{"id":100,"date":"2023-02-01T00:03:12","date_gmt":"2023-02-01T00:03:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/simplifying-and-evaluating-complex-fractions\/"},"modified":"2023-02-01T00:03:12","modified_gmt":"2023-02-01T00:03:12","slug":"simplifying-and-evaluating-complex-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/simplifying-and-evaluating-complex-fractions\/","title":{"raw":"Simplifying and Evaluating Complex Fractions","rendered":"Simplifying and Evaluating Complex Fractions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Simplify complex fractions that contain several different mathematical operations<\/li>\n \t<li>Evaluate variable expressions with fractions<\/li>\n<\/ul>\n<\/div>\nIn Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\LARGE{\\frac{3}{4}}}{\\LARGE{\\frac{5}{8}}}=\\Large\\frac{3}{4}\\div\\Large\\frac{5}{8}[\/latex]<\/p>\nNow we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.\n<div class=\"textbox shaded\">\n<h3>Simplify complex fractions<\/h3>\n<ol id=\"eip-id1168468461939\" class=\"stepwise\">\n \t<li>Simplify the numerator.<\/li>\n \t<li>Simplify the denominator.<\/li>\n \t<li>Divide the numerator by the denominator.<\/li>\n \t<li>Simplify if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nSimplify: [latex]\\Large\\frac{{\\Large{(\\LARGE\\frac{1}{2}})}^{2}}{4+{3}^{2}}[\/latex]\n\nSolution:\n<table id=\"eip-id1168467304832\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{{\\left(\\LARGE\\frac{1}{2}\\right)}^{2}}{4+{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{4+{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the term with the exponent in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{4+9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the terms in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{13}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator by the denominator.<\/td>\n<td>[latex]\\Large\\frac{1}{4}\\div \\normalsize13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as multiplication by the reciprocal.<\/td>\n<td>[latex]\\Large\\frac{1}{4}\\cdot\\Large\\frac{1}{13}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{1}{52}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]146431[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nSimplify: [latex]\\Large\\frac{\\LARGE\\frac{1}{2}+\\LARGE\\frac{2}{3}}{\\LARGE\\frac{3}{4}-\\LARGE\\frac{1}{6}}[\/latex]\n[reveal-answer q=\"429042\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"429042\"]\n\nSolution:\n<table id=\"eip-id1168468561600\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{2}+\\LARGE\\frac{2}{3}}{\\LARGE\\frac{3}{4}-\\LARGE\\frac{1}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite numerator with the LCD of [latex]6[\/latex] and denominator with LCD of [latex]12[\/latex].<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{3}{6}+\\LARGE\\frac{4}{6}}{\\LARGE\\frac{9}{12}-\\LARGE\\frac{2}{12}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the numerator. Subtract in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{7}{6}}{\\LARGE\\frac{7}{12}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator by the denominator.<\/td>\n<td>[latex]\\Large\\frac{7}{6}\\div\\Large\\frac{7}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as multiplication by the reciprocal.<\/td>\n<td>[latex]\\Large\\frac{7}{6}\\cdot\\Large\\frac{12}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite, showing common factors.<\/td>\n<td>[latex]\\Large\\frac{\\cancel{7}\\cdot \\cancel{6}\\cdot 2}{\\cancel{6}\\cdot \\cancel{7}\\cdot 1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]42079[\/ohm_question]\n\n[ohm_question]146435[\/ohm_question]\n\n<\/div>\nIn the following video we sow more examples of simplifying complex expressions.\n\nhttps:\/\/youtu.be\/lQCwze2w7OU\n<h2>Evaluate Variable Expressions with Fractions<\/h2>\nWe have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nEvaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when\n<ol>\n \t<li>[latex]x=-\\Large\\frac{1}{3}[\/latex]<\/li>\n \t<li>[latex]x=-\\Large\\frac{3}{4}[\/latex]<\/li>\n<\/ol>\nSolution\n1. To evaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when [latex]x=-\\Large\\frac{1}{3}[\/latex], substitute [latex]-\\Large\\frac{1}{3}[\/latex] for [latex]x[\/latex] in the expression.\n<table id=\"eip-id1168467081489\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says x plus one-third. The next line says, \">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{1}{3}}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{1}{3}} +\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n2. To evaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when [latex]x=-\\Large\\frac{3}{4}[\/latex], we substitute [latex]-\\Large\\frac{3}{4}[\/latex] for [latex]x[\/latex] in the expression.\n<table id=\"eip-id1168466833969\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says x plus one-third. The next line says, \">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{3}{4}}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{3}{4}} +\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD, [latex]12[\/latex].<\/td>\n<td>[latex]-\\Large\\frac{3\\cdot 3}{4\\cdot 3}+\\Large\\frac{1\\cdot 4}{3\\cdot 4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerators and denominators.<\/td>\n<td>[latex]-\\Large\\frac{9}{12}+\\Large\\frac{4}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]-\\Large\\frac{5}{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]146049[\/ohm_question]\n\n[ohm_question]146090[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nEvaluate [latex]y-\\Large\\frac{5}{6}[\/latex] when [latex]y=-\\Large\\frac{2}{3}[\/latex]\n[reveal-answer q=\"848965\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"848965\"]\n\nSolution:\nWe substitute [latex]-\\Large\\frac{2}{3}[\/latex] for [latex]y[\/latex] in the expression.\n<table id=\"eip-id1168469728183\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says y minus five-sixths. The next line says, \">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y-\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{2}{3}}[\/latex] for y.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{2}{3}} -\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD, [latex]6[\/latex].<\/td>\n<td>[latex]-\\Large\\frac{4}{6}-\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-\\Large\\frac{9}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-\\Large\\frac{3}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]146320[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nEvaluate [latex]2{x}^{2}y[\/latex] when [latex]x=\\Large\\frac{1}{4}[\/latex] and [latex]y=-\\Large\\frac{2}{3}[\/latex]\n[reveal-answer q=\"955267\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"955267\"]\n\nSolution:\nSubstitute the values into the expression. In [latex]2{x}^{2}y[\/latex], the exponent applies only to [latex]x[\/latex].\n<table id=\"eip-id1168467326105\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 2 times x squared times y. The next line says, \">\n<tbody>\n<tr style=\"height: 15.03125px;\">\n<td style=\"height: 15.03125px;\"><\/td>\n<td style=\"height: 15.03125px;\">[latex]2{x}^{2}y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px;\">Substitute [latex]\\color{red}{\\Large\\frac{1}{4}}[\/latex] for x and [latex]\\color{blue}{--\\Large\\frac{2}{3}}[\/latex] for y.<\/td>\n<td style=\"height: 46px;\">[latex]2(\\color{red}{\\Large\\frac{1}{4}})^{2}(\\color{blue}{--\\Large\\frac{2}{3}})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\">Simplify exponents first.<\/td>\n<td style=\"height: 43px;\">[latex]2(\\Large\\frac{1}{16})(--\\Large\\frac{2}{3})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px;\">Multiply. The product will be negative.<\/td>\n<td style=\"height: 44px;\">[latex]--\\Large\\frac{2}{1}\\cdot\\Large\\frac{1}{16}\\cdot\\Large\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\">Simplify.<\/td>\n<td style=\"height: 43px;\">[latex]--\\Large\\frac{4}{48}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px;\">Remove the common factors.<\/td>\n<td style=\"height: 46px;\">[latex]--\\Large\\frac{1\\cdot\\color{red}{4}}{\\color{red}{4} \\cdot\\ 12}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px;\">Simplify.<\/td>\n<td style=\"height: 44px;\">[latex]--\\Large\\frac{1}{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try It<\/h3>\n[ohm_question]146096[\/ohm_question]\n\n[ohm_question]146093[\/ohm_question]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nEvaluate [latex]\\Large\\frac{p+q}{r}[\/latex] when [latex]p=-4,q=-2[\/latex], and [latex]r=8[\/latex]\n[reveal-answer q=\"871534\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"871534\"]\n\nSolution:\nWe substitute the values into the expression and simplify.\n<table id=\"eip-id1168469803463\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says p plus q over r. The following line says, \">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{p+q}{r}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--4}[\/latex] for p, [latex]\\color{blue}{--2}[\/latex] for q, and [latex]\\color{red}{8}[\/latex] for r.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{--4} + \\color{blue}{(--2)}}{\\color{red}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the numerator first.<\/td>\n<td>[latex]-\\Large\\frac{6}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-\\Large\\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question]146106[\/ohm_question]\n\n[ohm_question]146108[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify complex fractions that contain several different mathematical operations<\/li>\n<li>Evaluate variable expressions with fractions<\/li>\n<\/ul>\n<\/div>\n<p>In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{\\LARGE{\\frac{3}{4}}}{\\LARGE{\\frac{5}{8}}}=\\Large\\frac{3}{4}\\div\\Large\\frac{5}{8}[\/latex]<\/p>\n<p>Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplify complex fractions<\/h3>\n<ol id=\"eip-id1168468461939\" class=\"stepwise\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator.<\/li>\n<li>Simplify if possible.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\Large\\frac{{\\Large{(\\LARGE\\frac{1}{2}})}^{2}}{4+{3}^{2}}[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168467304832\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{{\\left(\\LARGE\\frac{1}{2}\\right)}^{2}}{4+{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{4+{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the term with the exponent in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{4+9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the terms in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{4}}{13}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator by the denominator.<\/td>\n<td>[latex]\\Large\\frac{1}{4}\\div \\normalsize13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as multiplication by the reciprocal.<\/td>\n<td>[latex]\\Large\\frac{1}{4}\\cdot\\Large\\frac{1}{13}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\Large\\frac{1}{52}[\/latex]<\/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=\"ohm146431\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146431&theme=oea&iframe_resize_id=ohm146431&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\Large\\frac{\\LARGE\\frac{1}{2}+\\LARGE\\frac{2}{3}}{\\LARGE\\frac{3}{4}-\\LARGE\\frac{1}{6}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q429042\">Show Solution<\/span><\/p>\n<div id=\"q429042\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468561600\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{1}{2}+\\LARGE\\frac{2}{3}}{\\LARGE\\frac{3}{4}-\\LARGE\\frac{1}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite numerator with the LCD of [latex]6[\/latex] and denominator with LCD of [latex]12[\/latex].<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{3}{6}+\\LARGE\\frac{4}{6}}{\\LARGE\\frac{9}{12}-\\LARGE\\frac{2}{12}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the numerator. Subtract in the denominator.<\/td>\n<td>[latex]\\Large\\frac{\\LARGE\\frac{7}{6}}{\\LARGE\\frac{7}{12}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide the numerator by the denominator.<\/td>\n<td>[latex]\\Large\\frac{7}{6}\\div\\Large\\frac{7}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as multiplication by the reciprocal.<\/td>\n<td>[latex]\\Large\\frac{7}{6}\\cdot\\Large\\frac{12}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite, showing common factors.<\/td>\n<td>[latex]\\Large\\frac{\\cancel{7}\\cdot \\cancel{6}\\cdot 2}{\\cancel{6}\\cdot \\cancel{7}\\cdot 1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2[\/latex]<\/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=\"ohm42079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=42079&theme=oea&iframe_resize_id=ohm42079&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146435\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146435&theme=oea&iframe_resize_id=ohm146435&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we sow more examples of simplifying complex expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Simplify a Complex Fraction (No Variables)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lQCwze2w7OU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Evaluate Variable Expressions with Fractions<\/h2>\n<p>We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when<\/p>\n<ol>\n<li>[latex]x=-\\Large\\frac{1}{3}[\/latex]<\/li>\n<li>[latex]x=-\\Large\\frac{3}{4}[\/latex]<\/li>\n<\/ol>\n<p>Solution<br \/>\n1. To evaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when [latex]x=-\\Large\\frac{1}{3}[\/latex], substitute [latex]-\\Large\\frac{1}{3}[\/latex] for [latex]x[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168467081489\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says x plus one-third. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{1}{3}}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{1}{3}} +\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. To evaluate [latex]x+\\Large\\frac{1}{3}[\/latex] when [latex]x=-\\Large\\frac{3}{4}[\/latex], we substitute [latex]-\\Large\\frac{3}{4}[\/latex] for [latex]x[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168466833969\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says x plus one-third. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{3}{4}}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{3}{4}} +\\Large\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD, [latex]12[\/latex].<\/td>\n<td>[latex]-\\Large\\frac{3\\cdot 3}{4\\cdot 3}+\\Large\\frac{1\\cdot 4}{3\\cdot 4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify the numerators and denominators.<\/td>\n<td>[latex]-\\Large\\frac{9}{12}+\\Large\\frac{4}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]-\\Large\\frac{5}{12}[\/latex]<\/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=\"ohm146049\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146049&theme=oea&iframe_resize_id=ohm146049&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146090&theme=oea&iframe_resize_id=ohm146090&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]y-\\Large\\frac{5}{6}[\/latex] when [latex]y=-\\Large\\frac{2}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q848965\">Show Solution<\/span><\/p>\n<div id=\"q848965\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe substitute [latex]-\\Large\\frac{2}{3}[\/latex] for [latex]y[\/latex] in the expression.<\/p>\n<table id=\"eip-id1168469728183\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says y minus five-sixths. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y-\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--\\Large\\frac{2}{3}}[\/latex] for y.<\/td>\n<td>[latex]\\color{red}{--\\Large\\frac{2}{3}} -\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as equivalent fractions with the LCD, [latex]6[\/latex].<\/td>\n<td>[latex]-\\Large\\frac{4}{6}-\\Large\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-\\Large\\frac{9}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-\\Large\\frac{3}{2}[\/latex]<\/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=\"ohm146320\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146320&theme=oea&iframe_resize_id=ohm146320&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]2{x}^{2}y[\/latex] when [latex]x=\\Large\\frac{1}{4}[\/latex] and [latex]y=-\\Large\\frac{2}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955267\">Show Solution<\/span><\/p>\n<div id=\"q955267\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nSubstitute the values into the expression. In [latex]2{x}^{2}y[\/latex], the exponent applies only to [latex]x[\/latex].<\/p>\n<table id=\"eip-id1168467326105\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says 2 times x squared times y. The next line says,\">\n<tbody>\n<tr style=\"height: 15.03125px;\">\n<td style=\"height: 15.03125px;\"><\/td>\n<td style=\"height: 15.03125px;\">[latex]2{x}^{2}y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px;\">Substitute [latex]\\color{red}{\\Large\\frac{1}{4}}[\/latex] for x and [latex]\\color{blue}{--\\Large\\frac{2}{3}}[\/latex] for y.<\/td>\n<td style=\"height: 46px;\">[latex]2(\\color{red}{\\Large\\frac{1}{4}})^{2}(\\color{blue}{--\\Large\\frac{2}{3}})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\">Simplify exponents first.<\/td>\n<td style=\"height: 43px;\">[latex]2(\\Large\\frac{1}{16})(--\\Large\\frac{2}{3})[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px;\">Multiply. The product will be negative.<\/td>\n<td style=\"height: 44px;\">[latex]--\\Large\\frac{2}{1}\\cdot\\Large\\frac{1}{16}\\cdot\\Large\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\">Simplify.<\/td>\n<td style=\"height: 43px;\">[latex]--\\Large\\frac{4}{48}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td style=\"height: 46px;\">Remove the common factors.<\/td>\n<td style=\"height: 46px;\">[latex]--\\Large\\frac{1\\cdot\\color{red}{4}}{\\color{red}{4} \\cdot\\ 12}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px;\">Simplify.<\/td>\n<td style=\"height: 44px;\">[latex]--\\Large\\frac{1}{12}[\/latex]<\/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=\"ohm146096\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146096&theme=oea&iframe_resize_id=ohm146096&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146093\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146093&theme=oea&iframe_resize_id=ohm146093&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\Large\\frac{p+q}{r}[\/latex] when [latex]p=-4,q=-2[\/latex], and [latex]r=8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q871534\">Show Solution<\/span><\/p>\n<div id=\"q871534\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe substitute the values into the expression and simplify.<\/p>\n<table id=\"eip-id1168469803463\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says p plus q over r. The following line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{p+q}{r}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--4}[\/latex] for p, [latex]\\color{blue}{--2}[\/latex] for q, and [latex]\\color{red}{8}[\/latex] for r.<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{--4} + \\color{blue}{(--2)}}{\\color{red}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add in the numerator first.<\/td>\n<td>[latex]-\\Large\\frac{6}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-\\Large\\frac{3}{4}[\/latex]<\/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=\"ohm146106\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146106&theme=oea&iframe_resize_id=ohm146106&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146108\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146108&theme=oea&iframe_resize_id=ohm146108&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-100\">\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: 146431, 146435, 146090, 146049, 146320, 146096, 146093, 146106, 146108. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Simplify a Complex Fraction (No Variables). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lQCwze2w7OU\">https:\/\/youtu.be\/lQCwze2w7OU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 42079. <strong>Authored by<\/strong>: Matthew Lewis. <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":538461,"menu_order":8,"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\":\"cc\",\"description\":\"Ex 1: Simplify a Complex Fraction (No Variables)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/lQCwze2w7OU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 146431, 146435, 146090, 146049, 146320, 146096, 146093, 146106, 146108\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID: 42079\",\"author\":\"Matthew Lewis\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-100","chapter","type-chapter","status-publish","hentry"],"part":92,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/chapters\/100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/wp\/v2\/users\/538461"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/chapters\/100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/wp\/v2\/contributor?post=100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/wp-json\/wp\/v2\/license?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}