{"id":9119,"date":"2017-05-02T14:42:12","date_gmt":"2017-05-02T14:42:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9119"},"modified":"2024-04-30T21:27:20","modified_gmt":"2024-04-30T21:27:20","slug":"apply-a-problem-solving-strategy-to-basic-word-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/apply-a-problem-solving-strategy-to-basic-word-problems\/","title":{"raw":"Apply a Problem-Solving Strategy to Word Problems","rendered":"Apply a Problem-Solving Strategy to Word Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Approach word problems with a positive attitude<\/li>\r\n \t<li>Use a problem solving strategy for word problems<\/li>\r\n \t<li>Translate more complex word problems into algebraic expressions and equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>\u00a0Approach Word Problems with a Positive Attitude<\/h3>\r\nThe world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?\r\n\r\nNow that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"564\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223235\/CNX_BMath_Figure_09_01_001.png\" alt=\"A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles. They read, &quot;I don't know whether to add, subtract, multiply, or divide!&quot;, &quot;I don't understand word problems!&quot;, &quot;My teachers never explained this!&quot;, &quot;If I just skip all the word problems, I can probably still pass the class.&quot;, and &quot;I just can't do this!&quot;. \" width=\"564\" height=\"441\" \/> Negative thoughts about word problems can be barriers to success.[\/caption]\r\n\r\nWhen we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.\r\n\r\nStart with a fresh slate and begin to think positive thoughts, like the student in the cartoon below.\u00a0Read the positive thoughts and say them out loud.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"580\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223237\/CNX_BMath_Figure_09_01_002.png\" alt=\"A cartoon image of a girl with a confident expression holding some books is shown. There are 4 thought bubbles. They read, &quot;while word problems were hard in the past I think I can try them now.&quot;, &quot;I am better prepared now. I think I will begin to understand word problems.&quot;, &quot; I think I can! I think I can!&quot;, and &quot;It may take time, but I can begin to solve word problems.&quot;. \" width=\"580\" height=\"492\" \/> When it comes to word problems, a positive attitude is a big step toward success.[\/caption]\r\n\r\nIf we take control and believe we can be successful, we will be able to master word problems.\r\n\r\nThink of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!\r\n<h3>Use a Problem-Solving Strategy for Word Problems<\/h3>\r\nIn earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then, you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.\r\n\r\nYou have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.\r\n\r\nNow we'll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We'll demonstrate the strategy as we solve the following problem.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nPete bought a shirt on sale for $[latex]18[\/latex], which is one-half the original price. What was the original price of the shirt?\r\n\r\nSolution:\r\nStep 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the Internet.\r\n<ul id=\"fs-id1764631\">\r\n \t<li><em>In this problem, do you understand what is being discussed? Do you understand every word?<\/em><\/li>\r\n<\/ul>\r\nStep 2. <strong>Identify<\/strong> what you are looking for. It's hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!\r\n<ul id=\"fs-id1475126\">\r\n \t<li><em>In this problem, the words \"what was the original price of the shirt\" tell you what you are looking for: the original price of the shirt.<\/em><\/li>\r\n<\/ul>\r\nStep 3. <strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.\r\n<ul id=\"fs-id1171505228428\">\r\n \t<li><em>Let [latex]p=[\/latex] the original price of the shirt<\/em><\/li>\r\n<\/ul>\r\nStep 4. <strong>Translate<\/strong> into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223238\/CNX_BMath_Figure_09_01_002_img.png\" alt=\"The top line reads: &quot;18 is one half of the original price&quot;. The bottom line translates the top line from words to an algebraic equation. The word &quot;is&quot; translates to an equal sign. The phrase &quot;one half&quot; translates to &quot;1\/2&quot;. The word &quot;of&quot; translates to a multiplication symbol. The phrase &quot;the original price&quot; translates to &quot;p&quot;. This gives the full algebraic equation &quot;18 = 1\/2 times p&quot;. \" width=\"443\" height=\"76\" \/>\r\nStep 5. <strong>Solve<\/strong> the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.\r\n<table id=\"eip-id1168047368619\" class=\"unnumbered unstyled\" summary=\"The top line reads, \">\r\n<tbody>\r\n<tr>\r\n<td>Write the equation.<\/td>\r\n<td>[latex]18=\\Large\\frac{1}{2}p[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply both sides by 2.<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot18=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize p[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]36=p[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nStep 6. <strong>Check<\/strong> the answer in the problem and make sure it makes sense.\r\n<ul id=\"fs-id1594085\">\r\n \t<li><em>We found that<\/em> [latex]p=36[\/latex], <em>which means the original price was<\/em> [latex]\\text{\\$36}[\/latex]. <em>Does<\/em> [latex]\\text{\\$36}[\/latex] <em>make sense in the problem? Yes, because<\/em> [latex]18[\/latex] <em>is one-half of<\/em> [latex]36[\/latex], <em>and the shirt was on sale at half the original price.<\/em><\/li>\r\n<\/ul>\r\nStep 7. <strong>Answer<\/strong> the question with a complete sentence.\r\n<ul id=\"fs-id1495941\">\r\n \t<li><em>The problem asked \"What was the original price of the shirt?\" The answer to the question is: \"The original price of the shirt was<\/em> [latex]\\text{\\$36}[\/latex].\"<\/li>\r\n<\/ul>\r\nIf this were a homework exercise, our work might look like this:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223244\/CNX_BMath_Figure_09_01_022.png\" alt=\"An example of what a student's work might look like for the problem. Let p equal the original price. 18 is one half the original price. 18 equals one half p. 2 times 18 equals 2 times one half p. 36 equals p. Check: is $36 a reasonable price for a shirt? Yes. Is 18 one half of 36? Yes. The original price of the shirt was $36.\" width=\"542\" height=\"319\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142694&amp;amp;amp;amp;theme=oea&amp;amp;amp;amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nWe list the steps we took to solve the previous example.\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">Problem-Solving Strategy<\/h3>\r\n<ol id=\"eip-id1168469627809\" class=\"stepwise\">\r\n \t<li><strong>Read<\/strong> the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet.<\/li>\r\n \t<li><strong>Identify<\/strong> what you are looking for.\u00a0\u00a0Determine the constants and variables in the problem.\u00a0 A constant is a number in the problem that is not going to change.\u00a0 A variable is a number that you don't yet know its value.<\/li>\r\n \t<li><strong>Name<\/strong> what you are looking for. Choose a letter to represent that quantity.<\/li>\r\n \t<li><strong>Translate<\/strong> words into algebraic expressions and equations.\u00a0 Write an equation to represent the problem.\u00a0It may be helpful to first restate the problem in one sentence before translating.<\/li>\r\n \t<li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li>\r\n \t<li><strong>Check<\/strong> the answer in the problem. Make sure it makes sense.<\/li>\r\n \t<li><strong>Answer<\/strong> the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Translate word problems into expressions<\/h2>\r\nOne of the first steps to solving word problems is converting an English sentence into a mathematical sentence. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it's good to see what mathematical operators are associated with them.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Addition [latex]+[\/latex]<\/th>\r\n<th>Subtraction [latex]-[\/latex]<\/th>\r\n<th>Multiplication [latex]\\times[\/latex]<\/th>\r\n<th>Variable ?<\/th>\r\n<th>Equals [latex]=[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>More than<\/td>\r\n<td>Less than<\/td>\r\n<td>Double<\/td>\r\n<td>A number<\/td>\r\n<td>Is<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Together<\/td>\r\n<td>In the past<\/td>\r\n<td>Product<\/td>\r\n<td>Often, a value for which no information is given.<\/td>\r\n<td>The same as<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sum<\/td>\r\n<td>Slower than<\/td>\r\n<td>\u00a0times<\/td>\r\n<td>After how many hours?<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>The remainder of<\/td>\r\n<td><\/td>\r\n<td>How much will it cost?<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>In the future<\/td>\r\n<td>\u00a0Difference<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Faster than<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSome examples follow:\r\n<ul>\r\n \t<li>\"[latex]x\\text{ is }5[\/latex]\"\u00a0becomes [latex]x=5[\/latex]<\/li>\r\n \t<li>\"Three more than a number\" becomes [latex]x+3[\/latex]<\/li>\r\n \t<li>\"Four less than a number\" becomes [latex]x-4[\/latex]<\/li>\r\n \t<li>\"Double the cost\" becomes [latex]2\\cdot\\text{ cost }[\/latex]<\/li>\r\n \t<li>\"Groceries and gas together for the week cost $250\" means [latex]\\text{ groceries }+\\text{ gas }=250[\/latex]<\/li>\r\n \t<li>\"The difference of [latex]9[\/latex] and a number\" becomes [latex]9-x[\/latex]. Notice how [latex]9[\/latex] is first in the sentence and the expression.<\/li>\r\n<\/ul>\r\nLet's practice translating a few more English phrases into algebraic\u00a0expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate the table into algebraic expressions:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0some number<\/td>\r\n<td>\u00a0the sum of the number and [latex]3[\/latex]<\/td>\r\n<td>\u00a0twice the sum of the number and [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a length<\/td>\r\n<td>\u00a0double the length<\/td>\r\n<td>\u00a0double the length, decreased by [latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a cost<\/td>\r\n<td>\u00a0the difference of the cost and [latex]20[\/latex]<\/td>\r\n<td>\u00a0[latex]2[\/latex] times the difference of the cost and [latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0some quantity<\/td>\r\n<td>\u00a0the difference of [latex]5[\/latex] and the quantity<\/td>\r\n<td>\u00a0\u00a0the difference of [latex]5[\/latex] and the quantity, divided by [latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0an amount of time<\/td>\r\n<td>\u00a0triple the amount of time<\/td>\r\n<td>\u00a0triple the amount of time, increased by [latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a distance<\/td>\r\n<td>\u00a0the sum of [latex]-4[\/latex] and the distance<\/td>\r\n<td>\u00a0the sum of [latex]-4[\/latex] and the twice the distance<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"790402\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"790402\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0[latex]a[\/latex]<\/td>\r\n<td>\u00a0[latex]a+3[\/latex]<\/td>\r\n<td>\u00a0[latex]2\\left(a+3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]l[\/latex]<\/td>\r\n<td>\u00a0[latex]2l[\/latex]<\/td>\r\n<td>\u00a0[latex]2l-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]c[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]c-20[\/latex]<\/td>\r\n<td>\u00a0[latex]2\\left(c-20\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]q[\/latex]<\/td>\r\n<td>\u00a0[latex]5-q[\/latex]<\/td>\r\n<td>\u00a0[latex]\\frac{5-q}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]t[\/latex]<\/td>\r\n<td>\u00a0[latex]3t[\/latex]<\/td>\r\n<td>\u00a0[latex]3t+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]d[\/latex]<\/td>\r\n<td>\u00a0[latex]-4+d[\/latex]<\/td>\r\n<td>\u00a0[latex]-4+2d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this example video, we show how to translate more words into mathematical expressions.\r\n\r\nhttps:\/\/youtu.be\/uD_V5t-6Kzs\r\n\r\nFor another review of how to translate algebraic statements into words, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/Hub7ku7UHT4\r\n\r\nThe power of algebra is how it can help you model real situations in order to answer questions about them.\u00a0 Let's use this approach with another example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nYash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[\/latex] apples to the picnic. How many bananas did he bring?\r\n[reveal-answer q=\"15930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15930\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467372660\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>How many bananas did he bring?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name<\/strong> what you are looking for.Choose a variable to represent the number of bananas.<\/td>\r\n<td>Let [latex]b=\\text{number of bananas}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Restate the problem in one sentence with all the important information.Translate into an equation.<\/td>\r\n<td>[latex]11\\enspace\\Rightarrow[\/latex] The number of apples[latex]=\\enspace\\Rightarrow[\/latex] was\r\n\r\n[latex]3\\enspace\\Rightarrow[\/latex] three\r\n\r\n[latex]+\\enspace\\Rightarrow[\/latex] more than\r\n\r\n[latex]2b\\enspace\\Rightarrow[\/latex] twice the number of bananas<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]11=2b+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract 3 from each side.<\/td>\r\n<td>[latex]11\\color{red}{-3}=2b+3\\color{red}{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8=2b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide each side by 2.<\/td>\r\n<td>[latex]\\Large\\frac{8}{\\color{red}{2}}\\normalsize =\\Large\\frac{2b}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]4=b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice 4 bananas is 8. Three more than 8 is 11.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>Yash brought 4 bananas to the picnic.<\/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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142722&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTwenty-eight\u00a0less than five times a certain number is [latex]232[\/latex]. What is the number?\r\n\r\n[reveal-answer q=\"720402\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"720402\"]\r\n\r\nFollowing the steps provided:\r\n<ol>\r\n \t<li><strong>Read and understand:<\/strong> we are looking for a number.<\/li>\r\n \t<li><strong>Constants and variables:<\/strong>\u00a0[latex]28[\/latex] and [latex]232[\/latex] are constants, \"a certain number\" is our variable, because we don't know its value, and we are asked to find it. We will call it\u00a0[latex]x[\/latex].<\/li>\r\n \t<li><strong>Translate:\u00a0<\/strong>five times a certain number translates to [latex]5x[\/latex]\r\nTwenty-eight\u00a0less than five times a certain number translates to\u00a0[latex]5x-28[\/latex], because subtraction is built backward.\r\n\"is 232\" translates to \"[latex]=232\"[\/latex] since \"is\" is associated with equals.<\/li>\r\n \t<li><strong>Write an equation:<\/strong>\u00a0[latex]5x-28=232[\/latex]<\/li>\r\n \t<li><strong>Solve the equation using what you know about solving linear equations:<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5x-28=232\\\\5x=260\\\\x=52\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<\/li>\r\n \t<li><strong>Check and interpret:<\/strong> We can substitute [latex]52[\/latex] for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\left(52\\right)-28=232\\\\5\\left(52\\right)=260\\\\260=260\\end{array}[\/latex]<\/p>\r\nTRUE!<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.\r\n\r\nhttps:\/\/youtu.be\/izIIqOztUyI\r\n\r\n&nbsp;\r\n\r\nIn the next example, we will apply our Problem-Solving Strategy to applications of percent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nNga's car insurance premium increased by [latex]\\text{\\$60}[\/latex], which was [latex]\\text{8%}[\/latex] of the original cost. What was the original cost of the premium?\r\n[reveal-answer q=\"662772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"662772\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469833028\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 1. <strong>Read<\/strong> the problem. Remember, if there are words you don't understand, look them up.<\/td>\r\n<td style=\"width: 328.217px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"width: 328.217px;\">the original cost of the premium<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent the original cost of premium.<\/td>\r\n<td style=\"width: 328.217px;\">Let [latex]c=\\text{the original cost}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 4. <strong>Translate.<\/strong> Restate as one sentence. Translate into an equation.<\/td>\r\n<td style=\"width: 328.217px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223257\/CNX_BMath_Figure_09_01_024_img-01.png\" alt=\"The top line reads: &quot;$60 was 8% of the original cost&quot;. The bottom line translates the top line from words to an algebraic equation. The word &quot;was&quot; translates to an equal sign. The term &quot;8%&quot; translates to &quot;0.08&quot;. The word &quot;of&quot; translates to a multiplication symbol. The phrase &quot;the original cost&quot; translates to &quot;c&quot;. This gives the full algebraic equation &quot;60 = 0.08 times c&quot;. \" width=\"304\" height=\"71\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td style=\"width: 328.217px;\">[latex]60=0.08c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Divide both sides by [latex]0.08[\/latex].<\/td>\r\n<td style=\"width: 328.217px;\">[latex]\\Large\\frac{60}{\\color{red}{0.08}}\\normalsize =\\Large\\frac{0.08c}{\\color{red}{0.08}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Simplify.<\/td>\r\n<td style=\"width: 328.217px;\">[latex]c=750[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 6. <strong>Check:<\/strong> Is our answer reasonable? Yes, a [latex]\\text{\\$750}[\/latex] premium on auto insurance is reasonable. Now let's check our algebra. Is 8% of 750 equal to [latex]60[\/latex]?[latex]750=c[\/latex]\r\n\r\n[latex]0.08(750)=60[\/latex]\r\n\r\n[latex]60=60\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"width: 328.217px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 328.217px;\">The original cost of Nga's premium was [latex]\\text{\\$750}[\/latex].<\/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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142735&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142761&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Approach word problems with a positive attitude<\/li>\n<li>Use a problem solving strategy for word problems<\/li>\n<li>Translate more complex word problems into algebraic expressions and equations<\/li>\n<\/ul>\n<\/div>\n<h3>\u00a0Approach Word Problems with a Positive Attitude<\/h3>\n<p>The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?<\/p>\n<p>Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?<\/p>\n<div style=\"width: 574px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223235\/CNX_BMath_Figure_09_01_001.png\" alt=\"A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles. They read, &quot;I don't know whether to add, subtract, multiply, or divide!&quot;, &quot;I don't understand word problems!&quot;, &quot;My teachers never explained this!&quot;, &quot;If I just skip all the word problems, I can probably still pass the class.&quot;, and &quot;I just can't do this!&quot;.\" width=\"564\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\">Negative thoughts about word problems can be barriers to success.<\/p>\n<\/div>\n<p>When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.<\/p>\n<p>Start with a fresh slate and begin to think positive thoughts, like the student in the cartoon below.\u00a0Read the positive thoughts and say them out loud.<\/p>\n<div style=\"width: 590px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223237\/CNX_BMath_Figure_09_01_002.png\" alt=\"A cartoon image of a girl with a confident expression holding some books is shown. There are 4 thought bubbles. They read, &quot;while word problems were hard in the past I think I can try them now.&quot;, &quot;I am better prepared now. I think I will begin to understand word problems.&quot;, &quot; I think I can! I think I can!&quot;, and &quot;It may take time, but I can begin to solve word problems.&quot;.\" width=\"580\" height=\"492\" \/><\/p>\n<p class=\"wp-caption-text\">When it comes to word problems, a positive attitude is a big step toward success.<\/p>\n<\/div>\n<p>If we take control and believe we can be successful, we will be able to master word problems.<\/p>\n<p>Think of something that you can do now but couldn&#8217;t do three years ago. Whether it&#8217;s driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!<\/p>\n<h3>Use a Problem-Solving Strategy for Word Problems<\/h3>\n<p>In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then, you&#8217;ve increased your math vocabulary as you learned about more algebraic procedures, and you&#8217;ve had more practice translating from words into algebra.<\/p>\n<p>You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.<\/p>\n<p>Now we&#8217;ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We&#8217;ll demonstrate the strategy as we solve the following problem.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Pete bought a shirt on sale for $[latex]18[\/latex], which is one-half the original price. What was the original price of the shirt?<\/p>\n<p>Solution:<br \/>\nStep 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don&#8217;t understand, look them up in a dictionary or on the Internet.<\/p>\n<ul id=\"fs-id1764631\">\n<li><em>In this problem, do you understand what is being discussed? Do you understand every word?<\/em><\/li>\n<\/ul>\n<p>Step 2. <strong>Identify<\/strong> what you are looking for. It&#8217;s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!<\/p>\n<ul id=\"fs-id1475126\">\n<li><em>In this problem, the words &#8220;what was the original price of the shirt&#8221; tell you what you are looking for: the original price of the shirt.<\/em><\/li>\n<\/ul>\n<p>Step 3. <strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.<\/p>\n<ul id=\"fs-id1171505228428\">\n<li><em>Let [latex]p=[\/latex] the original price of the shirt<\/em><\/li>\n<\/ul>\n<p>Step 4. <strong>Translate<\/strong> into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223238\/CNX_BMath_Figure_09_01_002_img.png\" alt=\"The top line reads: &quot;18 is one half of the original price&quot;. The bottom line translates the top line from words to an algebraic equation. The word &quot;is&quot; translates to an equal sign. The phrase &quot;one half&quot; translates to &quot;1\/2&quot;. The word &quot;of&quot; translates to a multiplication symbol. The phrase &quot;the original price&quot; translates to &quot;p&quot;. This gives the full algebraic equation &quot;18 = 1\/2 times p&quot;.\" width=\"443\" height=\"76\" \/><br \/>\nStep 5. <strong>Solve<\/strong> the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.<\/p>\n<table id=\"eip-id1168047368619\" class=\"unnumbered unstyled\" summary=\"The top line reads,\">\n<tbody>\n<tr>\n<td>Write the equation.<\/td>\n<td>[latex]18=\\Large\\frac{1}{2}p[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply both sides by 2.<\/td>\n<td>[latex]\\color{red}{2}\\cdot18=\\color{red}{2}\\cdot\\Large\\frac{1}{2}\\normalsize p[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]36=p[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 6. <strong>Check<\/strong> the answer in the problem and make sure it makes sense.<\/p>\n<ul id=\"fs-id1594085\">\n<li><em>We found that<\/em> [latex]p=36[\/latex], <em>which means the original price was<\/em> [latex]\\text{\\$36}[\/latex]. <em>Does<\/em> [latex]\\text{\\$36}[\/latex] <em>make sense in the problem? Yes, because<\/em> [latex]18[\/latex] <em>is one-half of<\/em> [latex]36[\/latex], <em>and the shirt was on sale at half the original price.<\/em><\/li>\n<\/ul>\n<p>Step 7. <strong>Answer<\/strong> the question with a complete sentence.<\/p>\n<ul id=\"fs-id1495941\">\n<li><em>The problem asked &#8220;What was the original price of the shirt?&#8221; The answer to the question is: &#8220;The original price of the shirt was<\/em> [latex]\\text{\\$36}[\/latex].&#8221;<\/li>\n<\/ul>\n<p>If this were a homework exercise, our work might look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223244\/CNX_BMath_Figure_09_01_022.png\" alt=\"An example of what a student's work might look like for the problem. Let p equal the original price. 18 is one half the original price. 18 equals one half p. 2 times 18 equals 2 times one half p. 36 equals p. Check: is $36 a reasonable price for a shirt? Yes. Is 18 one half of 36? Yes. The original price of the shirt was $36.\" width=\"542\" height=\"319\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142694\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142694&#38;theme=oea&#38;iframe_resize_id=ohm142694&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We list the steps we took to solve the previous example.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Problem-Solving Strategy<\/h3>\n<ol id=\"eip-id1168469627809\" class=\"stepwise\">\n<li><strong>Read<\/strong> the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don&#8217;t understand, look them up in a dictionary or on the internet.<\/li>\n<li><strong>Identify<\/strong> what you are looking for.\u00a0\u00a0Determine the constants and variables in the problem.\u00a0 A constant is a number in the problem that is not going to change.\u00a0 A variable is a number that you don&#8217;t yet know its value.<\/li>\n<li><strong>Name<\/strong> what you are looking for. Choose a letter to represent that quantity.<\/li>\n<li><strong>Translate<\/strong> words into algebraic expressions and equations.\u00a0 Write an equation to represent the problem.\u00a0It may be helpful to first restate the problem in one sentence before translating.<\/li>\n<li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li>\n<li><strong>Check<\/strong> the answer in the problem. Make sure it makes sense.<\/li>\n<li><strong>Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<h2>Translate word problems into expressions<\/h2>\n<p>One of the first steps to solving word problems is converting an English sentence into a mathematical sentence. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it&#8217;s good to see what mathematical operators are associated with them.<\/p>\n<table>\n<thead>\n<tr>\n<th>Addition [latex]+[\/latex]<\/th>\n<th>Subtraction [latex]-[\/latex]<\/th>\n<th>Multiplication [latex]\\times[\/latex]<\/th>\n<th>Variable ?<\/th>\n<th>Equals [latex]=[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>More than<\/td>\n<td>Less than<\/td>\n<td>Double<\/td>\n<td>A number<\/td>\n<td>Is<\/td>\n<\/tr>\n<tr>\n<td>Together<\/td>\n<td>In the past<\/td>\n<td>Product<\/td>\n<td>Often, a value for which no information is given.<\/td>\n<td>The same as<\/td>\n<\/tr>\n<tr>\n<td>Sum<\/td>\n<td>Slower than<\/td>\n<td>\u00a0times<\/td>\n<td>After how many hours?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>The remainder of<\/td>\n<td><\/td>\n<td>How much will it cost?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>In the future<\/td>\n<td>\u00a0Difference<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Faster than<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Some examples follow:<\/p>\n<ul>\n<li>&#8220;[latex]x\\text{ is }5[\/latex]&#8221;\u00a0becomes [latex]x=5[\/latex]<\/li>\n<li>&#8220;Three more than a number&#8221; becomes [latex]x+3[\/latex]<\/li>\n<li>&#8220;Four less than a number&#8221; becomes [latex]x-4[\/latex]<\/li>\n<li>&#8220;Double the cost&#8221; becomes [latex]2\\cdot\\text{ cost }[\/latex]<\/li>\n<li>&#8220;Groceries and gas together for the week cost $250&#8221; means [latex]\\text{ groceries }+\\text{ gas }=250[\/latex]<\/li>\n<li>&#8220;The difference of [latex]9[\/latex] and a number&#8221; becomes [latex]9-x[\/latex]. Notice how [latex]9[\/latex] is first in the sentence and the expression.<\/li>\n<\/ul>\n<p>Let&#8217;s practice translating a few more English phrases into algebraic\u00a0expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate the table into algebraic expressions:<\/p>\n<table>\n<tbody>\n<tr>\n<td>\u00a0some number<\/td>\n<td>\u00a0the sum of the number and [latex]3[\/latex]<\/td>\n<td>\u00a0twice the sum of the number and [latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a length<\/td>\n<td>\u00a0double the length<\/td>\n<td>\u00a0double the length, decreased by [latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a cost<\/td>\n<td>\u00a0the difference of the cost and [latex]20[\/latex]<\/td>\n<td>\u00a0[latex]2[\/latex] times the difference of the cost and [latex]20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0some quantity<\/td>\n<td>\u00a0the difference of [latex]5[\/latex] and the quantity<\/td>\n<td>\u00a0\u00a0the difference of [latex]5[\/latex] and the quantity, divided by [latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0an amount of time<\/td>\n<td>\u00a0triple the amount of time<\/td>\n<td>\u00a0triple the amount of time, increased by [latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a distance<\/td>\n<td>\u00a0the sum of [latex]-4[\/latex] and the distance<\/td>\n<td>\u00a0the sum of [latex]-4[\/latex] and the twice the distance<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790402\">Show Solution<\/span><\/p>\n<div id=\"q790402\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>\u00a0[latex]a[\/latex]<\/td>\n<td>\u00a0[latex]a+3[\/latex]<\/td>\n<td>\u00a0[latex]2\\left(a+3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]l[\/latex]<\/td>\n<td>\u00a0[latex]2l[\/latex]<\/td>\n<td>\u00a0[latex]2l-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]c[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]c-20[\/latex]<\/td>\n<td>\u00a0[latex]2\\left(c-20\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]q[\/latex]<\/td>\n<td>\u00a0[latex]5-q[\/latex]<\/td>\n<td>\u00a0[latex]\\frac{5-q}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]t[\/latex]<\/td>\n<td>\u00a0[latex]3t[\/latex]<\/td>\n<td>\u00a0[latex]3t+5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]d[\/latex]<\/td>\n<td>\u00a0[latex]-4+d[\/latex]<\/td>\n<td>\u00a0[latex]-4+2d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In this example video, we show how to translate more words into mathematical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Writing Algebraic Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/uD_V5t-6Kzs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For another review of how to translate algebraic statements into words, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Write Algebraic Expressions from Statements: Form  ax+b and a(x+b)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hub7ku7UHT4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The power of algebra is how it can help you model real situations in order to answer questions about them.\u00a0 Let&#8217;s use this approach with another example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[\/latex] apples to the picnic. How many bananas did he bring?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15930\">Show Solution<\/span><\/p>\n<div id=\"q15930\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467372660\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem.\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>How many bananas did he bring?<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name<\/strong> what you are looking for.Choose a variable to represent the number of bananas.<\/td>\n<td>Let [latex]b=\\text{number of bananas}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Restate the problem in one sentence with all the important information.Translate into an equation.<\/td>\n<td>[latex]11\\enspace\\Rightarrow[\/latex] The number of apples[latex]=\\enspace\\Rightarrow[\/latex] was<\/p>\n<p>[latex]3\\enspace\\Rightarrow[\/latex] three<\/p>\n<p>[latex]+\\enspace\\Rightarrow[\/latex] more than<\/p>\n<p>[latex]2b\\enspace\\Rightarrow[\/latex] twice the number of bananas<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]11=2b+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract 3 from each side.<\/td>\n<td>[latex]11\\color{red}{-3}=2b+3\\color{red}{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8=2b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide each side by 2.<\/td>\n<td>[latex]\\Large\\frac{8}{\\color{red}{2}}\\normalsize =\\Large\\frac{2b}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]4=b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice 4 bananas is 8. Three more than 8 is 11.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>Yash brought 4 bananas to the picnic.<\/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=\"ohm142722\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142722&#38;theme=oea&#38;iframe_resize_id=ohm142722&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Twenty-eight\u00a0less than five times a certain number is [latex]232[\/latex]. What is the number?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q720402\">Show Solution<\/span><\/p>\n<div id=\"q720402\" class=\"hidden-answer\" style=\"display: none\">\n<p>Following the steps provided:<\/p>\n<ol>\n<li><strong>Read and understand:<\/strong> we are looking for a number.<\/li>\n<li><strong>Constants and variables:<\/strong>\u00a0[latex]28[\/latex] and [latex]232[\/latex] are constants, &#8220;a certain number&#8221; is our variable, because we don&#8217;t know its value, and we are asked to find it. We will call it\u00a0[latex]x[\/latex].<\/li>\n<li><strong>Translate:\u00a0<\/strong>five times a certain number translates to [latex]5x[\/latex]<br \/>\nTwenty-eight\u00a0less than five times a certain number translates to\u00a0[latex]5x-28[\/latex], because subtraction is built backward.<br \/>\n&#8220;is 232&#8221; translates to &#8220;[latex]=232\"[\/latex] since &#8220;is&#8221; is associated with equals.<\/li>\n<li><strong>Write an equation:<\/strong>\u00a0[latex]5x-28=232[\/latex]<\/li>\n<li><strong>Solve the equation using what you know about solving linear equations:<\/strong>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5x-28=232\\\\5x=260\\\\x=52\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/li>\n<li><strong>Check and interpret:<\/strong> We can substitute [latex]52[\/latex] for [latex]x[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\left(52\\right)-28=232\\\\5\\left(52\\right)=260\\\\260=260\\end{array}[\/latex]<\/p>\n<p>TRUE!<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Write and Solve a Linear Equations to Solve a Number Problem (1)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/izIIqOztUyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>In the next example, we will apply our Problem-Solving Strategy to applications of percent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Nga&#8217;s car insurance premium increased by [latex]\\text{\\$60}[\/latex], which was [latex]\\text{8%}[\/latex] of the original cost. What was the original cost of the premium?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q662772\">Show Solution<\/span><\/p>\n<div id=\"q662772\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469833028\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td style=\"width: 506.783px;\">Step 1. <strong>Read<\/strong> the problem. Remember, if there are words you don&#8217;t understand, look them up.<\/td>\n<td style=\"width: 328.217px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"width: 328.217px;\">the original cost of the premium<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent the original cost of premium.<\/td>\n<td style=\"width: 328.217px;\">Let [latex]c=\\text{the original cost}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 4. <strong>Translate.<\/strong> Restate as one sentence. Translate into an equation.<\/td>\n<td style=\"width: 328.217px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223257\/CNX_BMath_Figure_09_01_024_img-01.png\" alt=\"The top line reads: &quot;$60 was 8% of the original cost&quot;. The bottom line translates the top line from words to an algebraic equation. The word &quot;was&quot; translates to an equal sign. The term &quot;8%&quot; translates to &quot;0.08&quot;. The word &quot;of&quot; translates to a multiplication symbol. The phrase &quot;the original cost&quot; translates to &quot;c&quot;. This gives the full algebraic equation &quot;60 = 0.08 times c&quot;.\" width=\"304\" height=\"71\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td style=\"width: 328.217px;\">[latex]60=0.08c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Divide both sides by [latex]0.08[\/latex].<\/td>\n<td style=\"width: 328.217px;\">[latex]\\Large\\frac{60}{\\color{red}{0.08}}\\normalsize =\\Large\\frac{0.08c}{\\color{red}{0.08}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Simplify.<\/td>\n<td style=\"width: 328.217px;\">[latex]c=750[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 6. <strong>Check:<\/strong> Is our answer reasonable? Yes, a [latex]\\text{\\$750}[\/latex] premium on auto insurance is reasonable. Now let&#8217;s check our algebra. Is 8% of 750 equal to [latex]60[\/latex]?[latex]750=c[\/latex]<\/p>\n<p>[latex]0.08(750)=60[\/latex]<\/p>\n<p>[latex]60=60\\quad\\checkmark[\/latex]<\/td>\n<td style=\"width: 328.217px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"width: 328.217px;\">The original cost of Nga&#8217;s premium was [latex]\\text{\\$750}[\/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=\"ohm142735\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142735&#38;theme=oea&#38;iframe_resize_id=ohm142735&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm142761\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142761&#38;theme=oea&#38;iframe_resize_id=ohm142761&#38;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-9119\">\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>Write Algebraic Expressions from Statements: Form ax+b and a(x+b). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hub7ku7UHT4\">https:\/\/youtu.be\/Hub7ku7UHT4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 142694, 142722, 142735, 142761. <strong>Authored by<\/strong>: Lumen Learning. <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 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