{"id":1234,"date":"2021-12-02T19:37:04","date_gmt":"2021-12-03T00:37:04","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/3-7-numeric-word-problems\/"},"modified":"2023-08-30T12:47:12","modified_gmt":"2023-08-30T16:47:12","slug":"numeric-word-problems","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/numeric-word-problems\/","title":{"raw":"3.7 Numeric Word Problems","rendered":"3.7 Numeric Word Problems"},"content":{"raw":"Number-based word problems can be very confusing, and it takes practice to convert a word-based sentence into a mathematical equation. The best strategy to solve these problems is to identify keywords that can be pulled out of a sentence and use them to set up an algebraic equation.\r\n\r\nVariables that are to be solved for are often written as \u201ca number,\u201d \u201can unknown,\u201d or \u201ca value.\u201d\r\n\r\n\"Equal\" is generally represented by the words \u201cis,\u201d \u201cwas,\u201d \u201cwill be,\u201d or \u201care.\u201d\r\n\r\nAddition is often stated as \u201cmore than,\u201d \u201cthe sum of,\u201d \u201cadded to,\u201d \u201cincreased by,\u201d \u201cplus,\u201d \u201call,\u201d or \u201ctotal.\u201d Addition statements are quite often written backwards. An example of this is \"three more than an unknown number,\" which is written as [latex]x + 3.[\/latex]\r\n\r\nSubtraction is often written as \u201cless than,\u201d \u201cminus,\u201d \u201cdecreased by,\u201d \u201creduced by,\u201d \u201csubtracted from,\u201d or \u201cthe difference of.\u201d Subtraction statements are quite often written backwards. An example of this is \"three less than an unknown number,\" which is written as [latex]x - 3.[\/latex]\r\n\r\nMultiplication can be seen in written problems with the words \u201ctimes,\u201d \u201cthe product of,\u201d or \u201cmultiplied by.\u201d\r\n\r\nDivision is generally found by a statement such as \u201cdivided by,\u201d \u201cthe quotient of,\u201d or \u201cper.\u201d\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3.7.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n28 less than five times a certain number is 232. What is the number?\r\n<ul>\r\n \t<li><strong>28 less<\/strong> means that it is subtracted from the unknown number (write this as \u221228)<\/li>\r\n \t<li><strong>five times an unknown number<\/strong> is written as [latex]5x[\/latex]<\/li>\r\n \t<li><strong>is 232<\/strong> means it equals 232 (write this as = 232)<\/li>\r\n<\/ul>\r\nPutting these pieces together and solving gives:\r\n\r\n[latex]\\begin{array}{rrrrrr}\r\n5x&amp;-&amp;28&amp;=&amp;232&amp; \\\\\r\n&amp;+&amp;28&amp;&amp;+28&amp; \\\\\r\n\\hline\r\n&amp;&amp;5x&amp;=&amp;260&amp; \\\\ \\\\\r\n&amp;&amp;x&amp;=&amp;\\dfrac{260}{5}&amp;\\text{or }52\r\n\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3.7.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFifteen more than three times a number is the same as nine less than six times the number. What is the number?\r\n<ul>\r\n \t<li><strong>Fifteen more than three times a number<\/strong> is [latex]3x + 15[\/latex] or [latex]15 + 3x[\/latex]<\/li>\r\n \t<li><strong>is<\/strong> means =<\/li>\r\n \t<li><strong>nine less than six times the number<\/strong> is [latex]6x-9[\/latex]<\/li>\r\n<\/ul>\r\nPutting these parts together gives:\r\n\r\n[latex]\\begin{array}{rrrrrrr}\r\n3x&amp;+&amp;15&amp;=&amp;6x&amp;-&amp;9 \\\\\r\n-6x&amp;-&amp;15&amp;=&amp;-6x&amp;-&amp;15 \\\\\r\n\\hline\r\n&amp;&amp;-3x&amp;=&amp;-24&amp;&amp; \\\\ \\\\\r\n&amp;&amp;x&amp;=&amp;\\dfrac{-24}{-3}&amp;\\text{or }8&amp; \\\\\r\n\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\nAnother type of number problem involves consecutive integers, consecutive odd integers, or consecutive even integers. Consecutive integers are numbers that come one after the other, such as 3, 4, 5, 6, 7. The equation that relates consecutive integers is:\r\n\r\n[latex]x, x + 1, x + 2, x + 3, x + 4[\/latex]\r\n\r\nConsecutive odd integers and consecutive even integers both share the same equation, since every second number must be skipped to remain either odd (such as 3, 5, 7, 9) or even (2, 4, 6, 8). The equation that is used to represent consecutive odd or even integers is:\r\n\r\n[latex]x, x + 2, x + 4, x + 6, x + 8[\/latex]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3.7.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe sum of three consecutive integers is 93. What are the integers?\r\n\r\nThe relationships described in equation form are as follows:\r\n\r\n[latex]x + x + 1 + x + 2 = 93[\/latex]\r\n\r\nWhich reduces to:\r\n\r\n[latex]\\begin{array}{rrrrrr}\r\n3x&amp;+&amp;3&amp;=&amp;93&amp; \\\\\r\n&amp;-&amp;3&amp;&amp;-3&amp; \\\\\r\n\\hline\r\n&amp;&amp;3x&amp;=&amp;90&amp; \\\\ \\\\\r\n&amp;&amp;x&amp;=&amp;\\dfrac{90}{3}&amp;\\text{or }30 \\\\\r\n\\end{array}[\/latex]\r\n\r\nThis means that the three consecutive integers are 30, 31, and 32.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3.7.4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe sum of three consecutive even integers is 246. What are the integers?\r\n\r\nThe relationships described in equation form are as follows:\r\n\r\n[latex]x + x + 2 + x + 4 = 246[\/latex]\r\n\r\nWhich reduces to:\r\n\r\n[latex]\\begin{array}{rrrrrr}\r\n3x&amp;+&amp;6&amp;=&amp;246&amp; \\\\\r\n&amp;-&amp;6&amp;&amp;-6&amp; \\\\\r\n\\hline\r\n&amp;&amp;3x&amp;=&amp;240&amp; \\\\ \\\\\r\n&amp;&amp;x&amp;=&amp;\\dfrac{240}{3}&amp;\\text{ or }80 \\\\\r\n\\end{array}[\/latex]\r\n\r\nThis means that the three consecutive even integers are 80, 82, and 84.\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Questions<\/h1>\r\nFor questions 1 to 8, write the formula defining each relationship. <strong>Do not solve.<\/strong>\r\n<ol>\r\n \t<li>Five more than twice an unknown number is 25.<\/li>\r\n \t<li>Twelve more than 4 times an unknown number is 36.<\/li>\r\n \t<li>Three times an unknown number decreased by 8 is 22.<\/li>\r\n \t<li>Six times an unknown number less 8 is 22.<\/li>\r\n \t<li>When an unknown number is decreased by 8, the difference is half the unknown number.<\/li>\r\n \t<li>When an unknown number is decreased by 4, the difference is half the unknown number.<\/li>\r\n \t<li>The sum of three consecutive integers is 21.<\/li>\r\n \t<li>The sum of the first two of three odd consecutive integers, less the third, is 5.<\/li>\r\n<\/ol>\r\nFor questions 9 to 16, write and solve the equation describing each relationship.\r\n<ol start=\"9\">\r\n \t<li>When five is added to three times a certain number, the result is 17. What is the number?<\/li>\r\n \t<li>If five is subtracted from three times a certain number, the result is 10. What is the number?<\/li>\r\n \t<li>Sixty more than nine times a number is the same as two less than ten times the number. What is the number?<\/li>\r\n \t<li>Eleven less than seven times a number is five more than six times the number. Find the number.<\/li>\r\n \t<li>The sum of three consecutive integers is 108. What are the integers?<\/li>\r\n \t<li>The sum of three consecutive integers is \u2212126. What are the integers?<\/li>\r\n \t<li>Find three consecutive integers such that the sum of the first, twice the second, and three times the third is \u221276.<\/li>\r\n \t<li>Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70.<\/li>\r\n<\/ol>\r\n<a class=\"internal\" href=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-3-7\/\">Answer Key 3.7<\/a>","rendered":"<p>Number-based word problems can be very confusing, and it takes practice to convert a word-based sentence into a mathematical equation. The best strategy to solve these problems is to identify keywords that can be pulled out of a sentence and use them to set up an algebraic equation.<\/p>\n<p>Variables that are to be solved for are often written as \u201ca number,\u201d \u201can unknown,\u201d or \u201ca value.\u201d<\/p>\n<p>&#8220;Equal&#8221; is generally represented by the words \u201cis,\u201d \u201cwas,\u201d \u201cwill be,\u201d or \u201care.\u201d<\/p>\n<p>Addition is often stated as \u201cmore than,\u201d \u201cthe sum of,\u201d \u201cadded to,\u201d \u201cincreased by,\u201d \u201cplus,\u201d \u201call,\u201d or \u201ctotal.\u201d Addition statements are quite often written backwards. An example of this is &#8220;three more than an unknown number,&#8221; which is written as [latex]x + 3.[\/latex]<\/p>\n<p>Subtraction is often written as \u201cless than,\u201d \u201cminus,\u201d \u201cdecreased by,\u201d \u201creduced by,\u201d \u201csubtracted from,\u201d or \u201cthe difference of.\u201d Subtraction statements are quite often written backwards. An example of this is &#8220;three less than an unknown number,&#8221; which is written as [latex]x - 3.[\/latex]<\/p>\n<p>Multiplication can be seen in written problems with the words \u201ctimes,\u201d \u201cthe product of,\u201d or \u201cmultiplied by.\u201d<\/p>\n<p>Division is generally found by a statement such as \u201cdivided by,\u201d \u201cthe quotient of,\u201d or \u201cper.\u201d<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3.7.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>28 less than five times a certain number is 232. What is the number?<\/p>\n<ul>\n<li><strong>28 less<\/strong> means that it is subtracted from the unknown number (write this as \u221228)<\/li>\n<li><strong>five times an unknown number<\/strong> is written as [latex]5x[\/latex]<\/li>\n<li><strong>is 232<\/strong> means it equals 232 (write this as = 232)<\/li>\n<\/ul>\n<p>Putting these pieces together and solving gives:<\/p>\n<p>[latex]\\begin{array}{rrrrrr}  5x&-&28&=&232& \\\\  &+&28&&+28& \\\\  \\hline  &&5x&=&260& \\\\ \\\\  &&x&=&\\dfrac{260}{5}&\\text{or }52  \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3.7.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Fifteen more than three times a number is the same as nine less than six times the number. What is the number?<\/p>\n<ul>\n<li><strong>Fifteen more than three times a number<\/strong> is [latex]3x + 15[\/latex] or [latex]15 + 3x[\/latex]<\/li>\n<li><strong>is<\/strong> means =<\/li>\n<li><strong>nine less than six times the number<\/strong> is [latex]6x-9[\/latex]<\/li>\n<\/ul>\n<p>Putting these parts together gives:<\/p>\n<p>[latex]\\begin{array}{rrrrrrr}  3x&+&15&=&6x&-&9 \\\\  -6x&-&15&=&-6x&-&15 \\\\  \\hline  &&-3x&=&-24&& \\\\ \\\\  &&x&=&\\dfrac{-24}{-3}&\\text{or }8& \\\\  \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Another type of number problem involves consecutive integers, consecutive odd integers, or consecutive even integers. Consecutive integers are numbers that come one after the other, such as 3, 4, 5, 6, 7. The equation that relates consecutive integers is:<\/p>\n<p>[latex]x, x + 1, x + 2, x + 3, x + 4[\/latex]<\/p>\n<p>Consecutive odd integers and consecutive even integers both share the same equation, since every second number must be skipped to remain either odd (such as 3, 5, 7, 9) or even (2, 4, 6, 8). The equation that is used to represent consecutive odd or even integers is:<\/p>\n<p>[latex]x, x + 2, x + 4, x + 6, x + 8[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3.7.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The sum of three consecutive integers is 93. What are the integers?<\/p>\n<p>The relationships described in equation form are as follows:<\/p>\n<p>[latex]x + x + 1 + x + 2 = 93[\/latex]<\/p>\n<p>Which reduces to:<\/p>\n<p>[latex]\\begin{array}{rrrrrr}  3x&+&3&=&93& \\\\  &-&3&&-3& \\\\  \\hline  &&3x&=&90& \\\\ \\\\  &&x&=&\\dfrac{90}{3}&\\text{or }30 \\\\  \\end{array}[\/latex]<\/p>\n<p>This means that the three consecutive integers are 30, 31, and 32.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3.7.4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The sum of three consecutive even integers is 246. What are the integers?<\/p>\n<p>The relationships described in equation form are as follows:<\/p>\n<p>[latex]x + x + 2 + x + 4 = 246[\/latex]<\/p>\n<p>Which reduces to:<\/p>\n<p>[latex]\\begin{array}{rrrrrr}  3x&+&6&=&246& \\\\  &-&6&&-6& \\\\  \\hline  &&3x&=&240& \\\\ \\\\  &&x&=&\\dfrac{240}{3}&\\text{ or }80 \\\\  \\end{array}[\/latex]<\/p>\n<p>This means that the three consecutive even integers are 80, 82, and 84.<\/p>\n<\/div>\n<\/div>\n<h1>Questions<\/h1>\n<p>For questions 1 to 8, write the formula defining each relationship. <strong>Do not solve.<\/strong><\/p>\n<ol>\n<li>Five more than twice an unknown number is 25.<\/li>\n<li>Twelve more than 4 times an unknown number is 36.<\/li>\n<li>Three times an unknown number decreased by 8 is 22.<\/li>\n<li>Six times an unknown number less 8 is 22.<\/li>\n<li>When an unknown number is decreased by 8, the difference is half the unknown number.<\/li>\n<li>When an unknown number is decreased by 4, the difference is half the unknown number.<\/li>\n<li>The sum of three consecutive integers is 21.<\/li>\n<li>The sum of the first two of three odd consecutive integers, less the third, is 5.<\/li>\n<\/ol>\n<p>For questions 9 to 16, write and solve the equation describing each relationship.<\/p>\n<ol start=\"9\">\n<li>When five is added to three times a certain number, the result is 17. What is the number?<\/li>\n<li>If five is subtracted from three times a certain number, the result is 10. What is the number?<\/li>\n<li>Sixty more than nine times a number is the same as two less than ten times the number. What is the number?<\/li>\n<li>Eleven less than seven times a number is five more than six times the number. Find the number.<\/li>\n<li>The sum of three consecutive integers is 108. What are the integers?<\/li>\n<li>The sum of three consecutive integers is \u2212126. What are the integers?<\/li>\n<li>Find three consecutive integers such that the sum of the first, twice the second, and three times the third is \u221276.<\/li>\n<li>Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70.<\/li>\n<\/ol>\n<p><a class=\"internal\" href=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-3-7\/\">Answer Key 3.7<\/a><\/p>\n","protected":false},"author":90,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1234","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1189,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1234\/revisions"}],"predecessor-version":[{"id":2087,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1234\/revisions\/2087"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1189"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1234\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1234"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1234"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1234"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}