{"id":9572,"date":"2017-05-03T15:45:44","date_gmt":"2017-05-03T15:45:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9572"},"modified":"2020-07-02T02:46:32","modified_gmt":"2020-07-02T02:46:32","slug":"translating-words-to-equations-and-solving","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/chapter\/translating-words-to-equations-and-solving\/","title":{"raw":"Translating Words to Equations and Solving","rendered":"Translating Words to Equations and Solving"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Translate statements into equations and solve<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Translate Sentences to Equations and Solve<\/h2>\r\nRecall the four properties of equality\u2014subtraction, addition, division, and multiplication. We\u2019ll list them all together here for easy reference. We will use these to solve equations that contain fractions.\r\n<table id=\"fs-id908853\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"Table with two columns. The first column says \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>Subtraction Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\r\n<td><strong>Addition Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>Division Property of Equality:<\/strong>\r\n\r\nFor any numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex] where [latex]\\mathit{\\text{c}}\\ne \\mathit{0}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]\r\n\r\n\\Large\\frac{a}{c}=\r\n\r\n\\Large\\frac{b}{c}[\/latex]<\/td>\r\n<td><strong>Multiplication Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.\r\n\r\nIn the next few examples, we\u2019ll translate sentences that contain fractions into equations and then solve the equations.\u00a0 When we translate sentences into equations, we start by locating the\u00a0\u201cequals\u201d word or words and translating them into an equals sign.\u00a0 (Note: This could be the phrase \"is equal to\" or simply the word \"is\").\u00a0 We then translate the words to the left of the \u201cequals\u201d word(s) into an algebraic expression and the words to\u00a0the right of the \u201cequals\u201d word(s) into an algebraic expression.\r\n\r\nIn the next examples we will use the Multiplication Property of Equality to solve the equation.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate and solve: [latex]n[\/latex] divided by [latex]6[\/latex] is [latex]-24[\/latex].\r\n\r\nSolution:\r\n<table id=\"eip-id1168048345376\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">Translate.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221336\/CNX_BMath_Figure_04_07_037_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Multiply both sides by [latex]6[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{6}\\cdot\\Large\\frac{n}{6}\\normalsize=\\color{red}{6}(-24)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]n=-144[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>Is [latex]-144[\/latex] divided by [latex]6[\/latex] equal to [latex]-24[\/latex] ?<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]\\Large\\frac{-144}{6}\\normalsize\\stackrel{?}{=}-24[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify. It checks.<\/td>\r\n<td>[latex]-24=-24\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146166[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate and solve: The quotient of [latex]q[\/latex] and [latex]-5[\/latex] is [latex]70[\/latex].\r\n[reveal-answer q=\"545751\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"545751\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168047338066\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">Translate.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221341\/CNX_BMath_Figure_04_07_038_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Multiply both sides by [latex]-5[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{5}\\Large(\\frac{q}{-5}) \\normalsize= \\color{red}{-5}(70)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]q=-350[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>Is the quotient of [latex]-350[\/latex] and [latex]-5[\/latex] equal to [latex]70[\/latex] ?<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]\\Large\\frac{-350}{-5}\\normalsize\\stackrel{?}{=}70[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify. It checks.<\/td>\r\n<td>[latex]70=70\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146172[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate and solve: Two-thirds of [latex]f[\/latex] is [latex]18[\/latex].\r\n\r\n[reveal-answer q=\"896067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"896067\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168047349783\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">Translate.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221346\/CNX_BMath_Figure_04_07_039_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Multiply both sides by [latex]\\Large\\frac{3}{2}[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{\\Large\\frac{3}{2}}\\cdot\\Large\\frac{2}{3}\\normalsize f=\\color{red}{\\Large\\frac{3}{2}}\\cdot \\normalsize18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]f=27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>Is two-thirds of [latex]27[\/latex] equal to [latex]18[\/latex] ?<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]\\Large\\frac{2}{3}\\normalsize(27)\\normalsize\\stackrel{?}{=}18[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify. It checks.<\/td>\r\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try It<\/h3>\r\n[ohm_question height=\"270\"]146180[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate and solve: The quotient of [latex]m[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] is [latex]\\Large\\frac{3}{4}[\/latex].\r\n[reveal-answer q=\"383299\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383299\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466022434\" class=\"unnumbered unstyled\" style=\"width: 690px; height: 463px;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 440px;\" colspan=\"2\"><\/td>\r\n<td style=\"width: 227px;\">The quotient of [latex]m[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] is [latex]\\Large\\frac{3}{4}[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 440px;\" colspan=\"2\">Translate.<\/td>\r\n<td style=\"width: 227px;\">[latex]\\Large\\frac{m}{\\LARGE\\frac{5}{6}}=\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 440px;\" colspan=\"2\">Multiply both sides by [latex]\\Large\\frac{5}{6}[\/latex] to isolate [latex]m[\/latex] .<\/td>\r\n<td style=\"width: 227px;\">[latex]\\Large\\frac{5}{6}\\left(\\Large\\frac{m}{\\LARGE\\frac{5}{6}}\\right)=\\Large\\frac{5}{6}\\left(\\Large\\frac{3}{4}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 440px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"width: 227px;\">[latex]m=\\Large\\frac{5\\cdot 3}{6\\cdot 4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 440px;\" colspan=\"2\">Remove common factors and multiply.<\/td>\r\n<td style=\"width: 227px;\">[latex]m=\\Large\\frac{5}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 228px;\">Check:<\/td>\r\n<td style=\"width: 201px;\"><\/td>\r\n<td style=\"width: 227px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 228px;\">Is the quotient of [latex]\\Large\\frac{5}{8}[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] equal to [latex]\\Large\\frac{3}{4}[\/latex] ?<\/td>\r\n<td style=\"width: 201px;\">[latex]\\Large\\frac{\\LARGE\\frac{5}{8}}{\\LARGE\\frac{5}{6}}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<td style=\"width: 227px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 228px;\">Rewrite as division.<\/td>\r\n<td style=\"width: 201px;\">[latex]\\Large\\frac{5}{8}\\div\\Large\\frac{5}{6}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<td style=\"width: 227px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 228px;\">Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td style=\"width: 201px;\">[latex]\\Large\\frac{5}{8}\\cdot\\Large\\frac{6}{5}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\r\n<td style=\"width: 227px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 228px;\">Simplify.<\/td>\r\n<td style=\"width: 201px;\">[latex]\\Large\\frac{3}{4}=\\Large\\frac{3}{4}\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"width: 227px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOur solution checks.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146184[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next set of examples, we will use the Subtraction and Addition Properties of Equality to solve the equations.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate and solve: The sum of three-eighths and [latex]x[\/latex] is three and one-half.\r\n[reveal-answer q=\"288095\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288095\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467290209\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase, \">\r\n<tbody>\r\n<tr style=\"height: 85px;\">\r\n<td style=\"height: 85px;\">Translate.<\/td>\r\n<td style=\"height: 85px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221352\/CNX_BMath_Figure_04_07_040_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Use the Subtraction Property of Equality to subtract [latex]\\Large\\frac{3}{8}[\/latex] from both sides.<\/td>\r\n<td style=\"height: 30px;\">[latex]\\Large\\frac{3}{8}+\\normalsize x-\\Large\\frac{3}{8}=\\normalsize3\\Large\\frac{1}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Combine like terms on the left side.<\/td>\r\n<td style=\"height: 15px;\">[latex]x=3\\Large\\frac{1}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Convert mixed number to improper fraction.<\/td>\r\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{7}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Convert to equivalent fractions with LCD of [latex]8[\/latex].<\/td>\r\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{28}{8}-\\Large\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Subtract.<\/td>\r\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{25}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.40625px;\">\r\n<td style=\"height: 15.40625px;\">Write as a mixed number.<\/td>\r\n<td style=\"height: 15.40625px;\">[latex]x=3\\Large\\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe write the answer as a mixed number because the original problem used a mixed number.\r\n\r\nCheck:\r\nIs the sum of three-eighths and [latex]3\\Large\\frac{1}{8}[\/latex] equal to three and one-half?\r\n<table id=\"eip-id1168468659436\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{3}{8}\\normalsize+3\\Large\\frac{1}{8}\\normalsize\\stackrel{?}{=}3\\Large\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]3\\Large\\frac{4}{8}\\normalsize\\stackrel{?}{=}3\\Large\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]3\\Large\\frac{1}{2}\\normalsize=3\\Large\\frac{1}{2}\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe solution checks.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try It<\/h3>\r\n[ohm_question height=\"270\"]146189[\/ohm_question]\r\n\r\n[ohm_question height=\"270\"]146199[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/11FtuM35OgOD_jXuTHBgpGwASU6SY-iljINKsS8Hn85U\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Translate statements into equations and solve<\/li>\n<\/ul>\n<\/div>\n<h2>Translate Sentences to Equations and Solve<\/h2>\n<p>Recall the four properties of equality\u2014subtraction, addition, division, and multiplication. We\u2019ll list them all together here for easy reference. We will use these to solve equations that contain fractions.<\/p>\n<table id=\"fs-id908853\" class=\"unnumbered\" style=\"width: 85%;\" summary=\"Table with two columns. The first column says\">\n<tbody>\n<tr valign=\"top\">\n<td><strong>Subtraction Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\n<td><strong>Addition Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>Division Property of Equality:<\/strong><\/p>\n<p>For any numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex] where [latex]\\mathit{\\text{c}}\\ne \\mathit{0}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]\\Large\\frac{a}{c}=    \\Large\\frac{b}{c}[\/latex]<\/td>\n<td><strong>Multiplication Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.<\/p>\n<p>In the next few examples, we\u2019ll translate sentences that contain fractions into equations and then solve the equations.\u00a0 When we translate sentences into equations, we start by locating the\u00a0\u201cequals\u201d word or words and translating them into an equals sign.\u00a0 (Note: This could be the phrase &#8220;is equal to&#8221; or simply the word &#8220;is&#8221;).\u00a0 We then translate the words to the left of the \u201cequals\u201d word(s) into an algebraic expression and the words to\u00a0the right of the \u201cequals\u201d word(s) into an algebraic expression.<\/p>\n<p>In the next examples we will use the Multiplication Property of Equality to solve the equation.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate and solve: [latex]n[\/latex] divided by [latex]6[\/latex] is [latex]-24[\/latex].<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168048345376\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase,\">\n<tbody>\n<tr>\n<td colspan=\"2\">Translate.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221336\/CNX_BMath_Figure_04_07_037_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Multiply both sides by [latex]6[\/latex] .<\/td>\n<td>[latex]\\color{red}{6}\\cdot\\Large\\frac{n}{6}\\normalsize=\\color{red}{6}(-24)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]n=-144[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>Is [latex]-144[\/latex] divided by [latex]6[\/latex] equal to [latex]-24[\/latex] ?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]\\Large\\frac{-144}{6}\\normalsize\\stackrel{?}{=}-24[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify. It checks.<\/td>\n<td>[latex]-24=-24\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146166\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146166&theme=oea&iframe_resize_id=ohm146166&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate and solve: The quotient of [latex]q[\/latex] and [latex]-5[\/latex] is [latex]70[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q545751\">Show Solution<\/span><\/p>\n<div id=\"q545751\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168047338066\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase,\">\n<tbody>\n<tr>\n<td colspan=\"2\">Translate.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221341\/CNX_BMath_Figure_04_07_038_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Multiply both sides by [latex]-5[\/latex] .<\/td>\n<td>[latex]\\color{red}{5}\\Large(\\frac{q}{-5}) \\normalsize= \\color{red}{-5}(70)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]q=-350[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>Is the quotient of [latex]-350[\/latex] and [latex]-5[\/latex] equal to [latex]70[\/latex] ?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]\\Large\\frac{-350}{-5}\\normalsize\\stackrel{?}{=}70[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify. It checks.<\/td>\n<td>[latex]70=70\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146172\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146172&theme=oea&iframe_resize_id=ohm146172&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate and solve: Two-thirds of [latex]f[\/latex] is [latex]18[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q896067\">Show Solution<\/span><\/p>\n<div id=\"q896067\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168047349783\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase,\">\n<tbody>\n<tr>\n<td colspan=\"2\">Translate.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221346\/CNX_BMath_Figure_04_07_039_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Multiply both sides by [latex]\\Large\\frac{3}{2}[\/latex] .<\/td>\n<td>[latex]\\color{red}{\\Large\\frac{3}{2}}\\cdot\\Large\\frac{2}{3}\\normalsize f=\\color{red}{\\Large\\frac{3}{2}}\\cdot \\normalsize18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]f=27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>Is two-thirds of [latex]27[\/latex] equal to [latex]18[\/latex] ?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]\\Large\\frac{2}{3}\\normalsize(27)\\normalsize\\stackrel{?}{=}18[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify. It checks.<\/td>\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146180\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146180&theme=oea&iframe_resize_id=ohm146180&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate and solve: The quotient of [latex]m[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] is [latex]\\Large\\frac{3}{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q383299\">Show Solution<\/span><\/p>\n<div id=\"q383299\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466022434\" class=\"unnumbered unstyled\" style=\"width: 690px; height: 463px;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 440px;\" colspan=\"2\"><\/td>\n<td style=\"width: 227px;\">The quotient of [latex]m[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] is [latex]\\Large\\frac{3}{4}[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 440px;\" colspan=\"2\">Translate.<\/td>\n<td style=\"width: 227px;\">[latex]\\Large\\frac{m}{\\LARGE\\frac{5}{6}}=\\Large\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 440px;\" colspan=\"2\">Multiply both sides by [latex]\\Large\\frac{5}{6}[\/latex] to isolate [latex]m[\/latex] .<\/td>\n<td style=\"width: 227px;\">[latex]\\Large\\frac{5}{6}\\left(\\Large\\frac{m}{\\LARGE\\frac{5}{6}}\\right)=\\Large\\frac{5}{6}\\left(\\Large\\frac{3}{4}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 440px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"width: 227px;\">[latex]m=\\Large\\frac{5\\cdot 3}{6\\cdot 4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 440px;\" colspan=\"2\">Remove common factors and multiply.<\/td>\n<td style=\"width: 227px;\">[latex]m=\\Large\\frac{5}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 228px;\">Check:<\/td>\n<td style=\"width: 201px;\"><\/td>\n<td style=\"width: 227px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 228px;\">Is the quotient of [latex]\\Large\\frac{5}{8}[\/latex] and [latex]\\Large\\frac{5}{6}[\/latex] equal to [latex]\\Large\\frac{3}{4}[\/latex] ?<\/td>\n<td style=\"width: 201px;\">[latex]\\Large\\frac{\\LARGE\\frac{5}{8}}{\\LARGE\\frac{5}{6}}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\n<td style=\"width: 227px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 228px;\">Rewrite as division.<\/td>\n<td style=\"width: 201px;\">[latex]\\Large\\frac{5}{8}\\div\\Large\\frac{5}{6}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\n<td style=\"width: 227px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 228px;\">Multiply the first fraction by the reciprocal of the second.<\/td>\n<td style=\"width: 201px;\">[latex]\\Large\\frac{5}{8}\\cdot\\Large\\frac{6}{5}\\stackrel{?}{=}\\Large\\frac{3}{4}[\/latex]<\/td>\n<td style=\"width: 227px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 228px;\">Simplify.<\/td>\n<td style=\"width: 201px;\">[latex]\\Large\\frac{3}{4}=\\Large\\frac{3}{4}\\quad\\checkmark[\/latex]<\/td>\n<td style=\"width: 227px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Our solution checks.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146184\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146184&theme=oea&iframe_resize_id=ohm146184&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>In the next set of examples, we will use the Subtraction and Addition Properties of Equality to solve the equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate and solve: The sum of three-eighths and [latex]x[\/latex] is three and one-half.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288095\">Show Solution<\/span><\/p>\n<div id=\"q288095\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467290209\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says to translate. Underneath the phrase,\">\n<tbody>\n<tr style=\"height: 85px;\">\n<td style=\"height: 85px;\">Translate.<\/td>\n<td style=\"height: 85px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221352\/CNX_BMath_Figure_04_07_040_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Use the Subtraction Property of Equality to subtract [latex]\\Large\\frac{3}{8}[\/latex] from both sides.<\/td>\n<td style=\"height: 30px;\">[latex]\\Large\\frac{3}{8}+\\normalsize x-\\Large\\frac{3}{8}=\\normalsize3\\Large\\frac{1}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Combine like terms on the left side.<\/td>\n<td style=\"height: 15px;\">[latex]x=3\\Large\\frac{1}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Convert mixed number to improper fraction.<\/td>\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{7}{2}-\\Large\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Convert to equivalent fractions with LCD of [latex]8[\/latex].<\/td>\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{28}{8}-\\Large\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Subtract.<\/td>\n<td style=\"height: 15px;\">[latex]x=\\Large\\frac{25}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.40625px;\">\n<td style=\"height: 15.40625px;\">Write as a mixed number.<\/td>\n<td style=\"height: 15.40625px;\">[latex]x=3\\Large\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We write the answer as a mixed number because the original problem used a mixed number.<\/p>\n<p>Check:<br \/>\nIs the sum of three-eighths and [latex]3\\Large\\frac{1}{8}[\/latex] equal to three and one-half?<\/p>\n<table id=\"eip-id1168468659436\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{3}{8}\\normalsize+3\\Large\\frac{1}{8}\\normalsize\\stackrel{?}{=}3\\Large\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]3\\Large\\frac{4}{8}\\normalsize\\stackrel{?}{=}3\\Large\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]3\\Large\\frac{1}{2}\\normalsize=3\\Large\\frac{1}{2}\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The solution checks.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146189\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146189&theme=oea&iframe_resize_id=ohm146189&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146199\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146199&theme=oea&iframe_resize_id=ohm146199&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/11FtuM35OgOD_jXuTHBgpGwASU6SY-iljINKsS8Hn85U\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9572\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID: 146166, 146172, 146180, 146184, 146189, 146199. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID: 146166, 146172, 146180, 146184, 146189, 146199\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"160d148a7a4048f78f1b5c95e96ec4b9, 95bd1142fb2f4a0e920a606dcfd05204","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9572","chapter","type-chapter","status-publish","hentry"],"part":6633,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapters\/9572","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":32,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapters\/9572\/revisions"}],"predecessor-version":[{"id":18934,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapters\/9572\/revisions\/18934"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/parts\/6633"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapters\/9572\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/wp\/v2\/media?parent=9572"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/pressbooks\/v2\/chapter-type?post=9572"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/wp\/v2\/contributor?post=9572"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/wp-json\/wp\/v2\/license?post=9572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}