{"id":138,"date":"2021-06-23T00:53:49","date_gmt":"2021-06-23T00:53:49","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/add-and-subtract-fractions\/"},"modified":"2023-11-01T17:43:51","modified_gmt":"2023-11-01T17:43:51","slug":"add-and-subtract-fractions","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/add-and-subtract-fractions\/","title":{"raw":"1.3 Fractions","rendered":"1.3 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 it is expected that you will be able to:\r\n<ul>\r\n \t<li>Multiply and divide fractions<\/li>\r\n \t<li>Simplifying expressions with fraction bar<\/li>\r\n \t<li>Add or subtract fractions with a common denominator<\/li>\r\n \t<li>Add or subtract fractions with different denominators<\/li>\r\n \t<li>Use the order of operations to simplify complex fractions<\/li>\r\n<\/ul>\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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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=\"#example1\">Example 1<\/a>, we will multiply negative and a positive, so the product will be negative.<a id=\"example1\"><\/a><\/p>\r\n\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-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\" style=\"width: 100%;\" 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 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<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 2<\/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\" style=\"width: 100%;\" 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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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 2<\/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<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 3<\/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\" style=\"width: 100%;\" 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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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 3<\/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<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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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 4<\/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\" style=\"width: 100%;\" 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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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 4<\/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>\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 5<\/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\" style=\"width: 100%;\" 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 5<\/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>\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 6<\/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\" style=\"width: 100%;\" 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 6<\/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>\r\n<\/div>\r\n<h1>Add or Subtract Fractions with a Common Denominator<\/h1>\r\n<p id=\"fs-id1170655354165\">When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.<\/p>\r\n\r\n<div id=\"fs-id1170654919725\" 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 Addition and Subtraction<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170655188656\">If \\(a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c\\) are numbers where \\(c\\ne 0\\), then<\/p>\r\n\r\n<div id=\"fs-id1170655166745\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n\r\n\\(\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\frac{a}{c}-\\phantom{\\rule{0.2em}{0ex}}\\frac{b}{c}=\\frac{a-b}{c}\\)\r\n\r\nTo add or subtract fractions, add or subtract the numerators and place the result over the common denominator.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"manipulative-math\" data-type=\"note\">\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-id1170655202348\" data-type=\"problem\">\r\n<p id=\"fs-id1170655132751\">Find the sum: \\(\\frac{3}{7}+\\frac{2}{7}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655126354\" 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-899\" style=\"height: 65px;\" summary=\".\" width=\"100%\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 529.757px; height: 15px;\"><\/td>\r\n<td style=\"width: 216.424px; height: 15px;\">\\(\\frac{3}{7}+\\frac{2}{7}\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 50px;\">\r\n<td style=\"width: 529.757px; height: 50px;\">Add the numerators and place the sum over the common denominator.\r\n\r\nSimplify.<\/td>\r\n<td style=\"width: 216.424px; height: 50px;\">\\(\\frac{3+2}{7}\\)\r\n\r\n\\(\\frac{5}{7}\\)<\/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<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655171769\" data-type=\"problem\">\r\n<p id=\"fs-id1170655219838\">Find the sum: \\(\\frac{5}{9}+\\frac{2}{9}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655063055\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654965957\">\\(\\frac{7}{9}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170655113498\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170655174670\" data-type=\"exercise\">\r\n<div data-type=\"solution\">\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-id1170655083064\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654878890\" data-type=\"problem\">\r\n<p id=\"fs-id1170655029152\">Find the difference: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{23}{24}-\\phantom{\\rule{0.2em}{0ex}}\\frac{13}{24}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655120820\" 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-803\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{23}{24}-\\phantom{\\rule{0.2em}{0ex}}\\frac{13}{24}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract the numerators and place the difference over the common denominator.<\/td>\r\n<td>\\(\\frac{-23-13}{24}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(\\frac{-36}{24}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify. Remember, \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{a}{b}=\\frac{-a}{b}\\).<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{2}\\)<\/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 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655107412\" data-type=\"problem\">\r\n<p id=\"fs-id1170654982582\">Find the difference: \\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{19}{28}-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{28}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654984424\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654977278\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{26}{28}\\)<\/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 id=\"fs-id1170655117407\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170654861951\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654925186\" data-type=\"solution\">\r\n\r\nNow we will do an example that has both addition and subtraction.\r\n\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 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654958831\" data-type=\"problem\">\r\n<p id=\"fs-id1170655132925\">Simplify: \\(\\frac{3}{8}+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{8}\\right)-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{8}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655127415\" 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-89\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Add and subtract fractions\u2014do they have a common denominator? Yes.<\/td>\r\n<td>\\(\\frac{3}{8}+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{8}\\right)-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add and subtract the numerators and place the difference over the common denominator.<\/td>\r\n<td>\\(\\frac{3+\\left(-5\\right)-1}{8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify left to right.<\/td>\r\n<td>\\(\\frac{-2-1}{8}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}\\)<\/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 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655219867\" data-type=\"problem\">\r\n<p id=\"fs-id1170655197202\">Simplify: \\(\\frac{2}{5}+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{9}\\right)-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{9}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654972445\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654981205\">\\(-1\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Add or Subtract Fractions with Different Denominators<\/h1>\r\n<p id=\"fs-id1170655126325\">As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common <span class=\"no-emphasis\" data-type=\"term\">denominator<\/span> of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.<\/p>\r\n\r\n<div id=\"fs-id1170655194149\" 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\">Least Common Denominator<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655061300\">After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!<a id=\"example10\"><\/a><\/p>\r\n\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-id1170654983383\" data-type=\"exercise\">\r\n<div id=\"fs-id1170654938746\" data-type=\"problem\">\r\n<p id=\"fs-id1170655112854\">Add: \\(\\frac{7}{12}+\\frac{5}{18}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655216078\" 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-id1170655396298\" data-type=\"media\" data-alt=\"In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have \u201cStep 1. Do they have a common denominator? No \u2013 rewrite each fraction with the LCD (least common denominator).\u201d To the right of this, we have the statement \u201cNo. Find the LCD 12, 18.\u201d To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have \u201cChange into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!\u201d To the right of this, we have 7\/12 plus 5\/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21\/36 plus 10\/36.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001a_new.jpg\" alt=\"In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have \u201cStep 1. Do they have a common denominator? No \u2013 rewrite each fraction with the LCD (least common denominator).\u201d To the right of this, we have the statement \u201cNo. Find the LCD 12, 18.\u201d To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have \u201cChange into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!\u201d To the right of this, we have 7\/12 plus 5\/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21\/36 plus 10\/36.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170654914425\" data-type=\"media\" data-alt=\"The next step reads \u201cStep 2. Add or subtract the fractions.\u201d The hint reads \u201cAdd.\u201d And we have 31\/36.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001b_new.jpg\" alt=\"The next step reads \u201cStep 2. Add or subtract the fractions.\u201d The hint reads \u201cAdd.\u201d And we have 31\/36.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170654931846\" data-type=\"media\" data-alt=\"The final step reads \u201cStep 3. Simplify, if possible.\u201d The explanation reads \u201cBecause 31 is a prime number, it has no factors in common with 36. The answer is simplified.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001c_new.jpg\" alt=\"The final step reads \u201cStep 3. Simplify, if possible.\u201d The explanation reads \u201cBecause 31 is a prime number, it has no factors in common with 36. The answer is simplified.\u201d\" 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 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655123979\" data-type=\"problem\">\r\n<p id=\"fs-id1170655123981\">Add: \\(\\frac{7}{12}+\\frac{11}{15}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655223620\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655108732\">\\(\\frac{79}{60}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Add or Subtract Fractions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1166424871688\" class=\"stepwise\" type=\"1\">\r\n \t<li>Do they have a common denominator?\r\n<ul id=\"fs-id1170655108352\" data-bullet-style=\"open-circle\">\r\n \t<li>Yes\u2014go to step 2.<\/li>\r\n \t<li>No\u2014rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Add or subtract the fractions.<\/li>\r\n \t<li>Simplify, if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170655092253\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170654903279\" data-type=\"exercise\">\r\n<div id=\"fs-id1170655062834\" data-type=\"solution\">\r\n\r\nWhen finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654940613\">Look at the factors of the LCD and then at each column above those factors. The \u201cmissing\u201d factors of each denominator are the numbers we need.<\/p>\r\n<span id=\"fs-id1170655102403\" data-type=\"media\" data-alt=\"The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked \u201cmissing factors.\u201d The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_002_img_new.jpg\" alt=\"The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked \u201cmissing factors.\u201d The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654967245\">In <a class=\"autogenerated-content\" href=\"#example10\">(Example 10)<\/a>, the LCD, 36, has two factors of 2 and two factors of \\(3\\).<\/p>\r\n<p id=\"fs-id1170655175041\">The numerator 12 has two factors of 2 but only one of 3\u2014so it is \u201cmissing\u201d one 3\u2014we multiply the numerator and denominator by 3<\/p>\r\n<p id=\"fs-id1170655175044\">The numerator 18 is missing one factor of 2\u2014so we multiply the numerator and denominator by 2<\/p>\r\n<p id=\"fs-id1170654972409\">We will apply this method as we subtract the fractions in the next example.<\/p>\r\n\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-id1170655000686\" data-type=\"problem\">\r\n<p id=\"fs-id1170655081926\">Subtract: \\(\\frac{7}{15}-\\phantom{\\rule{0.2em}{0ex}}\\frac{19}{24}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655200456\" 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-id1170655150350\">Do the fractions have a common denominator? No, so we need to find the LCD.<\/p>\r\n\r\n<table id=\"eip-id1169754059819\" class=\"unnumbered unstyled can-break has-images\" summary=\"In the first line of this problem, the expression 7\/15 minus 19\/24 is given. On the following line, we have the direction \u201cFind the LCD.\u201d The number 15 is factored into 3 times 5 with three extra spaces before the 3, and the number 24 is factored into 2 times 2 times 2 times 3 with an extra space after the 3. The LCD is marked as 2 times 2 times 2 times 3 times 5, which is equal to 120. On the following line, we have the direction \u201cMultiply 8 in the first fraction and 5 in the second fraction to get the LCD.\u201d Under this reads, Rewrite as equivalent fractions with the LCD.\u201d We have the quantity (7 times 8) divided by the quantity (15 times 8) minus the fraction with (19 times 5) as the numerator and (24 times 5) as the denominator. The following line tells us to \u201cSimplify each numerator and denominator.\u201d Hence, we have 56\/120 minus 95\/120. The next lines tells us to \u201cSubtract.\u201d Hence, we have negative 39\/120. The next line tells us to \u201cRewrite showing the common factor of 3.\u201d We have negative (13 times 3) divided by the quantity (40 times 3). We are then instructed to \u201cRemove the common factor to simplify.\u201d We obtain negative 13\/40.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Find the LCD.\\(\\phantom{\\rule{5em}{0ex}}\\)<span id=\"eip-id1169754362252\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<td style=\"width: 441.641px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Notice, 15 is \u201cmissing\u201d three factors of 2 and 24 is \u201cmissing\u201d the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.<\/td>\r\n<td style=\"width: 441.641px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Rewrite as equivalent fractions with the LCD.<\/td>\r\n<td style=\"width: 441.641px;\"><span id=\"eip-id1169754058803\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Simplify.<\/td>\r\n<td style=\"width: 441.641px;\"><span id=\"eip-id1169754058820\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Subtract.<\/td>\r\n<td style=\"width: 441.641px;\">\\(\\phantom{\\rule{2em}{0ex}}-\\phantom{\\rule{0.2em}{0ex}}\\frac{39}{120}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Check to see if the answer can be simplified.<\/td>\r\n<td style=\"width: 441.641px;\">\\(\\phantom{\\rule{2em}{0ex}}-\\phantom{\\rule{0.2em}{0ex}}\\frac{13\\cdot 3}{40\\cdot 3}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Both 39 and 120 have a factor of 3.<\/td>\r\n<td style=\"width: 441.641px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 532.578px;\">Simplify.<\/td>\r\n<td style=\"width: 441.641px;\">\\(\\phantom{\\rule{2em}{0ex}}-\\phantom{\\rule{0.2em}{0ex}}\\frac{13}{40}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDo not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!\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 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654968270\" data-type=\"problem\">\r\n<p id=\"fs-id1170655175417\">Subtract: \\(\\frac{13}{24}-\\phantom{\\rule{0.2em}{0ex}}\\frac{17}{32}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655164359\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655164361\">\\(\\frac{1}{96}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655129917\">We now have all four operations for fractions. The table below summarizes <span class=\"no-emphasis\" data-type=\"term\">fraction operations<\/span>.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<table style=\"width: 100%;\" border=\"0\"><caption>Summary of Fraction Operations<\/caption>\r\n<tbody>\r\n<tr>\r\n<th scope=\"col\">Fraction Operation<\/th>\r\n<th scope=\"col\">Sample Equation<\/th>\r\n<th scope=\"col\">What to Do<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>Fraction multiplication<\/td>\r\n<td>\\(\\frac{a}{b} \\cdot \\frac{c}{d}=\\frac{ac}{bd}\\)<\/td>\r\n<td>Multiply the numerators and multiply the denominators<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Fraction division<\/td>\r\n<td>\\(\\frac{a}{b} \\div \\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}\\)<\/td>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Fraction addition<\/td>\r\n<td>\\(\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\)<\/td>\r\n<td>Add the numerators and place the sum over the common denominator.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Fraction subtraction<\/td>\r\n<td>\\(\\frac{a}{c}-\\phantom{\\rule{0.2em}{0ex}}\\frac{b}{c}=\\frac{a-b}{c}\\)<\/td>\r\n<td>Subtract the numerators and place the difference over the common denominator.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.\r\n\r\n<\/div>\r\n<h1>Use the Order of Operations to Simplify Complex Fractions<\/h1>\r\n<p id=\"fs-id1170654969636\">We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. We simplified the complex fraction \\(\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\) by dividing \\(\\frac{3}{4}\\) by \\(\\frac{5}{8}\\).<\/p>\r\n<p id=\"fs-id1170655111615\">Now we\u2019ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.<\/p>\r\n\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-id1170655189464\" data-type=\"exercise\">\r\n<div id=\"fs-id1170655189466\" data-type=\"problem\">\r\n<p id=\"fs-id1170654962386\">Simplify: \\(\\frac{{\\left(\\frac{1}{2}\\right)}^{2}}{4+{3}^{2}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654953233\" 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-id1170654968903\" data-type=\"media\" data-alt=\"In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have \u201cStep 1. Simplify the numerator. Remember one half squared means one half times one half.\u201d To the right of this, we have the quantity (1\/2) squared all over the quantity (4 plus 3 squared). Then, we have 1\/4 over the quantity (4 plus 3 squared).\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010a_new.jpg\" alt=\"In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have \u201cStep 1. Simplify the numerator. Remember one half squared means one half times one half.\u201d To the right of this, we have the quantity (1\/2) squared all over the quantity (4 plus 3 squared). Then, we have 1\/4 over the quantity (4 plus 3 squared).\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655096686\" data-type=\"media\" data-alt=\"The next line\u2019s direction reads \u201cStep 2. Simplify the denominator.\u201d To the right of this, we have 1\/4 over the quantity (4 plus 9), under which we have 1\/4 over 13.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010b_new.jpg\" alt=\"The next line\u2019s direction reads \u201cStep 2. Simplify the denominator.\u201d To the right of this, we have 1\/4 over the quantity (4 plus 9), under which we have 1\/4 over 13.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655090064\" data-type=\"media\" data-alt=\"The final step is \u201cStep 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.\u201d To the right we have 1\/4 divided by 13. Then we have 1\/4 times 1\/13, which equals 1\/52.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010c_new.jpg\" alt=\"The final step is \u201cStep 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.\u201d To the right we have 1\/4 divided by 13. Then we have 1\/4 times 1\/13, which equals 1\/52.\" 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 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655029722\" data-type=\"problem\">\r\n<p id=\"fs-id1170655029724\">Simplify: \\(\\frac{{\\left(\\frac{1}{3}\\right)}^{2}}{{2}^{3}+2}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655178937\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655178939\">\\(\\frac{1}{90}\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">HOW TO: Simplify Complex Fractions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol id=\"fs-id1166424845977\" class=\"stepwise\" type=\"1\">\r\n \t<li>Simplify the numerator.<\/li>\r\n \t<li>Simplify the denominator.<\/li>\r\n \t<li>Divide the numerator by the denominator. Simplify if possible.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654962204\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170654962207\" data-type=\"exercise\">\r\n<div id=\"fs-id1170655029722\" data-type=\"problem\">\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-id1170654968376\" data-type=\"problem\">\r\n<p id=\"fs-id1170655021220\">Simplify: \\(\\frac{\\frac{1}{2}+\\frac{2}{3}}{\\frac{3}{4}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{6}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655102612\" 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-id1170655102617\">It may help to put parentheses around the numerator and the denominator.<\/p>\r\n\r\n<table id=\"eip-20\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(\\frac{\\left(\\frac{1}{2}+\\frac{2}{3}\\right)}{\\left(\\frac{3}{4}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{6}\\right)}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).<\/td>\r\n<td>\\(\\frac{\\left(\\frac{3}{6}+\\frac{4}{6}\\right)}{\\left(\\frac{9}{12}-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{12}\\right)}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(\\frac{\\left(\\frac{7}{6}\\right)}{\\left(\\frac{7}{12}\\right)}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide the numerator by the denominator.<\/td>\r\n<td>\\(\\frac{7}{6}\\div \\frac{7}{12}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(\\frac{7}{6}\\cdot\\frac{12}{7}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide out common factors.<\/td>\r\n<td>\\(\\frac{7\\cdot6\\cdot2}{6\\cdot7}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>\\(2\\)<\/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 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655213970\" data-type=\"problem\">\r\n<p id=\"fs-id1170655213972\">Simplify: \\(\\frac{\\frac{1}{3}+\\frac{1}{2}}{\\frac{3}{4}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{3}}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655166560\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655166563\">2<\/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>Key Concepts<\/h1>\r\n<ul id=\"fs-id1170655194091\" data-bullet-style=\"bullet\">\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\">Fraction Addition and Subtraction:<\/strong> If \\(a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c\\) are numbers where \\(c\\ne 0\\), then\r\n\\(\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\) and \\(\\frac{a}{c}-\\phantom{\\rule{0.2em}{0ex}}\\frac{b}{c}=\\frac{a-b}{c}\\).\r\nTo add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/li>\r\n \t<li><strong data-effect=\"bold\">Strategy for Adding or Subtracting Fractions<\/strong>\r\n<ol id=\"fs-id1170654966823\" class=\"stepwise\" type=\"1\">\r\n \t<li>Do they have a common denominator?\r\nYes\u2014go to step 2.\r\nNo\u2014Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\r\n \t<li>Add or subtract the fractions.<\/li>\r\n \t<li>Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Simplify Complex Fractions<\/strong>\r\n<ol id=\"fs-id1170655260316\" class=\"stepwise\" type=\"1\">\r\n \t<li>Simplify the numerator.<\/li>\r\n \t<li>Simplify the denominator.<\/li>\r\n \t<li>Divide the numerator by the denominator. Simplify if possible.<\/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-id1166424922293\">\r\n \t<dt>least common denominator<\/dt>\r\n \t<dd id=\"fs-id1166424922299\">The least common denominator (LCD) of two fractions is the Least common multiple (LCM) of their denominators.<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h1>1.3 Exercise Set<\/h1>\r\n<p id=\"fs-id1166424880356\">In the following exercises, multiply.<\/p>\r\n\r\n<ol class=\"twocolumn\">\r\n \t<li>\\(\\frac{3}{4}\\cdot \\frac{9}{10}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}\\cdot\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}\\right)\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{9}\\cdot \\frac{3}{10}\\)<\/li>\r\n \t<li>\\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{14}{15}\\right) \\cdot \\left(\\frac{9}{20}\\right)\\)<\/li>\r\n \t<li>\\(\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{63}{84}\\right)\\cdot \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{44}{90}\\right)\\)<\/li>\r\n \t<li>\\(4\\cdot \\frac{5}{11}\\)<\/li>\r\n \t<li>\\(-8\\cdot \\left(\\frac{17}{4}\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166422785479\">In the following exercises, divide.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"8\">\r\n \t<li>\\(\\frac{3}{4}\\div \\frac{2}{3}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{9}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{4}\\right)\\)<\/li>\r\n \t<li>\\(\\frac{5}{18}\\div \\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{15}{24}\\right)\\)<\/li>\r\n \t<li>\\(-5\\div \\frac{1}{2}\\)<\/li>\r\n \t<li>\\(\\frac{3}{4}\\div \\left(-12\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170654215028\">In the following exercises, simplify.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"13\">\r\n \t<li>\\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{21}}{\\frac{12}{35}}\\)<\/li>\r\n \t<li>\\(\\frac{-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}}{2}\\)<\/li>\r\n \t<li>\\(\\frac{22+3}{10}\\)<\/li>\r\n \t<li>\\(\\frac{48}{24-15}\\)<\/li>\r\n \t<li>\\(\\frac{-6+6}{8+4}\\)<\/li>\r\n \t<li>\\(\\frac{4\\cdot3}{6\\cdot6}\\)<\/li>\r\n \t<li>\\(\\frac{{4}^{2}-1}{25}\\)<\/li>\r\n \t<li>\\(\\frac{8\\cdot3+2\\cdot9}{14+3}\\)<\/li>\r\n \t<li>\\(\\frac{5\\cdot6-3\\cdot4}{4\\cdot5-2\\cdot3}\\)<\/li>\r\n \t<li>\\(\\frac{{5}^{2}-{3}^{2}}{3-5}\\)<\/li>\r\n \t<li>\\(\\frac{7\\cdot4-2\\left(8-5\\right)}{9\\cdot3-3\\cdot5}\\)<\/li>\r\n \t<li>\\(\\frac{9\\left(8-2\\right)-3\\left(15-7\\right)}{6\\left(7-1\\right)-3\\left(17-9\\right)}\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166425099458\">In the following exercises, add.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"25\">\r\n \t<li>\\(\\frac{6}{13}+\\frac{5}{13}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{16}+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{16}\\right)\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{8}{17}+\\frac{15}{17}\\)<\/li>\r\n \t<li>\\(\\frac{6}{13}+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{13}\\right)+\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{13}\\right)\\)<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1170655121728\" data-type=\"exercise\">\r\n<div id=\"fs-id1170655121730\" data-type=\"problem\">In the following exercises, subtract.<\/div>\r\n<ol class=\"twocolumn\" start=\"29\">\r\n \t<li data-type=\"problem\">\\(\\frac{11}{15}-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{15}\\)<\/li>\r\n \t<li data-type=\"problem\">\\(\\frac{11}{12}-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{12}\\)<\/li>\r\n \t<li data-type=\"problem\">\\(\\frac{19}{21}-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{21}\\)<\/li>\r\n \t<li data-type=\"problem\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{5}-\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{5}\\right)\\)<\/li>\r\n \t<li data-type=\"problem\">\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{7}{9}-\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{9}\\right)\\)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1166422823699\">In the following exercises, add or subtract.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"34\">\r\n \t<li>\\(\\frac{1}{2}+\\frac{1}{7}\\)<\/li>\r\n \t<li>\\(\\frac{1}{3}-\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{9}\\right)\\)<\/li>\r\n \t<li>\\(\\frac{7}{12}+\\frac{5}{8}\\)<\/li>\r\n \t<li>\\(\\frac{7}{12}-\\phantom{\\rule{0.2em}{0ex}}\\frac{9}{16}\\)<\/li>\r\n \t<li>\\(\\frac{2}{3}-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{8}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{11}{30}+\\frac{27}{40}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{13}{30}+\\frac{25}{42}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{39}{56}-\\phantom{\\rule{0.2em}{0ex}}\\frac{22}{35}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}-\\left(-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{4}\\right)\\)<\/li>\r\n \t<li>\\(1+\\frac{7}{8}\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166424777198\">In the following exercises, simplify.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"44\">\r\n \t<li>\\(\\frac{{2}^{3}+{4}^{2}}{{\\left(\\frac{2}{3}\\right)}^{2}}\\)<\/li>\r\n \t<li>\\(\\frac{{\\left(\\frac{3}{5}\\right)}^{2}}{{\\left(\\frac{3}{7}\\right)}^{2}}\\)<\/li>\r\n \t<li>\\(\\frac{2}{\\frac{1}{3}+\\frac{1}{5}}\\)<\/li>\r\n \t<li>\\(\\frac{\\frac{7}{8}-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{3}}{\\frac{1}{2}+\\frac{3}{8}}\\)<\/li>\r\n \t<li>\\(\\frac{1}{2}+\\frac{2}{3}\\cdot \\frac{5}{12}\\)<\/li>\r\n \t<li>\\(1-\\phantom{\\rule{0.2em}{0ex}}\\frac{3}{5}\\div \\frac{1}{10}\\)<\/li>\r\n \t<li>\\(1-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{6}\\div \\frac{1}{12}\\)<\/li>\r\n \t<li>\\(\\frac{2}{3}+\\frac{1}{6}+\\frac{3}{4}\\)<\/li>\r\n \t<li>\\(\\frac{3}{8}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{6}+\\frac{3}{4}\\)<\/li>\r\n \t<li>\\(12\\left(\\frac{9}{20}-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{15}\\right)\\)<\/li>\r\n \t<li>\\(\\frac{\\frac{5}{8}+\\frac{1}{6}}{\\frac{19}{24}}\\)<\/li>\r\n \t<li>\\(\\left(\\frac{5}{9}+\\frac{1}{6}\\right)\\div \\left(\\frac{2}{3}-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{2}\\right)\\)<\/li>\r\n<\/ol>\r\n<ol start=\"56\">\r\n \t<li><strong data-effect=\"bold\">Decorating.<\/strong> Kayla is making covers for the throw pillows on her sofa. For each pillow cover, she needs \\(\\frac{1}{2}\\) yard of print fabric and \\(\\frac{3}{8}\\) yard of solid fabric. What is the total amount of fabric Kayla needs for each pillow cover?<\/li>\r\n \t<li><strong data-effect=\"bold\">Baking.<\/strong> A recipe for chocolate chip cookies calls for \\(\\frac{3}{4}\\) cup brown sugar. Leona wants to double the recipe. a) How much brown sugar will Leona 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 Leona could measure the brown sugar needed to double the cookie recipe.<\/li>\r\n \t<li><strong data-effect=\"bold\">Portions.<\/strong> Regin 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?<\/li>\r\n<\/ol>\r\n<h1>Answers:<\/h1>\r\n<ol class=\"twocolumn\">\r\n \t<li>\\(\\frac{27}{40}\\)<\/li>\r\n \t<li>\\(\\frac{1}{4}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{6}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{21}{50}\\)<\/li>\r\n \t<li>\\(\\frac{11}{30}\\)<\/li>\r\n \t<li>\\(\\frac{20}{11}\\)<\/li>\r\n \t<li>\\(-34\\)<\/li>\r\n \t<li>\\(\\frac{9}{8}\\)<\/li>\r\n \t<li>\\(\\frac{4}{9}\\).<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{4}{9}\\)<\/li>\r\n \t<li>\\(-10\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{16}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{10}{9}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{5}\\)<\/li>\r\n \t<li>\\(\\frac{5}{2}\\)<\/li>\r\n \t<li>\\(\\frac{16}{3}\\)<\/li>\r\n \t<li>0<\/li>\r\n \t<li>\\(\\frac{1}{3}\\)<\/li>\r\n \t<li>\\(\\frac{3}{5}\\)<\/li>\r\n \t<li>\\(2\\frac{8}{17}\\)<\/li>\r\n \t<li>\\(1\\frac{2}{7}\\)<\/li>\r\n \t<li>\\(-8\\)<\/li>\r\n \t<li>\\(\\frac{11}{6}\\)<\/li>\r\n \t<li>\\(\\frac{5}{2}\\)<\/li>\r\n \t<li>\\(\\frac{11}{13}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{5}{8}\\)<\/li>\r\n \t<li>\\(\\frac{7}{17}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{16}{13}\\)<\/li>\r\n \t<li>\\(\\frac{4}{15}\\)<\/li>\r\n \t<li>\\(\\frac{1}{2}\\)<\/li>\r\n \t<li>\\(\\frac{5}{7}\\)<\/li>\r\n \t<li>\\(\\frac{1}{5}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{2}{9}\\)<\/li>\r\n \t<li>\\(\\frac{9}{14}\\)<\/li>\r\n \t<li>\\(\\frac{4}{9}\\)<\/li>\r\n \t<li>\\(\\frac{29}{24}\\)<\/li>\r\n \t<li>\\(\\frac{1}{48}\\)<\/li>\r\n \t<li>\\(\\frac{7}{24}\\)<\/li>\r\n \t<li>\\(\\frac{37}{120}\\)<\/li>\r\n \t<li>\\(\\frac{17}{105}\\)<\/li>\r\n \t<li>\\(-\\phantom{\\rule{0.2em}{0ex}}\\frac{53}{40}\\)<\/li>\r\n \t<li>\\(\\frac{1}{12}\\)<\/li>\r\n \t<li>\\(\\frac{15}{8}\\)<\/li>\r\n \t<li>\\(54\\)<\/li>\r\n \t<li>\\(\\frac{49}{25}\\)<\/li>\r\n \t<li>\\(\\frac{15}{4}\\)<\/li>\r\n \t<li>\\(\\frac{5}{21}\\)<\/li>\r\n \t<li>\\(\\frac{7}{9}\\)<\/li>\r\n \t<li>\\(-5\\)<\/li>\r\n \t<li>\\(-9\\)<\/li>\r\n \t<li>\\(\\frac{19}{12}\\)<\/li>\r\n \t<li>\\(\\frac{23}{24}\\)<\/li>\r\n \t<li>\\(\\frac{11}{5}\\)<\/li>\r\n \t<li>1<\/li>\r\n \t<li>\\(\\frac{13}{3}\\)<\/li>\r\n \t<li>\\(\\frac{7}{8}\\) yard<\/li>\r\n \t<li>a) \\(1\\frac{1}{2}\\) cups b) answers will vary<\/li>\r\n \t<li>20 bags<\/li>\r\n<\/ol>\r\n<div data-type=\"glossary\"><\/div>\r\n<h1>Attributions<\/h1>\r\nThis chapter has been adapted from \u201cAdd and Subtract 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.","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 it is expected that you will be able to:<\/p>\n<ul>\n<li>Multiply and divide fractions<\/li>\n<li>Simplifying expressions with fraction bar<\/li>\n<li>Add or subtract fractions with a common denominator<\/li>\n<li>Add or subtract fractions with different denominators<\/li>\n<li>Use the order of operations to simplify complex fractions<\/li>\n<\/ul>\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\/businesstechnicalmath\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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=\"#example1\">Example 1<\/a>, we will multiply negative and a positive, so the product will be negative.<a id=\"example1\"><\/a><\/p>\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-id1170654025836\" data-type=\"problem\">\n<p id=\"fs-id1170653750622\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 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\/businesstechnicalmath\/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\/businesstechnicalmath\/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<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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 2<\/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\/businesstechnicalmath\/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\" style=\"width: 100%;\" 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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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 2<\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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<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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 3<\/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\/businesstechnicalmath\/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\" style=\"width: 100%;\" 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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 3<\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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<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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 4<\/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\/businesstechnicalmath\/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\" style=\"width: 100%;\" 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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/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\/businesstechnicalmath\/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 4<\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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>\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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 5<\/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\/businesstechnicalmath\/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\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 5<\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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>\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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 6<\/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\/businesstechnicalmath\/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\" style=\"width: 100%;\" summary=\"-\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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 6<\/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\/businesstechnicalmath\/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>\n<\/div>\n<h1>Add or Subtract Fractions with a Common Denominator<\/h1>\n<p id=\"fs-id1170655354165\">When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.<\/p>\n<div id=\"fs-id1170654919725\" 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 Addition and Subtraction<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170655188656\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-769150efa7aff4127e74e49dcde216ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#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;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"76\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c540d76c127a15e856d4b530b506abdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/>, then<\/p>\n<div id=\"fs-id1170655166745\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f95db0a42671c9b1bcaef124b493ce31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#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;&#125;&#123;&#99;&#125;&#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;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"252\" style=\"vertical-align: -6px;\" \/><\/p>\n<p>To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"manipulative-math\" data-type=\"note\">\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-id1170655202348\" data-type=\"problem\">\n<p id=\"fs-id1170655132751\">Find the sum: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c771245cec54742c798adbe93b04e301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655126354\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-899\" style=\"height: 65px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 529.757px; height: 15px;\"><\/td>\n<td style=\"width: 216.424px; height: 15px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c771245cec54742c798adbe93b04e301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 50px;\">\n<td style=\"width: 529.757px; height: 50px;\">Add the numerators and place the sum over the common denominator.<\/p>\n<p>Simplify.<\/td>\n<td style=\"width: 216.424px; height: 50px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-15653b1cceaa09e334d461206c50356e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#43;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca0b99a2ec22006d674eb42e47fe6367_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#55;&#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 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655171769\" data-type=\"problem\">\n<p id=\"fs-id1170655219838\">Find the sum: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60bb8dd0eac7073d62a666741a2266ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655063055\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654965957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7e8ce2f90e409bd9f90b29b7e9dacb23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#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>\n<div id=\"fs-id1170655113498\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170655174670\" data-type=\"exercise\">\n<div data-type=\"solution\">\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-id1170655083064\" data-type=\"exercise\">\n<div id=\"fs-id1170654878890\" data-type=\"problem\">\n<p id=\"fs-id1170655029152\">Find the difference: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d50a08bba6e0bb023a07b006f69564a_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;&#50;&#52;&#125;&#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;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655120820\" 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-803\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d50a08bba6e0bb023a07b006f69564a_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;&#50;&#52;&#125;&#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;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract the numerators and place the difference over the common denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-196a8fd7594cbdf760c9a9931ee75c34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#51;&#45;&#49;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-409e46f5f616fbe679274be8f7a23c11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#54;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify. Remember, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-68755c96a4e0bb43b3703ab433ca35da_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;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c6fec65f45c5eadc522bbe6950a4fdaf_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;&#50;&#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 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655107412\" data-type=\"problem\">\n<p id=\"fs-id1170654982582\">Find the difference: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-15466bcb7743fcd10dd072399939ed8f_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;&#57;&#125;&#123;&#50;&#56;&#125;&#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;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654984424\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654977278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b112ac1cc5bbe4d17389019558e9d7e7_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;&#54;&#125;&#123;&#50;&#56;&#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>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655117407\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654861951\" data-type=\"exercise\">\n<div id=\"fs-id1170654925186\" data-type=\"solution\">\n<p>Now we will do an example that has both addition and subtraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\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-id1170654958831\" data-type=\"problem\">\n<p id=\"fs-id1170655132925\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5f301f201845bc67c710ed13275bf01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#43;&#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;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"109\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655127415\" 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-89\" summary=\".\">\n<tbody>\n<tr>\n<td>Add and subtract fractions\u2014do they have a common denominator? Yes.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5f301f201845bc67c710ed13275bf01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#43;&#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;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"109\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add and subtract the numerators and place the difference over the common denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-af5b01808703954a8583590061f7d858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify left to right.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2106d14762d8bcb8a7f68bedb672df4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#45;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3573a0c03636e7c7e09ae5e613d5b6eb_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;\" 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>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655219867\" data-type=\"problem\">\n<p id=\"fs-id1170655197202\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7002cb1348c5ec13ba655e09cb76402d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#43;&#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;&#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=\"23\" width=\"109\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654972445\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654981205\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Add or Subtract Fractions with Different Denominators<\/h1>\n<p id=\"fs-id1170655126325\">As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common <span class=\"no-emphasis\" data-type=\"term\">denominator<\/span> of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.<\/p>\n<div id=\"fs-id1170655194149\" 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\">Least Common Denominator<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655061300\">After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!<a id=\"example10\"><\/a><\/p>\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-id1170654983383\" data-type=\"exercise\">\n<div id=\"fs-id1170654938746\" data-type=\"problem\">\n<p id=\"fs-id1170655112854\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-82968b0cace5472e682fd58974352443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655216078\" 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-id1170655396298\" data-type=\"media\" data-alt=\"In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have \u201cStep 1. Do they have a common denominator? No \u2013 rewrite each fraction with the LCD (least common denominator).\u201d To the right of this, we have the statement \u201cNo. Find the LCD 12, 18.\u201d To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have \u201cChange into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!\u201d To the right of this, we have 7\/12 plus 5\/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21\/36 plus 10\/36.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001a_new.jpg\" alt=\"In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have \u201cStep 1. Do they have a common denominator? No \u2013 rewrite each fraction with the LCD (least common denominator).\u201d To the right of this, we have the statement \u201cNo. Find the LCD 12, 18.\u201d To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have \u201cChange into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!\u201d To the right of this, we have 7\/12 plus 5\/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21\/36 plus 10\/36.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170654914425\" data-type=\"media\" data-alt=\"The next step reads \u201cStep 2. Add or subtract the fractions.\u201d The hint reads \u201cAdd.\u201d And we have 31\/36.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001b_new.jpg\" alt=\"The next step reads \u201cStep 2. Add or subtract the fractions.\u201d The hint reads \u201cAdd.\u201d And we have 31\/36.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170654931846\" data-type=\"media\" data-alt=\"The final step reads \u201cStep 3. Simplify, if possible.\u201d The explanation reads \u201cBecause 31 is a prime number, it has no factors in common with 36. The answer is simplified.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_001c_new.jpg\" alt=\"The final step reads \u201cStep 3. Simplify, if possible.\u201d The explanation reads \u201cBecause 31 is a prime number, it has no factors in common with 36. The answer is simplified.\u201d\" 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 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655123979\" data-type=\"problem\">\n<p id=\"fs-id1170655123981\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-27d7bf29b1b0917497367c3df4e4e05e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655223620\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655108732\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-64e25e8ede9f97da22579b3e05475e39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#57;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Add or Subtract Fractions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166424871688\" class=\"stepwise\" type=\"1\">\n<li>Do they have a common denominator?\n<ul id=\"fs-id1170655108352\" data-bullet-style=\"open-circle\">\n<li>Yes\u2014go to step 2.<\/li>\n<li>No\u2014rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\n<\/ul>\n<\/li>\n<li>Add or subtract the fractions.<\/li>\n<li>Simplify, if possible.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655092253\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654903279\" data-type=\"exercise\">\n<div id=\"fs-id1170655062834\" data-type=\"solution\">\n<p>When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654940613\">Look at the factors of the LCD and then at each column above those factors. The \u201cmissing\u201d factors of each denominator are the numbers we need.<\/p>\n<p><span id=\"fs-id1170655102403\" data-type=\"media\" data-alt=\"The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked \u201cmissing factors.\u201d The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_002_img_new.jpg\" alt=\"The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked \u201cmissing factors.\u201d The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654967245\">In <a class=\"autogenerated-content\" href=\"#example10\">(Example 10)<\/a>, the LCD, 36, has two factors of 2 and two factors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/>.<\/p>\n<p id=\"fs-id1170655175041\">The numerator 12 has two factors of 2 but only one of 3\u2014so it is \u201cmissing\u201d one 3\u2014we multiply the numerator and denominator by 3<\/p>\n<p id=\"fs-id1170655175044\">The numerator 18 is missing one factor of 2\u2014so we multiply the numerator and denominator by 2<\/p>\n<p id=\"fs-id1170654972409\">We will apply this method as we subtract the fractions in the next example.<\/p>\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-id1170655000686\" data-type=\"problem\">\n<p id=\"fs-id1170655081926\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c66e98aafdb72a32e0b7be7cc2845c1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#53;&#125;&#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;&#57;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655200456\" 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-id1170655150350\">Do the fractions have a common denominator? No, so we need to find the LCD.<\/p>\n<table id=\"eip-id1169754059819\" class=\"unnumbered unstyled can-break has-images\" summary=\"In the first line of this problem, the expression 7\/15 minus 19\/24 is given. On the following line, we have the direction \u201cFind the LCD.\u201d The number 15 is factored into 3 times 5 with three extra spaces before the 3, and the number 24 is factored into 2 times 2 times 2 times 3 with an extra space after the 3. The LCD is marked as 2 times 2 times 2 times 3 times 5, which is equal to 120. On the following line, we have the direction \u201cMultiply 8 in the first fraction and 5 in the second fraction to get the LCD.\u201d Under this reads, Rewrite as equivalent fractions with the LCD.\u201d We have the quantity (7 times 8) divided by the quantity (15 times 8) minus the fraction with (19 times 5) as the numerator and (24 times 5) as the denominator. The following line tells us to \u201cSimplify each numerator and denominator.\u201d Hence, we have 56\/120 minus 95\/120. The next lines tells us to \u201cSubtract.\u201d Hence, we have negative 39\/120. The next line tells us to \u201cRewrite showing the common factor of 3.\u201d We have negative (13 times 3) divided by the quantity (40 times 3). We are then instructed to \u201cRemove the common factor to simplify.\u201d We obtain negative 13\/40.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 532.578px;\">Find the LCD.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8b0c01809498b952e2c488a85b3a8da6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span id=\"eip-id1169754362252\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td style=\"width: 441.641px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Notice, 15 is \u201cmissing\u201d three factors of 2 and 24 is \u201cmissing\u201d the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.<\/td>\n<td style=\"width: 441.641px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Rewrite as equivalent fractions with the LCD.<\/td>\n<td style=\"width: 441.641px;\"><span id=\"eip-id1169754058803\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Simplify.<\/td>\n<td style=\"width: 441.641px;\"><span id=\"eip-id1169754058820\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_003c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Subtract.<\/td>\n<td style=\"width: 441.641px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3461138e507139438166c974b8439b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;&#57;&#125;&#123;&#49;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"44\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Check to see if the answer can be simplified.<\/td>\n<td style=\"width: 441.641px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7eecf79d479fe0a0972a173b3ce87a69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#125;&#123;&#52;&#48;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Both 39 and 120 have a factor of 3.<\/td>\n<td style=\"width: 441.641px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 532.578px;\">Simplify.<\/td>\n<td style=\"width: 441.641px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bad28bb3e0099f2a0f8c356c86ec7a33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#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;&#51;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"37\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Do not simplify the equivalent fractions! If you do, you\u2019ll get back to the original fractions and lose the common denominator!<\/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 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654968270\" data-type=\"problem\">\n<p id=\"fs-id1170655175417\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7651ab1dec433a430b6a82ff919cc113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#50;&#52;&#125;&#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;&#55;&#125;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655164359\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655164361\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc690b9489c0b92381876c1b2f812f67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655129917\">We now have all four operations for fractions. The table below summarizes <span class=\"no-emphasis\" data-type=\"term\">fraction operations<\/span>.<\/p>\n<div class=\"textbox shaded\">\n<table style=\"width: 100%;\">\n<caption>Summary of Fraction Operations<\/caption>\n<tbody>\n<tr>\n<th scope=\"col\">Fraction Operation<\/th>\n<th scope=\"col\">Sample Equation<\/th>\n<th scope=\"col\">What to Do<\/th>\n<\/tr>\n<tr>\n<td>Fraction multiplication<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/td>\n<td>Multiply the numerators and multiply the denominators<\/td>\n<\/tr>\n<tr>\n<td>Fraction division<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03e1dc6b979cd07e4177baeda8e75f7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#32;&#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;\" \/><\/td>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<\/tr>\n<tr>\n<td>Fraction addition<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e19c83ca68864bbe71a6b53b7a1ad55c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/td>\n<td>Add the numerators and place the sum over the common denominator.<\/td>\n<\/tr>\n<tr>\n<td>Fraction subtraction<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-898411226b6c7055181edbee3a9d6025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#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;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"94\" style=\"vertical-align: -6px;\" \/><\/td>\n<td>Subtract the numerators and place the difference over the common denominator.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.<\/p>\n<\/div>\n<h1>Use the Order of Operations to Simplify Complex Fractions<\/h1>\n<p id=\"fs-id1170654969636\">We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates <span class=\"no-emphasis\" data-type=\"term\">division<\/span>. We simplified the complex fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/> by dividing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/>.<\/p>\n<p id=\"fs-id1170655111615\">Now we\u2019ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.<\/p>\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-id1170655189464\" data-type=\"exercise\">\n<div id=\"fs-id1170655189466\" data-type=\"problem\">\n<p id=\"fs-id1170654962386\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5660e71714e4251f8a9bdf13903c4dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#43;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"32\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654953233\" 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-id1170654968903\" data-type=\"media\" data-alt=\"In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have \u201cStep 1. Simplify the numerator. Remember one half squared means one half times one half.\u201d To the right of this, we have the quantity (1\/2) squared all over the quantity (4 plus 3 squared). Then, we have 1\/4 over the quantity (4 plus 3 squared).\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010a_new.jpg\" alt=\"In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have \u201cStep 1. Simplify the numerator. Remember one half squared means one half times one half.\u201d To the right of this, we have the quantity (1\/2) squared all over the quantity (4 plus 3 squared). Then, we have 1\/4 over the quantity (4 plus 3 squared).\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655096686\" data-type=\"media\" data-alt=\"The next line\u2019s direction reads \u201cStep 2. Simplify the denominator.\u201d To the right of this, we have 1\/4 over the quantity (4 plus 9), under which we have 1\/4 over 13.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010b_new.jpg\" alt=\"The next line\u2019s direction reads \u201cStep 2. Simplify the denominator.\u201d To the right of this, we have 1\/4 over the quantity (4 plus 9), under which we have 1\/4 over 13.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1170655090064\" data-type=\"media\" data-alt=\"The final step is \u201cStep 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.\u201d To the right we have 1\/4 divided by 13. Then we have 1\/4 times 1\/13, which equals 1\/52.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_06_010c_new.jpg\" alt=\"The final step is \u201cStep 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.\u201d To the right we have 1\/4 divided by 13. Then we have 1\/4 times 1\/13, which equals 1\/52.\" 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 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655029722\" data-type=\"problem\">\n<p id=\"fs-id1170655029724\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-502938353edf4dc0e169c6e487b4edfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"32\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655178937\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655178939\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4ec3cb5149ae42207f9367831b4719a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO: Simplify Complex Fractions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1166424845977\" class=\"stepwise\" type=\"1\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator. Simplify if possible.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654962204\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170654962207\" data-type=\"exercise\">\n<div id=\"fs-id1170655029722\" data-type=\"problem\">\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-id1170654968376\" data-type=\"problem\">\n<p id=\"fs-id1170655021220\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3872361da263da168621b661906510f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"34\" style=\"vertical-align: -13px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655102612\" 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-id1170655102617\">It may help to put parentheses around the numerator and the denominator.<\/p>\n<table id=\"eip-20\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-45fdcb8e2e877ef30c85634d8e25afb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#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;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"48\" style=\"vertical-align: -14px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-189e2aac637ff556bbf5571cde433811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#50;&#125;&#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;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"60\" style=\"vertical-align: -15px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-78494bd88eaf4efc9c43f834577772be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"30\" style=\"vertical-align: -15px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator by the denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-74eb8da2b48679b9bd4adf2aafe34357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#100;&#105;&#118;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fa002f65b978e7f2582903efbf880a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#99;&#100;&#111;&#116;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide out common factors.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7202466dbd59b1e57dd8516ae0041c61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#99;&#100;&#111;&#116;&#54;&#92;&#99;&#100;&#111;&#116;&#50;&#125;&#123;&#54;&#92;&#99;&#100;&#111;&#116;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" 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 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655213970\" data-type=\"problem\">\n<p id=\"fs-id1170655213972\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d6e7a25e900ab86b50c2555c1c5acadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"34\" style=\"vertical-align: -13px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655166560\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655166563\">2<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1170655194091\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Fraction Division:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\/businesstechnicalmath\/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\">Fraction Addition and Subtraction:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-769150efa7aff4127e74e49dcde216ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#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;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"76\" style=\"vertical-align: -4px;\" \/> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c540d76c127a15e856d4b530b506abdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/>, then<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e19c83ca68864bbe71a6b53b7a1ad55c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-898411226b6c7055181edbee3a9d6025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#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;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"94\" style=\"vertical-align: -6px;\" \/>.<br \/>\nTo add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/li>\n<li><strong data-effect=\"bold\">Strategy for Adding or Subtracting Fractions<\/strong>\n<ol id=\"fs-id1170654966823\" class=\"stepwise\" type=\"1\">\n<li>Do they have a common denominator?<br \/>\nYes\u2014go to step 2.<br \/>\nNo\u2014Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\n<li>Add or subtract the fractions.<\/li>\n<li>Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Simplify Complex Fractions<\/strong>\n<ol id=\"fs-id1170655260316\" class=\"stepwise\" type=\"1\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator. Simplify if possible.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1166424922293\">\n<dt>least common denominator<\/dt>\n<dd id=\"fs-id1166424922299\">The least common denominator (LCD) of two fractions is the Least common multiple (LCM) of their denominators.<\/dd>\n<\/dl>\n<\/div>\n<h1>1.3 Exercise Set<\/h1>\n<p id=\"fs-id1166424880356\">In the following exercises, multiply.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166422785479\">In the following exercises, divide.<\/p>\n<ol class=\"twocolumn\" start=\"8\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1170654215028\">In the following exercises, simplify.<\/p>\n<ol class=\"twocolumn\" start=\"13\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166425099458\">In the following exercises, add.<\/p>\n<ol class=\"twocolumn\" start=\"25\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-10d19c67b47914577890bb95b5a612a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-19951288c53bb963de219c8bb3bbbb68_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;&#43;&#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;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e780b0bb7d913f8732f74726fbee6a02_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;&#56;&#125;&#123;&#49;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-69ec1c0b0fe1a0cfb02bb1959a92db41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#51;&#125;&#43;&#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;&#48;&#125;&#123;&#49;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#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;&#50;&#125;&#123;&#49;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"159\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<div id=\"fs-id1170655121728\" data-type=\"exercise\">\n<div id=\"fs-id1170655121730\" data-type=\"problem\">In the following exercises, subtract.<\/div>\n<ol class=\"twocolumn\" start=\"29\">\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-77a86e02ef4987189152697cdb79f81b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#53;&#125;&#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;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -7px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4d7253b4ae849975306c5421ba3de1cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#50;&#125;&#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;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -7px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b08189dedd1fecedb1539319a115ab28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#125;&#123;&#50;&#49;&#125;&#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=\"57\" style=\"vertical-align: -7px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e9f66887c35d5da7cde26a2d3b36cdf8_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;&#45;&#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;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3a2edb9742623da3752e7bb893c7953e_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;&#45;&#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;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1166422823699\">In the following exercises, add or subtract.<\/p>\n<ol class=\"twocolumn\" start=\"34\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-77cb88f5ad02b403e31d15a4e57c4bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8b8aead1e5694d0b84a416f453d4fb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#45;&#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;&#125;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-45141b729bd349147e7f675d38d6428a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-951a5329baf1b2ebb834ffdfa42d0229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-43acd3d307d28ea97aee3b73a6dcdbcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2b76926b81d9ca6ae5fa6e76aff2bab_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;&#51;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9161c28fd7e44470fc2fc0cd4565849b_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;&#51;&#125;&#123;&#51;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#52;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0e45591696260fa10b23c2eafa9e6958_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;&#57;&#125;&#123;&#53;&#54;&#125;&#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;&#50;&#125;&#123;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c720cec9e1df78eaa2223026d546940_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;&#45;&#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;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c26c45fbe2aa0e9c0e14f7eebefab5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166424777198\">In the following exercises, simplify.<\/p>\n<ol class=\"twocolumn\" start=\"44\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5b985e9cb87c770bb5e4856b70ad405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"38\" style=\"vertical-align: -16px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b151b8a502ef6fe85908b804124b29dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"31\" style=\"vertical-align: -16px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9d7e6a70f6fadd1069804f71df0cedc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f677f7c8b86f675b8539c6ed9194a385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#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;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"34\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-606d3129dde3df8b29d82fcf2dc01b13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#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;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c6ebf1e0819bd1ffba68e5de59845029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#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;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-679fc2a3c0905a84a1c0b1e506a40a82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#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;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f8b5fda4e2fac2384cab766598e1492d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fd0b734b7b979723fbe6244f5a9a636f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#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;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-96f4d421731b2582dd9ccdd3bdb738ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#48;&#125;&#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;&#49;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e501375f1dd409ba7b728c0939b66c2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#125;&#123;&#50;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-177da65c190429f63971cc8f13a37da2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#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;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<ol start=\"56\">\n<li><strong data-effect=\"bold\">Decorating.<\/strong> Kayla is making covers for the throw pillows on her sofa. For each pillow cover, she needs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/> yard of print fabric and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/> yard of solid fabric. What is the total amount of fabric Kayla needs for each pillow cover?<\/li>\n<li><strong data-effect=\"bold\">Baking.<\/strong> A recipe for chocolate chip cookies calls for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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. Leona wants to double the recipe. a) How much brown sugar will Leona need? Show your calculation. b) Measuring cups usually come in sets of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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 Leona could measure the brown sugar needed to double the cookie recipe.<\/li>\n<li><strong data-effect=\"bold\">Portions.<\/strong> Regin purchased a bulk package of candy that weighs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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\/businesstechnicalmath\/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?<\/li>\n<\/ol>\n<h1>Answers:<\/h1>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03691bec54827131718c68dfdca523bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/>.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li>0<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ead7644658dd4cf7462bf004ff9e8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"17\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9a8f8569ba9809e691d7f079685394f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-748d68e99bb811b28ea187f3c6a96de6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bb43a9902bcf4b53249f5a54cac21105_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;&#54;&#125;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca0b99a2ec22006d674eb42e47fe6367_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-981e93e7451df8c7df5cf2463503ebc1_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;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2ef4a27eb4ba14d0fff8bf471a95f785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03691bec54827131718c68dfdca523bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-51b5de452d11711f395ef24be1c2a813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#57;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b66a5ee1ae0ac052a0c18be3ae508b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a5494832ebe694eef919295182c3816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea02fa8b7e253484b125b9843d50da72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#55;&#125;&#123;&#49;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-06778df6698326d2490a23fb2ad5ac61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#49;&#48;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cb57b8c541fc4f79a432efe8f559ede1_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;&#51;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9e697137c2fc49524ec52982e9a9dc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a8a56158fa379dbc0126fa65adcee37f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-66ceeedc341151a1adecf7dd22c17553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ad3eef5a6b9153f5169702e6f3b6482e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#57;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-51269b0555b78cd51fc1dea9fa3b0890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6941d24b8eac415464621657779b5b7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7e8ce2f90e409bd9f90b29b7e9dacb23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0fe79a067d120e67573e4a3118e1f2b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-458b8ef89dedd7cf464d6aa24d1f60c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1239857d5f8f2f1f3625981723d94b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>1<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a8dc9e92eaa09a95e5c650903ec1cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-27a06edc5fe1932bd0c3274d9af93920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> yard<\/li>\n<li>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/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<\/li>\n<li>20 bags<\/li>\n<\/ol>\n<div data-type=\"glossary\"><\/div>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cAdd and Subtract 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.<\/p>\n","protected":false},"author":125,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-138","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":6,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions"}],"predecessor-version":[{"id":2154,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions\/2154"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}