0.2em}{0ex}}\\text{is}\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}\\).<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">81. Explain how you find the reciprocal of a fraction.<\/td>\r\n<td style=\"width: 50%;\">82. Explain how you find the reciprocal of a negative number.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Answers<\/h1>\r\n<div data-type=\"glossary\">\r\n<table style=\"border-collapse: collapse; width: 100%; height: 154px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 35px;\">\r\n<td style=\"width: 33.3333%; height: 35px;\">1. \\(\\frac{6}{16},\\frac{9}{24},\\frac{12}{32}\\) answers may vary<\/td>\r\n<td style=\"width: 33.3333%; height: 35px;\">3. \\(\\frac{10}{18},\\frac{15}{27},\\frac{20}{36}\\) answers may vary<\/td>\r\n<td style=\"width: 33.3333%; height: 35px;\">5. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{11}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">7. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{7}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">9. \\(\\frac{10}{21}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">11. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{x}{4y}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">13. \\(\\frac{2{x}^{2}}{3y}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">15. \\(\\frac{27}{40}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">17. \\(\\frac{1}{4}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">19. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{6}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">21. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{21}{50}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">23. \\(\\frac{11}{30}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">25. \\(\\frac{20}{11}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">27. 9<em data-effect=\"italics\">n<\/em><\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">29. \\(-34\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">31. \\(\\frac{9}{8}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">33. \\(1\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">35. \\(\\frac{33}{4x}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">37. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{9}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">39. \\(\\frac{10u}{9v}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">41. \\(-10\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 33.3333%; height: 17px;\">43. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{16}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">45. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{9}\\)<\/td>\r\n<td style=\"width: 33.3333%; height: 17px;\">47. \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">49. \\(\\frac{2m}{3n}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">51. \\(\\frac{5}{2}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">53. \\(\\frac{16}{3}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">55. 0<\/td>\r\n<td style=\"width: 33.3333%;\">57. \\(\\frac{1}{3}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">59. \\(\\frac{3}{5}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">61. \\(2\\frac{8}{17}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">63. \\(\\frac{3}{5}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">65. \\(-8\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">67. \\(\\frac{11}{6}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">69. \\(\\frac{5}{2}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">71. \\(\\frac{r}{s+10}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">73. \\(\\frac{x-y}{-3}\\)<\/td>\r\n<td style=\"width: 33.3333%;\">75. a) \\(1\\frac{1}{2}\\) cups b) answers will vary<\/td>\r\n<td style=\"width: 33.3333%;\">77. 20 bags<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">79. Answers may vary.<\/td>\r\n<td style=\"width: 33.3333%;\">81.\u00a0Answers may vary.<\/td>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Attributions<\/h1>\r\nThis chapter has been adapted from \u201cVisualize Fractions\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.\r\n\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Find equivalent fractions<\/li>\n<li>Simplify fractions<\/li>\n<li>Multiply fractions<\/li>\n<li>Divide fractions<\/li>\n<li>Simplify expressions written with a fraction bar<\/li>\n<li>Translate phrases to expressions with fractions<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Find Equivalent Fractions<\/h1>\n<p><strong data-effect=\"bold\">Fractions<\/strong> are a way to represent parts of a whole. The fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_001\">(Figure 1)<\/a>. The fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> represents two of three equal parts. In the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>, the 2 is called the numerator and the 3 is called the denominator.<\/p>\n<figure style=\"width: 230px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_ElemAlg_Figure_01_05_001_img_new.jpg\" alt=\"Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled \u201cone third\u201d in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.\" width=\"230\" height=\"106\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1.<\/figcaption><\/figure>\n<p>The circle on the left has been divided into 3 equal parts. Each part is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the 3 equal parts. In the circle on the right, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the circle is shaded (2 of the 3 equal parts).<\/p>\n<div id=\"fs-id1170653797209\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Fraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170653894126\">A <strong data-effect=\"bold\">fraction<\/strong> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8459b97fa73367952be2ca8cf9d1326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1edd53778d94abb1dc1e19acff79e9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<ul id=\"fs-id1166424908857\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">a<\/em> is the <strong data-effect=\"bold\">numerator<\/strong> and <em data-effect=\"italics\">b<\/em> is the <strong data-effect=\"bold\">denominator<\/strong>.<\/li>\n<\/ul>\n<p>A fraction represents parts of a whole. The denominator <em data-effect=\"italics\">b<\/em> is the number of equal parts the whole has been divided into, and the numerator <em data-effect=\"italics\">a<\/em> indicates how many parts are included.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653931062\">If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ccc0c204e595c09ce538eab73745899a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> pieces, or, in other words, one whole pie.<\/p>\n<p><span id=\"fs-id1170654015188\" data-type=\"media\" data-alt=\"A circle is shown and is divided into six section. All sections are shaded.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_002_new.jpg\" alt=\"A circle is shown and is divided into six section. All sections are shaded.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170653905316\">So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-262014b69a980b5c81990687cc9776a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/>. This leads us to the property of one that tells us that any number, except zero, divided by itself is 1<\/p>\n<div id=\"fs-id1170654116040\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Property of One<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c414eeb51ec6bd1895d4524d2a6bd8ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#97;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#110;&#101;&#32;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"145\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>Any number, except zero, divided by itself is one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654211143\">If a pie was cut in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> pieces and we ate all 6, we ate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ccc0c204e595c09ce538eab73745899a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6f2ea1e2966af14aa3a36d2ac5f3fd95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> pieces, or one whole pie. We ate the same amount\u2014one whole pie.<\/p>\n<p id=\"fs-id1170653885603\">The fractions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ccc0c204e595c09ce538eab73745899a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6f2ea1e2966af14aa3a36d2ac5f3fd95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> have the same value, 1, and so they are called equivalent fractions. <strong data-effect=\"bold\">Equivalent fractions<\/strong> are fractions that have the same value.<\/p>\n<p>Let\u2019s think of pizzas this time. <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_003\">(Figure 2)<\/a> shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9895c721c562367fb4b8dbdb7769533b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>. In other words, they are <span class=\"no-emphasis\" data-type=\"term\">equivalent fractions<\/span>.<\/p>\n<figure style=\"width: 258px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_003_img_new.jpg\" alt=\"A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.\" width=\"258\" height=\"165\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2.<\/figcaption><\/figure>\n<p>Since the same amount is of each pizza is shaded, we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9895c721c562367fb4b8dbdb7769533b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>. They are equivalent fractions.<\/p>\n<div id=\"fs-id1170653736114\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Equivalent Fractions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong data-effect=\"bold\">Equivalent fractions<\/strong> are fractions that have the same value.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>How can we use mathematics to change <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d2ecee3681f5171567b577d834a9003a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"17\" style=\"vertical-align: -6px;\" \/> How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23212a120eb92d226d44696a0b80bead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> pieces instead of just 2. Mathematically, what we\u2019ve described could be written like this as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8da05f3b007409e67d5b44c262ffc5e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/>. See <a class=\"autogenerated-content\" href=\"#CNX_ElemAlg_Figure_01_05_004_new\">(Figure 3)<\/a>.<\/p>\n<figure style=\"width: 119px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_004_new.jpg\" alt=\"A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.\" width=\"119\" height=\"119\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3.<\/figcaption><\/figure>\n<p>Cutting each half of the pizza into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/> pieces, gives us pizza cut into 8 pieces: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-be4dd1bbbb63b551760bc7ed0cf8da47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#99;&#100;&#111;&#116;&#52;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div id=\"CNX_ElemAlg_Figure_01_05_004_new\" class=\"bc-figure figure\"><\/div>\n<p id=\"fs-id1170654116758\">This model leads to the following property:<\/p>\n<div id=\"fs-id1170654068263\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Equivalent Fractions Property<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654016289\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58797fcd980ddcdad97f6b6f5260b5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c95b1fc74be8e9624bc66bfb3e8c0da4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\" \/>, then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f247503cc1742a06a64d47e95c5a77dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#123;&#98;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653751355\">If we had cut the pizza differently, we could get<\/p>\n<p><span id=\"fs-id1170653800497\" data-type=\"media\" data-alt=\"An image shows three rows of fractions. In the first row are the fractions \u201c1, times 2, divided by 2, times 2, equals two fourths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half, equals two fourths. The second row reads \u201c1, times 3, divided by 2 times 3, equals three sixths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, three sixths\u201d. The third row reads \u201c1 times 10, divided by 2 times 10, ten twentieths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, ten twentieths\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_005_img_new.jpg\" alt=\"An image shows three rows of fractions. In the first row are the fractions \u201c1, times 2, divided by 2, times 2, equals two fourths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half, equals two fourths. The second row reads \u201c1, times 3, divided by 2 times 3, equals three sixths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, three sixths\u201d. The third row reads \u201c1 times 10, divided by 2 times 10, ten twentieths\u201d. Next to this is the word \u201cso\u201d and the fraction \u201cone half equals, ten twentieths\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654114644\">So, we say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a5324c8c95634b8fa8059ee95f76149c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#54;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\" \/> are equivalent fractions.<\/p>\n<div id=\"fs-id1166424997376\" class=\"manipulative-math\" data-type=\"note\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654115770\" data-type=\"problem\">\n<p id=\"fs-id1170653910320\">Find three fractions equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0700364398eac9320f4d6eebefa6d6b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654069037\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654125509\">To find a fraction equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0700364398eac9320f4d6eebefa6d6b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>, we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let\u2019s multiply them by 2, 3, and then 5.<\/p>\n<p><span id=\"fs-id1170654014939\" data-type=\"media\" data-alt=\"A row of fractions reads \u201c2 times 2, divided by 5 times 2, equals four tenths\u201d. Next to this is \u201c2, times 3, divided by 5 times 3, equals six fifteenths\u201d. Next to this is \u201c2 times 5, divided by 5 times 5, equals ten twenty-fifths\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_006_img_new.jpg\" alt=\"A row of fractions reads \u201c2 times 2, divided by 5 times 2, equals four tenths\u201d. Next to this is \u201c2, times 3, divided by 5 times 3, equals six fifteenths\u201d. Next to this is \u201c2 times 5, divided by 5 times 5, equals ten twenty-fifths\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p>So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-130d114f2a404a99bdae960414a82ec6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#53;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/> are equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0700364398eac9320f4d6eebefa6d6b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653906562\" data-type=\"problem\">\n<p id=\"fs-id1170653832505\">Find three fractions equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653881636\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-159918bc0d6844a4bba8eb273f714a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#50;&#48;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -7px;\" \/> answers may vary<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654015429\" data-type=\"problem\">\n<p id=\"fs-id1170654013584\">Find three fractions equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654061051\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653905408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-264be8c54f03c1b160e1fe0cb09e9b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#49;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#50;&#48;&#125;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -7px;\" \/> answers may vary<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Simplify Fractions<\/h1>\n<p id=\"fs-id1170654116001\">A fraction is considered <strong data-effect=\"bold\">simplified<\/strong> if there are no common factors, other than 1, in its numerator and denominator.<\/p>\n<p id=\"fs-id1170654027609\">For example,<\/p>\n<ul id=\"fs-id1166422535686\" data-bullet-style=\"bullet\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is simplified because there are no common factors of 2 and 3.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-80f51ab0207944543454dc1900a35d61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> is not simplified because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is a common factor of 10 and 15.<\/li>\n<\/ul>\n<div id=\"fs-id1170654047581\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Simplified Fraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A fraction is considered simplified if there are no common factors in its numerator and denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654203066\">The phrase <em data-effect=\"italics\">reduce a fraction<\/em> means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.<\/p>\n<p id=\"fs-id1170654016508\">In <a class=\"autogenerated-content\" href=\"#fs-id1170654115770\">Example 1<\/a>, we used the equivalent fractions property to find equivalent fractions. Now we\u2019ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.<\/p>\n<div id=\"fs-id1170653882056\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Equivalent Fractions Property<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170653904200\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58797fcd980ddcdad97f6b6f5260b5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c95b1fc74be8e9624bc66bfb3e8c0da4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\" \/>,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c850fa20be87f18878e036dfa2d95f34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#123;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#123;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653886029\" data-type=\"problem\">\n<p id=\"fs-id1170654075477\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-64a923f60003bc80f0182a086be888ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#125;&#123;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653897505\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1169750893704\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction \u201cnegative thirty two fifty-sixths\u201d. The second row reads \u201cRewrite the numerator and denominator showing the common factors.\u201d Next to this in the right column reads \u201cnegative four times 8, divided by 7 times 8\u201d. The third row reads \u201cSimplify using the equivalent fractions properly\u201d. Next to this in the right column, it reads and \u201cnegative four sevenths\u201d.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-64a923f60003bc80f0182a086be888ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#125;&#123;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator showing the common factors.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169750893744\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_007_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify using the equivalent fractions property.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-41786e009b6d13df153d73109ddad2db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-41786e009b6d13df153d73109ddad2db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/> is simplified because there are no more common factors.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1166424799823\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653897688\" data-type=\"problem\">\n<p id=\"fs-id1170654061444\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cb7ed348634ba01e2faa2b47b93c2dbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#50;&#125;&#123;&#53;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654148628\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-87264f02c0fce02695c00e44ffad76dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653741575\" data-type=\"problem\">\n<p id=\"fs-id1170653898228\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8e37293bd680a3a9a3223332d82f82fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#53;&#125;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653808872\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653886874\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7f5198a4d35dc32f67c06390951aef68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654060626\">Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into <span class=\"no-emphasis\" data-type=\"term\">prime number<\/span>s. Then divide out the common factors using the equivalent fractions property.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How to Simplify a Fraction<\/div>\n<div id=\"fs-id1170653749775\" data-type=\"exercise\">\n<div id=\"fs-id1170654190237\" data-type=\"problem\">\n<p id=\"fs-id1170654147077\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e91f7694fb33d276fba310c34d66a00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#48;&#125;&#123;&#51;&#56;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654117064\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p><span id=\"fs-id1170653880713\" data-type=\"media\" data-alt=\"A table is shown with three columns and three rows. The first row of the left column reads \u201cStep 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree\u201d. Next to this in the middle column, it reads \u201crewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads \u201cnegative 210 divided by 385.\u201d Under this, is the equation \u201ctwo times three times five times seven.\u201d The five and 7 are blue and red respectively.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008a_new.jpg\" alt=\"A table is shown with three columns and three rows. The first row of the left column reads \u201cStep 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree\u201d. Next to this in the middle column, it reads \u201crewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads \u201cnegative 210 divided by 385.\u201d Under this, is the equation \u201ctwo times three times five times seven.\u201d The five and 7 are blue and red respectively.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655026820\" data-type=\"media\" data-alt=\"The next row down reads \u201cStep 2. Simplify using the equivalent fractions property by dividing out common factors.\u201d Next to this in the middle column, it reads, \u201cMark the common factors 5 and 7.\u201d Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation \u201cnegative two times 3 divided by 11.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008b_new.jpg\" alt=\"The next row down reads \u201cStep 2. Simplify using the equivalent fractions property by dividing out common factors.\u201d Next to this in the middle column, it reads, \u201cMark the common factors 5 and 7.\u201d Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation \u201cnegative two times 3 divided by 11.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655164362\" data-type=\"media\" data-alt=\"The next row reads, \u201cStep 3. Multiply the remaining factors, if necessary.\u201d Next to this in the right column is negative six elevenths.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_008c_new.jpg\" alt=\"The next row reads, \u201cStep 3. Multiply the remaining factors, if necessary.\u201d Next to this in the right column is negative six elevenths.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654221756\" data-type=\"problem\">\n<p id=\"fs-id1170653892448\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b3e42e7fd5223b09b1f2d09038467cb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#57;&#125;&#123;&#49;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654125391\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653742729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3a35b5a35eabf343f7956d0d793cb96b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654010172\" data-type=\"problem\">\n<p id=\"fs-id1170653906336\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-19f5480f9f47bafa2aeb0ce4e827057b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#48;&#125;&#123;&#49;&#57;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654073679\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-01dd0e273e62ac56a29d412e5f5d345d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653768282\">We now summarize the steps you should follow to simplify fractions.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Simplify a Fraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166424912916\" class=\"stepwise\" type=\"1\">\n<li>Rewrite the numerator and denominator to show the common factors.<br \/>\nIf needed, factor the numerator and denominator into prime numbers first.<\/li>\n<li>Simplify using the equivalent fractions property by dividing out common factors.<\/li>\n<li>Multiply any remaining factors, if needed.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170653905375\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653758697\" data-type=\"problem\">\n<p id=\"fs-id1170654031464\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4dea3a8b685eac19e62bfb050fdb1019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#53;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"15\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654007558\" data-type=\"solution\">\n<div data-type=\"title\"><strong><br \/>\nSolution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-id1169753205881\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction 5x divided by 5y\u201d. The second row reads \u201cRewrite showing the common factors, then divide out the factors.\u201d Next to this in the right column is a fraction equation \u201c5 times \u201cx\u201d, over five times \u201cy\u201d. The fives are crossed out in both the numerator and the denominator. The third row reads \u201cSimplify\u201d. Next to this in the right column, it reads \u201cx\u201d over \u201cy\u201d.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4dea3a8b685eac19e62bfb050fdb1019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#53;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"15\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing the common factors, then divide out the common factors.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169753269750\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_009_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5156593683f791fcfc08d98d0045ea64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653888423\" data-type=\"problem\">\n<p id=\"fs-id1170653912078\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3457cb541ad0db330df164c4a660ade5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#120;&#125;&#123;&#55;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"15\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653902694\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653879192\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5156593683f791fcfc08d98d0045ea64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653888423\" data-type=\"problem\">\n<p id=\"fs-id1170653912078\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3457cb541ad0db330df164c4a660ade5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#120;&#125;&#123;&#55;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"15\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653902694\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653879192\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5156593683f791fcfc08d98d0045ea64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Multiply Fractions<\/h1>\n<p id=\"fs-id1170653834744\">Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.<\/p>\n<p id=\"fs-id1170653909812\">We\u2019ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let\u2019s start with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p><span id=\"fs-id1170653895041\" data-type=\"media\" data-alt=\"A rectangle made up of four squares in a row. The first three squares are shaded.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_010_img_new.jpg\" alt=\"A rectangle made up of four squares in a row. The first three squares are shaded.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654080788\">Now we\u2019ll take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p><span id=\"fs-id1170653889891\" data-type=\"media\" data-alt=\"A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_011_img_new.jpg\" alt=\"A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170653869811\">Notice that now, the whole is divided into 8 equal parts. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f72ac1e7d5fdfc6821b4b6165546597f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p>To multiply fractions, we multiply the numerators and multiply the denominators.<\/p>\n<div id=\"fs-id1170653801922\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Fraction Multiplication<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654115792\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73702c1e5b3f33e81b906e7f7f1638a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a5d0819daabc21e0c119ecfddaddf296_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -4px;\" \/>, then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6f5098ce2f7429ea1490205015b313a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#99;&#125;&#123;&#98;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>To multiply fractions, multiply the numerators and multiply the denominators.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654066406\">When <span class=\"no-emphasis\" data-type=\"term\">multiplying fractions<\/span>, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the <span class=\"no-emphasis\" data-type=\"term\">product<\/span> as the first step. In <a class=\"autogenerated-content\" href=\"#fs-id1170654025836\">Example 5<\/a>, we will multiply negative and a positive, so the product will be negative.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654025836\" data-type=\"problem\">\n<p id=\"fs-id1170653750622\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-83ef616cc23eb51bafdd1f80b11d07d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654012761\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654030468\">The first step is to find the sign of the product. Since the signs are the different, the product is negative.<\/p>\n<table id=\"eip-201\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-175072bc8b0da3f8187ba627334b4096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-59fc717abe7e6da275d3d873e772982f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#92;&#99;&#100;&#111;&#116;&#53;&#125;&#123;&#49;&#50;&#92;&#99;&#100;&#111;&#116;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Are there any common factors in the numerator and the demoninator? No.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0c30b7db2a3a6a761201eaa2391512b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#53;&#125;&#123;&#56;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653750640\" data-type=\"problem\">\n<p id=\"fs-id1170653887277\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-924d29959a2bd08b964435def6ccc076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#56;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654008561\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654202807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-18186f2481f4673c380a854f5770bb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654061006\" data-type=\"problem\">\n<p id=\"fs-id1170654061283\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-003fcd23e8796bb0ce0a3db2db443836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#48;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653900368\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654285392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fd2eadbbc62b6e8b193be230da5e0939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654026520\">When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, <em data-effect=\"italics\">a<\/em>, can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ddf384f4f92d68f90e72962b32a2548c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/>. So, for example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d2a628a220aa4d3e86869f16af5b52ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653870152\" data-type=\"problem\">\n<p id=\"fs-id1170653907777\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-efbd77ec269f919f26a76c528fb3980e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654080945\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1166420388500\">Determine the sign of the product. The signs are the same, so the product is positive.<\/p>\n<table id=\"eip-id1169753124719\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction twelve fifths times negative twenty \u201cx\u201d in parentheses. The second row reads \u201cWrite negative twenty \u201cx\u201d as a fraction.\u201d Next to this in the right column is a fraction equation 12 times 20 times x over five times 1. The third row reads \u201cShow the common factors and then remove them\u201d. Next to this in the right column, it shows negative twelve times 4 times 5 over 5 times 1. The fives are crossed out; one in the numerator and one in the denominator. The last row reads, \u201cMultiply the remaining factors; simplify. To the right of this is negative 48 \u201cx\u201d.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-efbd77ec269f919f26a76c528fb3980e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7bebb1f042ed3c25ca510bdcd09a21c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/> as a fraction.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9167e31a7358a4235eb4d052547a9011_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#120;&#125;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"68\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Rewrite 20 to show the common factor 5 and divide it out.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169747225310\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_012_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-04e75d7e24bbe18fed6587fdf422e583_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"28\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654113920\" data-type=\"problem\">\n<p id=\"fs-id1170654148643\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-33746f9ae46fdbece4aa6cbbe57a80df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#51;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653761249\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653743318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-07517dccd8dab6bb9283a93771400624_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#51;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654165872\" data-type=\"problem\">\n<p id=\"fs-id1170652633105\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e70ce19e01130be3314d72ae2e94168c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#55;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654185150\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654129788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-30aef779fbcdc7fb770efa58fb781384_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Divide Fractions<\/h1>\n<p id=\"fs-id1170653750930\">Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.<\/p>\n<p id=\"fs-id1170654185172\">The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-eadb3de75c7c00057d0aede78f9ebd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p>Notice that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b0f11bb6367a83a5471e330aca8b13e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/>. A number and its reciprocal multiply to 1.<\/p>\n<p id=\"fs-id1170653770746\">To get a <span class=\"no-emphasis\" data-type=\"term\">product<\/span> of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.<\/p>\n<p id=\"fs-id1170654128489\">The reciprocal of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-30ca9fc96c72f0d68b97ca491d0705f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ae583658a24e27ae188a2b24b6c032c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/>, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e97c2c5f489db0c367940e5dd4d056e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#55;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"130\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<div id=\"fs-id1170654286462\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Reciprocal<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <strong data-effect=\"bold\">reciprocal<\/strong> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8459b97fa73367952be2ca8cf9d1326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fba554c641d5e412cb199aeb51095854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p>A number and its reciprocal multiply to one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1ae9a1af9c6f9af4910cbfa0f2e41976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654235623\">To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/p>\n<div id=\"fs-id1170652619282\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Fraction Division<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654189237\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73702c1e5b3f33e81b906e7f7f1638a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5961de2b3e5557b76eb16af1f1efa741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"166\" style=\"vertical-align: -4px;\" \/>, then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4daa90f239dd63228bde4541dbc3683c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653745038\">We need to say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5961de2b3e5557b76eb16af1f1efa741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"166\" style=\"vertical-align: -4px;\" \/> to be sure we don\u2019t divide by zero!<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654125521\" data-type=\"problem\">\n<p id=\"fs-id1170653786596\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a0d2cf9b56bfaafe17595697ab4dfd6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654239446\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-381\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a0d2cf9b56bfaafe17595697ab4dfd6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>To divide, multiply the first fraction by the reciprocal of the second.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c2b20697244039d39b9731d7fde16af4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b0c8ef08d5f430a95d59d67c33402a79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"34\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654068819\" data-type=\"problem\">\n<p id=\"fs-id1170653756083\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-878c0af1b38cbefd097b1e9b9424d832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653924388\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653879363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-587bdf0cbb8dc6ca02b0cf183bf94492_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#53;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"32\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654114180\" data-type=\"problem\">\n<p id=\"fs-id1170654116769\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-64523b6c1d0d65ea8466ceb1a82286b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654284311\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654239718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b4105c6b5d0df0a065a61880f115bc71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#56;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"32\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654150092\" data-type=\"problem\">\n<p id=\"fs-id1170654022376\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-10ea442995e22e5bcbc75b1540cbd60c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654010810\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-id1169747240983\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation seven eighteenths divided by fourteen twenty sevenths. The second row reads \u201cMultiply the first fraction by the reciprocal of the second.\u201d Next to this in the right column is a fraction equation seven eighteenths times twenty seven fourteenths. The third row reads \u201cMultiply\u201d. Next to this in the right column, is a fraction equation seven times twenty seven over eighteen times fourteen. The next row reads \u201crewrite showing common factors. Next to this in the right column is seven times nine times three over nine times two times seven times two. The seven and the nines are crossed out in the numerator and denominator. The next row reads \u201crewrite showing common factors.\u201d Next to this is \u201c3 over 2 times 2.\u201d The last row reads \u201csimplify.\u201d Next to this in the right column is the fraction three fourths.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-13f163ef2b97aa12ad8b8dd611df540c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>To divide, multiply the first fraction by the reciprocal of the second.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ee45c97cd60d6445906075ef781b58ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product, and then multiply..<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-aefb09d88f189665fcdbb581902b2e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#55;&#125;&#123;&#49;&#56;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing common factors.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169750795691\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_013_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c72be02998a4c4a761149230e2fc6c8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653800619\" data-type=\"problem\">\n<p id=\"fs-id1170654114238\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7a1c11fa3fd7acf090f70bb47b583783_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#55;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#53;&#125;&#123;&#51;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653898026\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653898028\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b6061ea4d4da1a30eadbc5bfff4712dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653790948\" data-type=\"problem\">\n<p id=\"fs-id1170653991340\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-05edbf5f3ab79d9877893f56fdfa7875_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653767637\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654166274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653930534\">There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.<\/p>\n<ul id=\"fs-id1166424848300\" data-bullet-style=\"bullet\">\n<li>\u201cTo multiply fractions, multiply the numerators and multiply the denominators.\u201d<\/li>\n<li>\u201cTo divide fractions, multiply the first fraction by the reciprocal of the second.\u201d<\/li>\n<\/ul>\n<p id=\"fs-id1170653795085\">Another way is to keep two examples in mind:<\/p>\n<p><span id=\"fs-id1170654007613\" data-type=\"media\" data-alt=\"This is an image with two columns. The first column reads \u201cOne fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the centre of each one representing one half. The halves are labeled \u201cone half\u201d. Under this is the equation \u201c2 times 1 fourth\u201d. Under this is another equation \u201ctwo over 1 times 1 fourth.\u201d Under this is the fraction two fourths and under this is the fraction one half. The next column reads \u201cthere are eight quarters in two dollars.\u201d Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation \u201ctwo over one divided by one fourth.\u201d Under this is two over one times four over one. Under this is the answer \u201c8\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_016_img_new.jpg\" alt=\"This is an image with two columns. The first column reads \u201cOne fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the centre of each one representing one half. The halves are labeled \u201cone half\u201d. Under this is the equation \u201c2 times 1 fourth\u201d. Under this is another equation \u201ctwo over 1 times 1 fourth.\u201d Under this is the fraction two fourths and under this is the fraction one half. The next column reads \u201cthere are eight quarters in two dollars.\u201d Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation \u201ctwo over one divided by one fourth.\u201d Under this is two over one times four over one. Under this is the answer \u201c8\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654286468\">The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a <strong data-effect=\"bold\">complex fraction<\/strong>.<\/p>\n<div id=\"fs-id1170653887308\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Complex Fraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A complex fraction is a fraction in which the numerator or the denominator contains a fraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653914822\">Some examples of complex fractions are:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5f81fd97feb081269b17018d94e7de33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"74\" style=\"vertical-align: -13px;\" \/><\/p>\n<p id=\"fs-id1170652622719\">To simplify a complex fraction, we remember that the fraction bar means <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. For example, the complex fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2684b90a3d8fb0811f29c1b1886d4a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"10\" style=\"vertical-align: -13px;\" \/> means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-115c89970dd090b92a41e26f86f7e97d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654076126\" data-type=\"problem\">\n<p id=\"fs-id1170653897982\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2684b90a3d8fb0811f29c1b1886d4a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"10\" style=\"vertical-align: -13px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653896387\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-id1169752928919\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation three fourths divided by five eighths. The second row on the left reads, \u201cRewrite as division.\u201d Next to this in the right column is a fraction equation three fourths divided by, using the division symbol, five eighths. The third row reads \u201cMultiply the first fraction by the reciprocal of the second\u201d. Next to this in the right column, is the fraction equation three fourths times five eighths. The next row down in the left column it has \u201cMultiply.\u201d Next to this in the right column, is the fraction equation three times eight divided by four times six. The next row reads \u201cLook for common factors. Next to this in the right column is the fraction equations three times four times two divided by four times six. The fours are crossed out in the numerator and denominator. The next row reads \u201cdivide out common factors and simplify.\u201d Next to this shows the fraction six fifths.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2684b90a3d8fb0811f29c1b1886d4a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"10\" style=\"vertical-align: -13px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite as division.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-115c89970dd090b92a41e26f86f7e97d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f48d923b89a83ea45c74ec69120c94ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-211d67aa01a628757bb9236d4528eaf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#56;&#125;&#123;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169753189361\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_014_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Divide out common factors and simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-05094d976698f8323b4ee8760d5e2dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654034116\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654239534\" data-type=\"exercise\">\n<div id=\"fs-id1170654122792\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654122792\" data-type=\"problem\">\n<p id=\"fs-id1170654122795\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2b4b1127c5e232cc83e9e8a807c2c8d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"10\" style=\"vertical-align: -13px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654080854\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653901056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653902067\" data-type=\"problem\">\n<p id=\"fs-id1170654128462\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f5deeb48bfcb289420fd5f06bcd02c62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"16\" style=\"vertical-align: -13px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654061219\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170652622093\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58b071736c8a86f28493a944743f0709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652618856\" data-type=\"problem\">\n<p id=\"fs-id1170654007229\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1caf198fadb80c706507eea6ddbaeee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"18\" style=\"vertical-align: -12px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170652631581\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-id1169754096098\" class=\"unnumbered unstyled\" summary=\"This solution image has two columns. The first column has the fraction equation \u201cx\u201d over 2 divided by \u201cxy\u201d over six. The second row on the left reads, \u201cRewrite as division.\u201d Next to this in the right column is a fraction equation \u201cx over 2 divided by, using the division symbol, \u201cxy\u201d over 6. The third row reads \u201cMultiply the first fraction by the reciprocal of the second\u201d. Next to this in the right column, is the fraction equation \u201cx\u201d over 2 times six over \u201cxy\u201d. The next row down in the left column it has \u201cMultiply.\u201d Next to this in the right column, is the fraction equation \u201cx\u201d times six divided by two times \u201cxy\u201d. The next row reads \u201cLook for common factors\u201d. Next to this in the right column is the fraction equation \u201cx\u201d times three times two divided by two times x times y\u201d The \u201cx\u201d\u2019s and 2\u2019s are crossed out in the numerator and denominator. The next row reads \u201cdivide out common factors and simplify.\u201d Next to this shows the fraction 3 over y.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1caf198fadb80c706507eea6ddbaeee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"18\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite as division.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b39858dd8d4b39c5917b8d2aa2b22991_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#92;&#100;&#105;&#118;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2b2cbf43dc14a03e9ae166837cb78d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"40\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-26f941f8e7a5347a2da2ce9e4c5300af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#92;&#99;&#100;&#111;&#116;&#32;&#54;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#120;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td data-align=\"center\"><span id=\"eip-id1169748377665\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_01_05_015_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Divide out common factors and simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-66e31551db10647b0a630e067487830e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170653902065\" data-type=\"exercise\">\n<div id=\"fs-id1170653902067\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654238021\" data-type=\"problem\">\n<p id=\"fs-id1170654238023\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9c49f69c7e925a73affbb436321e07ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#98;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"16\" style=\"vertical-align: -14px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170652622839\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170652622841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-24cd9963d965f100ffa8a449f0b0aaf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654150057\" data-type=\"problem\">\n<p id=\"fs-id1170654150060\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-eaf0077bace7b82a61d322afb3675f80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#112;&#113;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"22\" style=\"vertical-align: -30px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654234015\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654234017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b4bb2b2f896bea9c920023fc4025013e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"7\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Simplify Expressions with a Fraction Bar<\/h1>\n<p id=\"fs-id1170653878324\">The line that separates the numerator from the denominator in a fraction is called a fraction bar. A <span class=\"no-emphasis\" data-type=\"term\">fraction bar<\/span> acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.<\/p>\n<p id=\"fs-id1170654032774\">To simplify the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3b4aa83ac9381e5bc20357c106425e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#45;&#51;&#125;&#123;&#55;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/>, we first simplify the numerator and the denominator separately. Then we divide.<\/p>\n<div id=\"fs-id1170653894422\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3b4aa83ac9381e5bc20357c106425e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#45;&#51;&#125;&#123;&#55;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/div>\n<div id=\"fs-id1170654068842\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7048ed51598ce06a7b8220724a177c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/div>\n<div id=\"fs-id1170654150959\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-83dc4a903812856a369855508d0b2637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Simplify an Expression with a Fraction Bar<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166424867009\" class=\"stepwise\" type=\"1\">\n<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\n<li>Simplify the fraction.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654286084\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654032705\" data-type=\"exercise\">\n<div id=\"fs-id1170653910175\" data-type=\"problem\">\n<p id=\"fs-id1170653910178\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c21be9fa38b52254e7836de34f4962c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"43\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653752520\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<\/div>\n<\/div>\n<table id=\"eip-149\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c21be9fa38b52254e7836de34f4962c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"43\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Use the order of operations to simpliy the numerator and the denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-aa46c00f1daa27c2636cc68ef8adcdfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#54;&#125;&#123;&#52;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator and the denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-872211bd058c44d4b219f6379a6d41e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify. A negative divided by a positive is negative.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5e38fbbffc028ff17d3685089e222a37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654283295\" data-type=\"problem\">\n<p id=\"fs-id1170654283297\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-94251a01e048816f64e31c0b51786f08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"43\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654061370\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654061372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e8ab9ebbace69ceb967d69cd3d13f20c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653797687\" data-type=\"problem\">\n<p id=\"fs-id1170654154932\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f817ebc4d973e35ec9e74174dac4c256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"43\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653990815\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653990817\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-78d694220a2417a558c269b7ab05bb21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654237820\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654283293\" data-type=\"exercise\">\n<div id=\"fs-id1170654283295\" data-type=\"problem\">Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654185949\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-993c4643f92473f0ab3da28a95f09bb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#51;&#125;&#61;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#45;&#51;&#125;&#61;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"294\" style=\"vertical-align: -21px;\" \/><\/div>\n<div id=\"fs-id1170653878972\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Placement of Negative Sign in a Fraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654286245\">For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fe76ae3d93a5e4470de30644cbd9abff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#45;&#98;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654066922\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652633027\" data-type=\"problem\">\n<p id=\"fs-id1170652633029\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a7ed77e1faeec239917c087abb9a9302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"82\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654286156\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-599\" summary=\"-\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-31d3afa6f9efec350dcc54c57570db8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"82\" style=\"vertical-align: -10px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-90df7540f722c4071776ee41e25a4f52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#54;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9e072402dfc80c69c0f4b0c80b41e0c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#52;&#125;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654235874\" data-type=\"problem\">\n<p id=\"fs-id1170653885349\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3521b17e176acf288d2cfb8bb3322083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"82\" style=\"vertical-align: -10px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654038199\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654283278\">4<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654073995\" data-type=\"problem\">\n<p id=\"fs-id1170654073997\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-aeaf6724371038e8daacdbdda502ec08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"82\" style=\"vertical-align: -10px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654060112\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654060114\">2<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Translate Phrases to Expressions with Fractions<\/h1>\n<p id=\"fs-id1170654164343\">Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.<\/p>\n<p id=\"fs-id1170653890410\">The English words <span class=\"no-emphasis\" data-type=\"term\">quotient<\/span> and <span class=\"no-emphasis\" data-type=\"term\">ratio<\/span> are often used to describe fractions. Remember that \u201cquotient\u201d means <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. The quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the result we get from dividing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8459b97fa73367952be2ca8cf9d1326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654281372\" data-type=\"problem\">\n<p id=\"fs-id1170654281374\">Translate the English phrase into an algebraic expression: the quotient of the difference of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>, and <em data-effect=\"italics\">p<\/em>.<\/p>\n<\/div>\n<div id=\"fs-id1170654154539\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654235420\">We are looking for the <em data-effect=\"italics\">quotient<\/em> <em data-effect=\"italics\">of<\/em> the difference of <em data-effect=\"italics\">m<\/em> and <em data-effect=\"italics\">n<\/em>, <em data-effect=\"italics\">and<\/em> <em data-effect=\"italics\">p<\/em>. This means we want to divide the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4209e2c5e7be8636226f80db9ed3ddbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div id=\"fs-id1166425083474\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bbad94f59d00c9fd18c65f24995cd381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#45;&#110;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170652623005\">Translate the English phrase into an algebraic expression: the quotient of the difference of <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">cd<\/em>.<\/p>\n<\/div>\n<div id=\"fs-id1170654292142\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654292144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2af4fa018567055fb025dae005ded699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"24\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653794596\" data-type=\"problem\">\n<p id=\"fs-id1170654150368\">Translate the English phrase into an algebraic expression: the quotient of the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170653904815\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653904817\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b120ee9284be94f32711295d09a7ba57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#113;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1170654149151\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Equivalent Fractions Property:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58797fcd980ddcdad97f6b6f5260b5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c95b1fc74be8e9624bc66bfb3e8c0da4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\" \/>, then<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-da29c009e6864386f919e53e6f44d201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#123;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5cf0d0864d900eb4ea38357fc29e4f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#123;&#98;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -6px;\" \/>.<\/li>\n<li><strong data-effect=\"bold\">Fraction Division:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73702c1e5b3f33e81b906e7f7f1638a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bb6a1fc7edd46d1a8109269dc8193bc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"174\" style=\"vertical-align: -4px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4daa90f239dd63228bde4541dbc3683c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/>. To divide fractions, multiply the first fraction by the reciprocal of the second.<\/li>\n<li><strong data-effect=\"bold\">Fraction Multiplication:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73702c1e5b3f33e81b906e7f7f1638a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d3e61de7c99d1832171a1bffd2a17569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -4px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-322755e0cb73a49a7f62bb528e9d4cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#99;&#125;&#123;&#98;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -6px;\" \/>. To multiply fractions, multiply the numerators and multiply the denominators.<\/li>\n<li><strong data-effect=\"bold\">Placement of Negative Sign in a Fraction:<\/strong> For any positive numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4649a8b659da898542795396b27208da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-028a297d6c58b5b256ea55860cf9a2a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#45;&#98;&#125;&#61;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -6px;\" \/>.<\/li>\n<li><strong data-effect=\"bold\">Property of One:<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-12e892711dcf9a41efea4bd38d71cd0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#97;&#125;&#61;&#49;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"46\" style=\"vertical-align: -6px;\" \/> Any number, except zero, divided by itself is one.<\/li>\n<li><strong data-effect=\"bold\">Simplify a Fraction<\/strong>\n<ol class=\"stepwise\" type=\"1\">\n<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.<\/li>\n<li>Simplify using the equivalent fractions property by dividing out common factors.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Simplify an Expression with a Fraction Bar<\/strong>\n<ol id=\"fs-id1170654237200\" class=\"stepwise\" type=\"1\">\n<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\n<li>Simplify the fraction.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1166424866332\">\n<dt>complex fraction<\/dt>\n<dd id=\"fs-id1166424866337\">A complex fraction is a fraction in which the numerator or the denominator contains a fraction.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424866347\">\n<dt>denominator<\/dt>\n<dd id=\"fs-id1166425223004\">The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166425223009\">\n<dt>equivalent fractions<\/dt>\n<dd id=\"fs-id1166425223015\">Equivalent fractions are fractions that have the same value.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166425223019\">\n<dt>fraction<\/dt>\n<dd id=\"fs-id1166422580993\">A fraction is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8459b97fa73367952be2ca8cf9d1326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1edd53778d94abb1dc1e19acff79e9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the numerator and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the denominator. A fraction represents parts of a whole. The denominator <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the number of equal parts the whole has been divided into, and the numerator <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> indicates how many parts are included.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424896994\">\n<dt>numerator<\/dt>\n<dd id=\"fs-id1166424897000\">The numerator is the value on the top part of the fraction that indicates how many parts of the whole are included.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424844292\">\n<dt>reciprocal<\/dt>\n<dd id=\"fs-id1166424844298\">The reciprocal of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8459b97fa73367952be2ca8cf9d1326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"8\" style=\"vertical-align: -6px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fba554c641d5e412cb199aeb51095854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/>. A number and its reciprocal multiply to one: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1ae9a1af9c6f9af4910cbfa0f2e41976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166422601976\">\n<dt>simplified fraction<\/dt>\n<dd id=\"fs-id1166422601981\">A fraction is considered simplified if there are no common factors in its numerator and denominator.<\/dd>\n<\/dl>\n<\/div>\n<h1>Practice Makes Perfect<\/h1>\n<h2>Find Equivalent Fractions<\/h2>\n<p id=\"fs-id1166424799887\">In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a50f9f96c608e65c404d41e0ef6f8a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">2. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-44cf49c06a351926440bb2ad833776c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0f4ccf3e077500e691259205b8b0b08e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">4. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9de931d332a83dd3da0c1ba21347996c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Simplify Fractions<\/h2>\n<p id=\"fs-id1166422693579\">In the following exercises, simplify.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 90px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">5. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9908e84affbbe7afb66114aeaec60675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#125;&#123;&#56;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%; height: 15px;\">6. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e8fa5318ef7cdeb2e0a2fd57346d6974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#51;&#125;&#123;&#57;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">7. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-878f810a85a18cad88333361d016ab1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#56;&#125;&#123;&#54;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%; height: 15px;\">8. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bd421a8c8ce50fff7e2ec0e34ac1d11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#52;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">9. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5ee9238e528605b5556823d46c9ace98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#48;&#125;&#123;&#50;&#53;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%; height: 15px;\">10. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ae0d0f21ced7705df2bcd5d1a4d7cf7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#50;&#125;&#123;&#50;&#57;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d51dc4b7765de6a0d2c1a9e3f66a7722_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#49;&#50;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"40\" style=\"vertical-align: -9px;\" \/><\/td>\n<td style=\"width: 50%; height: 15px;\">12. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fd3cb1ce81d82d89608ff8db4efd8f3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#51;&#50;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"40\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">13. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3c62f1df1f12e1a65108a3bb8bd03514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#49;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"29\" style=\"vertical-align: -9px;\" \/><\/td>\n<td style=\"width: 50%; height: 15px;\">14. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c20d1231bdb86a218db79e10fbb87ca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#97;&#125;&#123;&#51;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"27\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Multiply Fractions<\/h2>\n<p id=\"fs-id1166424880356\">In the following exercises, multiply.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">15. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-27c816a95e0bc5f35e8254c2eec031f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">16. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bd8b7e6a359840b8e5eb3ccd1be5ef17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bdd3fa2b3b0ddd3b8dd0bc71d384035e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">18. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d2bbfdd505a361c6b70849cb39821071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8ce4e15c0a45afd746453d6dc658b369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">20. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-095c6d1046b947ece082b0346973f1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">21. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-17a94b1096f23855e13ce0984f3b7f33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#49;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">22. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5506ce7db864f167006ff3fd883f095_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#51;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">23. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f273dc2f26d8a519743739a0f345b907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#51;&#125;&#123;&#56;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#52;&#125;&#123;&#57;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">24. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-93b4dea7981c654646f5b49d1e54c7ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#51;&#125;&#123;&#54;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#125;&#123;&#56;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">25. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5f94511bd30a89f3c6a61a5e24d0130e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">26. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-106aafcede1517b9bfa47f0e903e2550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">27. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-52f6c8e5b4fb5dc49ff4ca7bdc3c57fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">28. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3dfd8df04d042f4dc033abe66a41f124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#48;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">29. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-28a5aa6f585f8b2834a4f2f15e390b17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">30. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-80010165a17f6edf0aec8f35535e65bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#100;&#111;&#116;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"90\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Divide Fractions<\/h2>\n<p id=\"fs-id1166422785479\">In the following exercises, divide.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">31. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-148862a9a4d76965e31d47ce6e1eb062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">32. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e7ec3fbc0c9d8dcaf7495036304d4adf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">33. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-447d39c40089f6397e139eb61a86cf9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">34. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d64926e069f5d5e922bb95471822abe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">35. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-871055f7c63e58973502808d77123983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">36. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1900cc255ca8a0b14b1bf57124cfc04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">37. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8a505e4c0c8f22e117d132b7b6a204cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"86\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">38. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-90bf2a7e46c63fe143bd54cd0dd80bfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"86\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">39. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-671a97f9a497c5817252375d686bb690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#117;&#125;&#123;&#49;&#53;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#118;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">40. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-21faa583ce9ae8a6e7961e9313c7e2a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#114;&#125;&#123;&#50;&#53;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#115;&#125;&#123;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">41. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e35517c61193fbad59de440ecfd10987_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">42. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9a116816403abb8103c6c737d1a29166_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">43. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6b0bb525c60a8f697688916c1a9ab98e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">44. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-44d95bac6b8a038186b1c17a1846cde3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170654215028\">In the following exercises, simplify.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">45. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1b4410ba9b18493ad34573a730cf841d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#50;&#49;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#51;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/td>\n<td style=\"width: 50%;\">46. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3e1e0c098043237985804363dcbf55a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#52;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">47. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5373d46394a175adc912b09b3ee0772e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">48. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dbc38a1d86e4891a29a70be591e61d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"16\" style=\"vertical-align: -14px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">49. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-19c021c5d733dd5b43a4f886d9492b35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"15\" style=\"vertical-align: -11px;\" \/><\/td>\n<td style=\"width: 50%;\">50. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5fe97fa92dbcada1a2f1325a80626799_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#125;&#123;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Simplify Expressions Written with a Fraction Bar<\/h2>\n<p id=\"fs-id1166424900706\">In the following exercises, simplify.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">51. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8ca3c60ca5d83142f3ba34518ce75541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#50;&#43;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">52. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9b024530e12cbbd7cc6138d7d49de669_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#45;&#52;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">53. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d9261c614eedbe7af9ef3d4a93d77b66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#56;&#125;&#123;&#50;&#52;&#45;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"39\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">54. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5ee7c2cde2388889aa6c6516129b9afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#54;&#125;&#123;&#52;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">55. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-83da4032f9f84ea909764e366d6becef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#43;&#54;&#125;&#123;&#56;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -8px;\" \/><\/td>\n<td style=\"width: 50%;\">56. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7e7e223656553649639a9a706b4a6bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#43;&#51;&#125;&#123;&#49;&#55;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"36\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">57. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-01ef04243f694b36cfa28210958c66df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#99;&#100;&#111;&#116;&#51;&#125;&#123;&#54;&#92;&#99;&#100;&#111;&#116;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">58. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73cebb2828d804994fb206d5017b75e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#92;&#99;&#100;&#111;&#116;&#54;&#125;&#123;&#57;&#92;&#99;&#100;&#111;&#116;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">59. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-362b032523ca3a0211796ed71e6b2bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">60. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4c76d694c80d515906f31027cdd46163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#55;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">61. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ab6b151d3902dfb855e61b9a972a1be3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#99;&#100;&#111;&#116;&#51;&#43;&#50;&#92;&#99;&#100;&#111;&#116;&#57;&#125;&#123;&#49;&#52;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"47\" style=\"vertical-align: -8px;\" \/><\/td>\n<td style=\"width: 50%;\">62. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-58bb00c3d9c94736d59e50fae503387a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#99;&#100;&#111;&#116;&#54;&#45;&#52;&#92;&#99;&#100;&#111;&#116;&#55;&#125;&#123;&#50;&#50;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"47\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">63. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-438267c8037953e63abce4b13684363f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#99;&#100;&#111;&#116;&#54;&#45;&#51;&#92;&#99;&#100;&#111;&#116;&#52;&#125;&#123;&#52;&#92;&#99;&#100;&#111;&#116;&#53;&#45;&#50;&#92;&#99;&#100;&#111;&#116;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">64. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d8341c8ca0c25062b33848fec2e44453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#99;&#100;&#111;&#116;&#57;&#45;&#55;&#92;&#99;&#100;&#111;&#116;&#54;&#125;&#123;&#53;&#92;&#99;&#100;&#111;&#116;&#54;&#45;&#57;&#92;&#99;&#100;&#111;&#116;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">65. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5941371979474f963757bac3384812a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">66. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6df711cce06c2a7943cd73c9fc62a456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#54;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">67. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-95f51545d7b5717c4e4318922a1d80d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#99;&#100;&#111;&#116;&#52;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#57;&#92;&#99;&#100;&#111;&#116;&#51;&#45;&#51;&#92;&#99;&#100;&#111;&#116;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 50%;\">68. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-32a379977f9154f15fd421ce8503ba6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#99;&#100;&#111;&#116;&#55;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#56;&#92;&#99;&#100;&#111;&#116;&#55;&#45;&#54;&#92;&#99;&#100;&#111;&#116;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"78\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">69. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-468532a0852ffe469ec0a17cfcedc527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#55;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"102\" style=\"vertical-align: -9px;\" \/><\/td>\n<td style=\"width: 50%;\">70. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8d988df0be5cf18655d6a38b0da4265d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"102\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Translate Phrases to Expressions with Fractions<\/h2>\n<p id=\"fs-id1166425080667\">In the following exercises, translate each English phrase into an algebraic expression.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">71. the quotient of <em data-effect=\"italics\">r<\/em> and the sum of <em data-effect=\"italics\">s<\/em> and 10<\/td>\n<td style=\"width: 50%;\">72. the quotient of <em data-effect=\"italics\">A<\/em> and the difference of 3 and <em data-effect=\"italics\">B<\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">73. the quotient of the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c5d758bfb955d91224eab44625f62504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%;\">74. the quotient of the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3c595ab01e133250e08befbb9f372308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">75. <strong data-effect=\"bold\">Baking.<\/strong> A recipe for chocolate chip cookies calls for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> cup brown sugar. Imelda wants to double the recipe. a) How much brown sugar will Imelda need? Show your calculation. b) Measuring cups usually come in sets of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8e4ad67ce53fd2074e1496ae740a3f2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/> cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.<\/td>\n<td style=\"width: 50%;\">76. <strong data-effect=\"bold\">Baking.<\/strong> Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> cup of condensed milk. a) How much condensed milk will Nina need? Show your calculation. b) Measuring cups usually come in sets of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8e4ad67ce53fd2074e1496ae740a3f2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/> cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/> pans of fudge.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">77. <strong data-effect=\"bold\">Portions<\/strong> Don purchased a bulk package of candy that weighs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> pounds. He wants to sell the candy in little bags that hold <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-83dc4a903812856a369855508d0b2637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> pound. How many little bags of candy can he fill from the bulk package?<\/td>\n<td style=\"width: 50%;\">78. <strong data-effect=\"bold\">Portions<\/strong> Kristen has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> yards of ribbon that she wants to cut into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> equal parts to make hair ribbons for her daughter\u2019s 6 dolls. How long will each doll\u2019s hair ribbon be?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn\u2019t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael\u2019s reasoning.<\/td>\n<td style=\"width: 50%;\">80. Give an example from everyday life that demonstrates how <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8462608ed8179b49da8eb958ea56e219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">81. Explain how you find the reciprocal of a fraction.<\/td>\n<td style=\"width: 50%;\">82. Explain how you find the reciprocal of a negative number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Answers<\/h1>\n<div data-type=\"glossary\">\n<table style=\"border-collapse: collapse; width: 100%; height: 154px;\">\n<tbody>\n<tr style=\"height: 35px;\">\n<td style=\"width: 33.3333%; height: 35px;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-71b51af85df79f9323e5fc4311eeee44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"65\" style=\"vertical-align: -7px;\" \/> answers may vary<\/td>\n<td style=\"width: 33.3333%; height: 35px;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-564fc5f127fbdd701a7a656c3f9268d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#49;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"65\" style=\"vertical-align: -7px;\" \/> answers may vary<\/td>\n<td style=\"width: 33.3333%; height: 35px;\">5. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-51af606f03c0870a2d400ba8bcb32195_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">7. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ca4eb9cdc48a777cd4b978abeaaa30ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">9. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5dd5aeceb6caade5b6ce45cb93b3c000_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3c42063364d6e69cec965bcc4a120d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#52;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"33\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">13. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9e36cb9195f6da25b62a152db746fcdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"22\" style=\"vertical-align: -9px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">15. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d9d8882c558418c2c92346f1754f8bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-83dc4a903812856a369855508d0b2637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4c21671a162310bcecf310c8286a47e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">21. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c802e352325d304b72a633c3046b7301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#53;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">23. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4dc0d5b1b056d8ad35b3581c25971d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">25. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cc5d38918aba1777b399f7a0b8a9d684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">27. 9<em data-effect=\"italics\">n<\/em><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">29. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2ba600f80daa65c3d14ce4938e8b8e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">31. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8f2060800bb1aad8b0bc1a0c0bdcfa81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">33. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">35. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-282742f04238f02cbdbd823c12e1946a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#52;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">37. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-54e8eca1d373be7b2834658db82759da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">39. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b1ee5bee25f61a30fa7519873db9938e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#117;&#125;&#123;&#57;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">41. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 33.3333%; height: 17px;\">43. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-48c8883107b11edc86176cb3205c0843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">45. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-339989763e29f0dbefee22f05f25617e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%; height: 17px;\">47. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-63b9ebc61a0809c7b62bdde075decd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">49. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-180c1565231d608028cdd2154a4e8b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#109;&#125;&#123;&#51;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"20\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">51. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3c8a34714cd9e14f96438eaca16625df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">53. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e7be055a86dc72d43ac2d84c9076941f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">55. 0<\/td>\n<td style=\"width: 33.3333%;\">57. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">59. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">61. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-703cf597fbebe4d241fbb6774c5943a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">63. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">65. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">67. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f60306c12e0b1c2107d7595d78b20a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">69. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3c8a34714cd9e14f96438eaca16625df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">71. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5af36ceca5e0a89ea8e843d254529081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#125;&#123;&#115;&#43;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"31\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">73. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cdccc50f2fd06b1c9ac9b8421b9a3155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#121;&#125;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/td>\n<td style=\"width: 33.3333%;\">75. a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2bb20d9986d47c316031bc35dc35a37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"17\" style=\"vertical-align: -6px;\" \/> cups b) answers will vary<\/td>\n<td style=\"width: 33.3333%;\">77. 20 bags<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">79. Answers may vary.<\/td>\n<td style=\"width: 33.3333%;\">81.\u00a0Answers may vary.<\/td>\n<td style=\"width: 33.3333%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cVisualize Fractions\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. 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