{"id":4496,"date":"2020-04-13T14:40:25","date_gmt":"2020-04-13T14:40:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4496"},"modified":"2021-02-06T00:04:17","modified_gmt":"2021-02-06T00:04:17","slug":"solving-problems-involving-coins","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/solving-problems-involving-coins\/","title":{"raw":"Solving Problems Involving Coins","rendered":"Solving Problems Involving Coins"},"content":{"raw":"<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Determine the value of a given number of coins<\/li><li>Apply the problem-solving method to solve word problems involving coins<\/li><\/ul><\/div>Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?\r\n\r\nIf you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.\r\n\r\nOne way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.\r\n\r\n[caption id=\"attachment_12001\" align=\"aligncenter\" width=\"526\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/13222853\/16441865195_fc34001728_o.jpg\"><img class=\"wp-image-12001 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/13222853\/16441865195_fc34001728_o-1024x678.jpg\" alt=\"A small stack of dimes, a slightly larger stack of nickels, and a slightly larger stack of quarters, all lined up on a glossy table with their reflections showing\" width=\"526\" height=\"348\"><\/a> To determine the total value of a stack of nickels, multiply the number of nickels times the value of one nickel.[\/caption]\r\n\r\nHow would you determine the value of each pile? Think about the dime pile\u2014how much is it worth? If you count the number of dimes, you'll know how many you have\u2014the <em>number<\/em> of dimes.\r\n\r\nBut this does not tell you the <em>value<\/em> of all the dimes. Say you counted [latex]17[\/latex] dimes; how much are they worth? Each dime is worth [latex]\\text{\\$0.10}[\/latex] \u2014that is the <em>value<\/em> of one dime. To find the total value of the pile of [latex]17[\/latex] dimes, multiply [latex]17[\/latex] by [latex]\\text{\\$0.10}[\/latex] to get [latex]\\text{\\$1.70}[\/latex]. This is the total value of all [latex]17[\/latex] dimes.\r\n\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\hfill 17\\cdot \\text{\\$0.10}=\\text{\\$1.70}\\hfill \\\\ \\hfill \\text{number}\\cdot \\text{value}=\\text{total value}\\hfill \\end{array}[\/latex]\r\n\r\n<div class=\"textbox shaded\"><h3>Finding the Total Value for Coins of the Same Type<\/h3>For coins of the same type, the total value can be found as follows:\r\n\r\n[latex]\\text{number}\\cdot \\text{value}=\\text{total value}[\/latex]\r\nwhere <em>number<\/em> is the number of coins, <em>value<\/em> is the value of each coin, and <em>total value<\/em> is the total value of all the coins.\r\n\r\n<\/div>You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of <em>all<\/em> the coins, add the total value of each type of coin.\r\n\r\nLet's look at a specific case. Suppose there are [latex]14[\/latex] quarters, [latex]17[\/latex] dimes, [latex]21[\/latex] nickels, and [latex]39[\/latex] pennies. We'll make a table to organize the information \u2013 the type of coin, the number of each, and the value.\r\n\r\n<table id=\"fs-id1801596\" summary=\"A table is shown with 6 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Quarters<\/td><td>[latex]14[\/latex]<\/td><td>[latex]0.25[\/latex]<\/td><td>[latex]3.50[\/latex]<\/td><\/tr><tr valign=\"top\"><td>Dimes<\/td><td>[latex]17[\/latex]<\/td><td>[latex]0.10[\/latex]<\/td><td>[latex]1.70[\/latex]<\/td><\/tr><tr valign=\"top\"><td>Nickels<\/td><td>[latex]21[\/latex]<\/td><td>[latex]0.05[\/latex]<\/td><td>[latex]1.05[\/latex]<\/td><\/tr><tr valign=\"top\"><td>Pennies<\/td><td>[latex]39[\/latex]<\/td><td>[latex]0.01[\/latex]<\/td><td>[latex]0.39[\/latex]<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]6.64[\/latex]<\/td><\/tr><\/tbody><\/table>The total value of all the coins is [latex]\\text{\\$6.64}[\/latex]. Notice how the above table helped us organize all the information.\r\n\r\nLet's review the problem-solving method for word problems.\r\n\r\n<div class=\"textbox shaded\"><h3 class=\"title\">PROBLEM-SOLVING STRATEGY<\/h3>Step 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas.\r\n\r\nStep 2. <strong>Identify<\/strong> what you are looking for.\r\n\r\nStep 3. <strong>Name<\/strong> what you are looking for.\r\n\r\nStep 4. <strong>Translate<\/strong> into an equation. Restate the problem in one sentence. Then translate into an equation.\r\n\r\nStep 5. <strong>Solve<\/strong> the equation using good algebra techniques.\r\n\r\nStep 6. <strong>Check.<\/strong>\r\n\r\nStep 7. <strong>Answer<\/strong> the question.\r\n\r\n<\/div>Let's see how this method is used to solve a coin word problem.\r\n\r\n<div class=\"textbox exercises\"><h3>Example<\/h3>Adalberto has [latex]\\text{\\$2.25}[\/latex] in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have?\r\n\r\nSolution:\r\nStep 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas.\r\n\r\n<ul id=\"fs-id1495439\"><li>Determine the types of coins involved.<\/li><\/ul>Think about the strategy we used to find the value of the handful of coins. The first thing you need is to notice what types of coins are involved. Adalberto has dimes and nickels.\r\n\r\n<ul id=\"fs-id1377147\"><li><strong>Create a table<\/strong> to organize the information.<ul id=\"fs-id1548751\"><li>Label the columns \"type\", \"number\", \"value\", \"total value\".<\/li><li>List the types of coins.<\/li><li>Write in the value of each type of coin.<\/li><li>Write in the total value of all the coins.<\/li><\/ul><\/li><\/ul>We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) in the table as a reminder.\r\nThe value of a dime is [latex]\\text{\\$0.10}[\/latex] and the value of a nickel is [latex]\\text{\\$0.05}[\/latex]. The total value of all the coins is [latex]\\text{\\$2.25}[\/latex].\r\n\r\n<table id=\"fs-id1775190\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Dimes<\/td><td><\/td><td>[latex]0.10[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td>Nickels<\/td><td><\/td><td>[latex]0.05[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.25[\/latex]<\/td><\/tr><\/tbody><\/table>Step 2. <strong>Identify<\/strong> what you are looking for.\r\n\r\n<ul id=\"fs-id1600799\"><li>We are asked to find the number of dimes and nickels Adalberto has.<\/li><\/ul>Step 3. <strong>Name<\/strong> what you are looking for.\r\n\r\n<ul id=\"fs-id1576495\"><li>Use variable expressions to represent the number of each type of coin.<\/li><li>Multiply the number times the value to get the total value of each type of coin. In this problem you cannot count each type of coin\u2014that is what you are looking for\u2014but you have a clue. There are nine more nickels than dimes. The number of nickels is nine more than the number of dimes. [latex]\\text{Let }d=\\text{number of dimes.}[\/latex] [latex]\\text{Let }d+9=\\text{number of nickels.}[\/latex] Fill in the \"number\" column to help get everything organized.<\/li><\/ul><table id=\"fs-id1685351\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Dimes<\/td><td>[latex]d[\/latex]<\/td><td>[latex]0.10[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td>Nickels<\/td><td>[latex]d+9[\/latex]<\/td><td>[latex]0.05[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.25[\/latex]<\/td><\/tr><\/tbody><\/table>Now we have all the information we need from the problem!\r\nYou multiply the number times the value to get the total value of each type of coin. While you do not know the actual number, you do have an expression to represent it.\r\nAnd so now multiply [latex]{\\text{number}}\\cdot {\\text{value}}[\/latex] and write the results in the Total Value column.\r\n\r\n<table id=\"fs-id1602602\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Dimes<\/td><td>[latex]d[\/latex]<\/td><td>[latex]0.10[\/latex]<\/td><td>[latex]0.10d[\/latex]<\/td><\/tr><tr valign=\"top\"><td>Nickels<\/td><td>[latex]d+9[\/latex]<\/td><td>[latex]0.05[\/latex]<\/td><td>[latex]0.05\\left(d+9\\right)[\/latex]<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.25[\/latex]<\/td><\/tr><\/tbody><\/table>Step 4. <strong>Translate<\/strong> into an equation. Restate the problem in one sentence. Then translate 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\/24223458\/CNX_BMath_Figure_09_02_001_img.png\" alt=\"The sentence \">\r\nStep 5. <strong>Solve<\/strong> the equation using good algebra techniques.\r\n\r\n<table id=\"eip-id1168466046372\" class=\"unnumbered unstyled\" summary=\"The top line says, \"><tbody><tr><td>Write the equation.<\/td><td>[latex]0.10d+0.05(d+9)=2.25[\/latex]<\/td><\/tr><tr><td>Distribute.<\/td><td>[latex]0.10d+0.05d+0.45=2.25[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]0.15d+0.45=2.25[\/latex]<\/td><\/tr><tr><td>Subtract 0.45 from each side.<\/td><td>[latex]0.15d=1.80[\/latex]<\/td><\/tr><tr><td>Divide to find the number of dimes.<\/td><td>[latex]d=12[\/latex]<\/td><\/tr><tr><td>The number of nickels is <em>d<\/em> + 9<\/td><td>[latex]d+9[\/latex][latex]\\color{red}{12}+9[\/latex]\r\n\r\n[latex]21[\/latex]\r\n\r\n<\/td><\/tr><\/tbody><\/table>Step 6. <strong>Check.<\/strong>\r\n[latex]\\begin{array}{ccc}12\\text{ dimes: }12\\left(0.10\\right)\\hfill &amp; =\\hfill &amp; 1.20\\hfill \\\\ 21\\text{ nickels: }21\\left(0.05\\right)\\hfill &amp; =\\hfill &amp; \\underset{\\text{_____}}{1.05}\\hfill \\\\ &amp; &amp; \\text{\\$2.25}\\quad\\checkmark \\hfill \\end{array}[\/latex]\r\n\r\nStep 7. <strong>Answer<\/strong> the question.\r\n[latex]\\mathit{\\text{Adalberto has twelve dimes and twenty-one nickels.}}[\/latex]\r\nIf this were a homework exercise, our work might look like this:\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223507\/CNX_BMath_Figure_09_02_010.png\" alt=\"How many of each type does he have?\">\r\n\r\n<\/div>&nbsp;\r\n\r\n<div class=\"textbox key-takeaways\"><h3>Try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145537&amp;theme=oea&amp;iframe_resize_id=mom200[\/embed]\r\n\r\n\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145542&amp;theme=oea&amp;iframe_resize_id=mom700[\/embed]\r\n\r\n\r\n\r\n<\/div><div class=\"textbox shaded\"><h3>Solve a coin word problem<\/h3><ol id=\"eip-id1168467391830\" class=\"stepwise\"><li><strong>Read<\/strong> the problem. Make sure you understand all the words and ideas, and create a table to organize the information.<\/li><li><strong>Identify<\/strong> what you are looking for.<\/li><li><strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity.<ul id=\"eip-id1168469604810\"><li>Use variable expressions to represent the number of each type of coin and write them in the table.<\/li><li>Multiply the number times the value to get the total value of each type of coin.<\/li><\/ul><\/li><li><strong>Translate<\/strong> into an equation. Write the equation by adding the total values of all the types of coins.<\/li><li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li><li><strong>Check<\/strong> the answer in the problem and make sure it makes sense.<\/li><li><strong>Answer<\/strong> the question with a complete sentence.<\/li><\/ol>You may find it helpful to put all the numbers into the table to make sure they check.\r\n\r\n<table id=\"eip-429\" summary=\"...\"><thead><tr><th>Type<\/th><th>Number<\/th><th>Value ($)<\/th><th>Total Value<\/th><\/tr><\/thead><tbody><tr><td><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td><\/td><td><\/td><td><\/td><td><\/td><\/tr><tr><td colspan=\"3\"><\/td><td><\/td><\/tr><\/tbody><\/table><\/div>&nbsp;\r\n\r\n<div class=\"textbox exercises\"><h3>Example<\/h3>Maria has [latex]\\text{\\$2.43}[\/latex] in quarters and pennies in her wallet. She has twice as many pennies as quarters. How many coins of each type does she have?\r\n\r\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\r\n\r\n<p class=\"p1\">[hidden-answer a=\"190834\"]\r\n\r\nSolution:\r\nStep 1.&nbsp;<strong>Read<\/strong> the problem carefully.\r\n\r\n<ul id=\"fs-id1737044\"><li>Determine the types of coins involved. We know that Maria has quarters and pennies.<\/li><li>Create a table to organize the information.<ul id=\"fs-id1429247\"><li>Label the columns\u201ctype\u201d, \u201cnumber\u201d, \u201cvalue\u201d, \u201ctotal value\u201d.<\/li><li>List the types of coins.<\/li><li>Write in the value of each type of coin.<\/li><li>Write in the total value of all the coins.<\/li><\/ul><\/li><\/ul><table id=\"fs-id1662723\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Quarters<\/td><td><\/td><td>[latex]0.25[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td>Pennies<\/td><td><\/td><td>[latex]0.01[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.43[\/latex]<\/td><\/tr><\/tbody><\/table>Step 2. <strong>Identify<\/strong> what you are looking for.\r\n[latex]\\text{We are looking for the number of quarters and pennies.}[\/latex]\r\n\r\nStep 3. <strong>Name:<\/strong> Represent the number of quarters and pennies using variables.\r\n[latex]\\text{We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.}[\/latex]\r\n[latex]\\text{Let }q\\text{ represent the number of quarters.}[\/latex]\r\n[latex]\\text{Then the number of pennies is }2q[\/latex].\r\n\r\n<table id=\"fs-id1876422\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Quarters<\/td><td>[latex]q[\/latex]<\/td><td>[latex]0.25[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td>Pennies<\/td><td>[latex]2q[\/latex]<\/td><td>[latex]0.01[\/latex]<\/td><td><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.43[\/latex]<\/td><\/tr><\/tbody><\/table>Multiply the \"number\" and the \"value\" to get the \"total value\" of each type of coin.\r\n\r\n<table id=\"fs-id1749816\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>Quarters<\/td><td>[latex]q[\/latex]<\/td><td>[latex]0.25[\/latex]<\/td><td>[latex]0.25q[\/latex]<\/td><\/tr><tr valign=\"top\"><td>Pennies<\/td><td>[latex]2q[\/latex]<\/td><td>[latex]0.01[\/latex]<\/td><td>[latex]0.01\\left(2q\\right)[\/latex]<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]2.43[\/latex]<\/td><\/tr><\/tbody><\/table>Step 4. <strong>Translate.<\/strong> Write the equation by adding the \"total value\" of all the types of coins.\r\n\r\nStep 5. <strong>Solve<\/strong> the equation.\r\n\r\n<table id=\"eip-id1168468405585\" class=\"unnumbered unstyled\" summary=\"The top line says, \"><tbody><tr><td>Write the equation.<\/td><td>[latex]0.25q+0.01(2q)=2.43[\/latex]<\/td><\/tr><tr><td>Multiply.<\/td><td>[latex]0.25q+0.02q=2.43[\/latex]<\/td><\/tr><tr><td>Combine like terms.<\/td><td>[latex]0.27q=2.43[\/latex]<\/td><\/tr><tr><td>Divide by [latex]0.27[\/latex].<\/td><td>[latex]q=9[\/latex] quarters<\/td><\/tr><tr><td>The number of pennies is [latex]2q[\/latex].<\/td><td>[latex]2q[\/latex][latex]2\\cdot\\color{red}{9}[\/latex]\r\n\r\n[latex]18[\/latex] pennies\r\n\r\n<\/td><\/tr><\/tbody><\/table>Step 6. <strong>Check<\/strong> the answer in the problem.\r\nMaria has [latex]9[\/latex] quarters and [latex]18[\/latex] pennies. Does this make [latex]\\text{\\$2.43}?[\/latex]\r\n[latex]\\begin{array}{cccccc}\\text{9 quarters}\\hfill &amp; &amp; &amp; 9\\left(0.25\\right)\\hfill &amp; =\\hfill &amp; 2.25\\hfill \\\\ \\text{18 pennies}\\hfill &amp; &amp; &amp; 18\\left(0.01\\right)&amp; =\\hfill &amp; \\underset{\\text{_____}}{0.18}\\hfill \\\\ \\text{Total}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{\\$2.43}\\quad\\checkmark \\end{array}[\/latex]\r\n\r\nStep 7. <strong>Answer<\/strong> the question. Maria has nine quarters and eighteen pennies.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>&nbsp;\r\n\r\n<div class=\"textbox key-takeaways\"><h3>Try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142830&amp;theme=oea&amp;iframe_resize_id=mom20[\/embed]\r\n\r\n\r\n\r\n<\/div>In the next example, we'll show only the completed table\u2014make sure you understand how to fill it in step by step.\r\n\r\n<div class=\"textbox exercises\"><h3>Example<\/h3>Danny has [latex]\\text{\\$2.14}[\/latex] worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?\r\n\r\n<p class=\"p1\">[reveal-answer q=\"878656\"]Show Solution[\/reveal-answer]\r\n\r\n<p class=\"p1\">[hidden-answer a=\"878656\"]\r\n\r\nSolution:\r\n\r\n<table id=\"eip-id1168466711024\" class=\"unnumbered unstyled\" summary=\".\"><tbody><tr><td>Step 1: <strong>Read<\/strong> the problem.<\/td><td><\/td><\/tr><tr><td>Determine the types of coins involved.Create a table.\r\n\r\n<\/td><td>Pennies and nickels<\/td><\/tr><tr><td>Write in the value of each type of coin.<\/td><td>Pennies are worth [latex]$0.01[\/latex].Nickels are worth [latex]$0.05[\/latex].\r\n\r\n<\/td><\/tr><tr><td>Step 2: <strong>Identify<\/strong> what you are looking for.<\/td><td>the number of pennies and nickels<\/td><\/tr><tr><td>Step 3: <strong>Name.<\/strong> Represent the number of each type of coin using variables.The number of nickels is defined in terms of the number of pennies, so start with pennies.\r\n\r\n<\/td><td>Let [latex]p=\\text{number of pennies}[\/latex]<\/td><\/tr><tr><td>The number of nickels is two more than ten times the number of pennies.<\/td><td>[latex]10p+2=\\text{number of nickels}[\/latex]<\/td><\/tr><\/tbody><\/table>Multiply the number and the value to get the total value of each type of coin.\r\n\r\n<table id=\"fs-id1748710\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are, \"><thead><tr valign=\"top\"><th>Type<\/th><th>[latex]\\text{Number}[\/latex]<\/th><th>[latex]\\text{Value (\\$)}[\/latex]<\/th><th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>pennies<\/td><td>[latex]p[\/latex]<\/td><td>[latex]0.01[\/latex]<\/td><td>[latex]0.01p[\/latex]<\/td><\/tr><tr valign=\"top\"><td>nickels<\/td><td>[latex]10p+2[\/latex]<\/td><td>[latex]0.05[\/latex]<\/td><td>[latex]0.05\\left(10p+2\\right)[\/latex]<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\"><\/td><td>[latex]\\text{\\$2.14}[\/latex]<\/td><\/tr><\/tbody><\/table>Step 4. <strong>Translate:<\/strong> Write the equation by adding the total value of all the types of coins.\r\nStep 5. <strong>Solve<\/strong> the equation.\r\n\r\n<table id=\"eip-id1168048546793\" class=\"unnumbered unstyled\" summary=\"The top line gives the equation 0.01p plus 0.50p plus 0.10 equals 2.14. The next line shows 0.51p plus 0.10 equals 2.14. The next line shows 0.51p equals 2.04. The next line says p equals 4 pennies. Below this, it asks, \"><tbody><tr><td><\/td><td>[latex]0.01p+0.50p+0.10=2.14[\/latex]<\/td><\/tr><tr><td><\/td><td>[latex]0.51p+0.10=2.14[\/latex]<\/td><\/tr><tr><td><\/td><td>[latex]0.51p=2.04[\/latex]<\/td><\/tr><tr><td><\/td><td>[latex]p=4[\/latex] pennies<\/td><\/tr><tr><td>How many nickels?<\/td><td>[latex]10p+2[\/latex]<\/td><\/tr><tr><td><\/td><td>[latex]10(\\color{red}{4})+2[\/latex]<\/td><\/tr><tr><td><\/td><td>[latex]42[\/latex] nickels<\/td><\/tr><\/tbody><\/table>Step 6. <strong>Check.<\/strong> Is the total value of [latex]4[\/latex] pennies and [latex]42[\/latex] nickels equal to [latex]\\text{\\$2.14}?[\/latex]\r\n\r\n[latex]\\begin{array}{}\\\\ 4\\left(0.01\\right)+42\\left(0.05\\right)\\stackrel{?}{=}2.14\\\\ 2.14=2.14\\quad\\checkmark \\end{array}[\/latex]\r\nStep 7. <strong>Answer<\/strong> the question. Danny has [latex]4[\/latex] pennies and [latex]42[\/latex] nickels.\r\n\r\n<p class=\"p1\">[\/hidden-answer]\r\n\r\n<\/div>&nbsp;\r\n\r\n<div class=\"textbox key-takeaways\"><h3>Try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142831&amp;theme=oea&amp;iframe_resize_id=mom50[\/embed]\r\n\r\n\r\n\r\n<\/div>In the following video we show another example of how to solve a word problem that involves finding&nbsp;an amount of coins.\r\n\r\nhttps:\/\/youtu.be\/vOA7SGQCpr8\r\n\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine the value of a given number of coins<\/li>\n<li>Apply the problem-solving method to solve word problems involving coins<\/li>\n<\/ul>\n<\/div>\n<p>Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?<\/p>\n<p>If you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.<\/p>\n<p>One way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.<\/p>\n<div id=\"attachment_12001\" style=\"width: 536px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/13222853\/16441865195_fc34001728_o.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-12001\" class=\"wp-image-12001\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/05\/13222853\/16441865195_fc34001728_o-1024x678.jpg\" alt=\"A small stack of dimes, a slightly larger stack of nickels, and a slightly larger stack of quarters, all lined up on a glossy table with their reflections showing\" width=\"526\" height=\"348\" \/><\/a><\/p>\n<p id=\"caption-attachment-12001\" class=\"wp-caption-text\">To determine the total value of a stack of nickels, multiply the number of nickels times the value of one nickel.<\/p>\n<\/div>\n<p>How would you determine the value of each pile? Think about the dime pile\u2014how much is it worth? If you count the number of dimes, you&#8217;ll know how many you have\u2014the <em>number<\/em> of dimes.<\/p>\n<p>But this does not tell you the <em>value<\/em> of all the dimes. Say you counted [latex]17[\/latex] dimes; how much are they worth? Each dime is worth [latex]\\text{\\$0.10}[\/latex] \u2014that is the <em>value<\/em> of one dime. To find the total value of the pile of [latex]17[\/latex] dimes, multiply [latex]17[\/latex] by [latex]\\text{\\$0.10}[\/latex] to get [latex]\\text{\\$1.70}[\/latex]. This is the total value of all [latex]17[\/latex] dimes.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\hfill 17\\cdot \\text{\\$0.10}=\\text{\\$1.70}\\hfill \\\\ \\hfill \\text{number}\\cdot \\text{value}=\\text{total value}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Finding the Total Value for Coins of the Same Type<\/h3>\n<p>For coins of the same type, the total value can be found as follows:<\/p>\n<p>[latex]\\text{number}\\cdot \\text{value}=\\text{total value}[\/latex]<br \/>\nwhere <em>number<\/em> is the number of coins, <em>value<\/em> is the value of each coin, and <em>total value<\/em> is the total value of all the coins.<\/p>\n<\/div>\n<p>You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of <em>all<\/em> the coins, add the total value of each type of coin.<\/p>\n<p>Let&#8217;s look at a specific case. Suppose there are [latex]14[\/latex] quarters, [latex]17[\/latex] dimes, [latex]21[\/latex] nickels, and [latex]39[\/latex] pennies. We&#8217;ll make a table to organize the information \u2013 the type of coin, the number of each, and the value.<\/p>\n<table id=\"fs-id1801596\" summary=\"A table is shown with 6 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Quarters<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]3.50[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Dimes<\/td>\n<td>[latex]17[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]1.70[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Nickels<\/td>\n<td>[latex]21[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td>[latex]1.05[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Pennies<\/td>\n<td>[latex]39[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.39[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]6.64[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The total value of all the coins is [latex]\\text{\\$6.64}[\/latex]. Notice how the above table helped us organize all the information.<\/p>\n<p>Let&#8217;s review the problem-solving method for word problems.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">PROBLEM-SOLVING STRATEGY<\/h3>\n<p>Step 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas.<\/p>\n<p>Step 2. <strong>Identify<\/strong> what you are looking for.<\/p>\n<p>Step 3. <strong>Name<\/strong> what you are looking for.<\/p>\n<p>Step 4. <strong>Translate<\/strong> into an equation. Restate the problem in one sentence. Then translate into an equation.<\/p>\n<p>Step 5. <strong>Solve<\/strong> the equation using good algebra techniques.<\/p>\n<p>Step 6. <strong>Check.<\/strong><\/p>\n<p>Step 7. <strong>Answer<\/strong> the question.<\/p>\n<\/div>\n<p>Let&#8217;s see how this method is used to solve a coin word problem.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Adalberto has [latex]\\text{\\$2.25}[\/latex] in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have?<\/p>\n<p>Solution:<br \/>\nStep 1. <strong>Read<\/strong> the problem. Make sure you understand all the words and ideas.<\/p>\n<ul id=\"fs-id1495439\">\n<li>Determine the types of coins involved.<\/li>\n<\/ul>\n<p>Think about the strategy we used to find the value of the handful of coins. The first thing you need is to notice what types of coins are involved. Adalberto has dimes and nickels.<\/p>\n<ul id=\"fs-id1377147\">\n<li><strong>Create a table<\/strong> to organize the information.\n<ul id=\"fs-id1548751\">\n<li>Label the columns &#8220;type&#8221;, &#8220;number&#8221;, &#8220;value&#8221;, &#8220;total value&#8221;.<\/li>\n<li>List the types of coins.<\/li>\n<li>Write in the value of each type of coin.<\/li>\n<li>Write in the total value of all the coins.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) in the table as a reminder.<br \/>\nThe value of a dime is [latex]\\text{\\$0.10}[\/latex] and the value of a nickel is [latex]\\text{\\$0.05}[\/latex]. The total value of all the coins is [latex]\\text{\\$2.25}[\/latex].<\/p>\n<table id=\"fs-id1775190\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Dimes<\/td>\n<td><\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Nickels<\/td>\n<td><\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 2. <strong>Identify<\/strong> what you are looking for.<\/p>\n<ul id=\"fs-id1600799\">\n<li>We are asked to find the number of dimes and nickels Adalberto has.<\/li>\n<\/ul>\n<p>Step 3. <strong>Name<\/strong> what you are looking for.<\/p>\n<ul id=\"fs-id1576495\">\n<li>Use variable expressions to represent the number of each type of coin.<\/li>\n<li>Multiply the number times the value to get the total value of each type of coin. In this problem you cannot count each type of coin\u2014that is what you are looking for\u2014but you have a clue. There are nine more nickels than dimes. The number of nickels is nine more than the number of dimes. [latex]\\text{Let }d=\\text{number of dimes.}[\/latex] [latex]\\text{Let }d+9=\\text{number of nickels.}[\/latex] Fill in the &#8220;number&#8221; column to help get everything organized.<\/li>\n<\/ul>\n<table id=\"fs-id1685351\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Dimes<\/td>\n<td>[latex]d[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Nickels<\/td>\n<td>[latex]d+9[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we have all the information we need from the problem!<br \/>\nYou multiply the number times the value to get the total value of each type of coin. While you do not know the actual number, you do have an expression to represent it.<br \/>\nAnd so now multiply [latex]{\\text{number}}\\cdot {\\text{value}}[\/latex] and write the results in the Total Value column.<\/p>\n<table id=\"fs-id1602602\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Dimes<\/td>\n<td>[latex]d[\/latex]<\/td>\n<td>[latex]0.10[\/latex]<\/td>\n<td>[latex]0.10d[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Nickels<\/td>\n<td>[latex]d+9[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td>[latex]0.05\\left(d+9\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 4. <strong>Translate<\/strong> into an equation. Restate the problem in one sentence. Then translate into an equation.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223458\/CNX_BMath_Figure_09_02_001_img.png\" alt=\"The sentence\" \/><br \/>\nStep 5. <strong>Solve<\/strong> the equation using good algebra techniques.<\/p>\n<table id=\"eip-id1168466046372\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>Write the equation.<\/td>\n<td>[latex]0.10d+0.05(d+9)=2.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]0.10d+0.05d+0.45=2.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]0.15d+0.45=2.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract 0.45 from each side.<\/td>\n<td>[latex]0.15d=1.80[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide to find the number of dimes.<\/td>\n<td>[latex]d=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The number of nickels is <em>d<\/em> + 9<\/td>\n<td>[latex]d+9[\/latex][latex]\\color{red}{12}+9[\/latex]<\/p>\n<p>[latex]21[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 6. <strong>Check.<\/strong><br \/>\n[latex]\\begin{array}{ccc}12\\text{ dimes: }12\\left(0.10\\right)\\hfill & =\\hfill & 1.20\\hfill \\\\ 21\\text{ nickels: }21\\left(0.05\\right)\\hfill & =\\hfill & \\underset{\\text{_____}}{1.05}\\hfill \\\\ & & \\text{\\$2.25}\\quad\\checkmark \\hfill \\end{array}[\/latex]<\/p>\n<p>Step 7. <strong>Answer<\/strong> the question.<br \/>\n[latex]\\mathit{\\text{Adalberto has twelve dimes and twenty-one nickels.}}[\/latex]<br \/>\nIf this were a homework exercise, our work might look like this:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223507\/CNX_BMath_Figure_09_02_010.png\" alt=\"How many of each type does he have?\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145537\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145537&#38;theme=oea&#38;iframe_resize_id=ohm145537&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145542\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145542&#38;theme=oea&#38;iframe_resize_id=ohm145542&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Solve a coin word problem<\/h3>\n<ol id=\"eip-id1168467391830\" class=\"stepwise\">\n<li><strong>Read<\/strong> the problem. Make sure you understand all the words and ideas, and create a table to organize the information.<\/li>\n<li><strong>Identify<\/strong> what you are looking for.<\/li>\n<li><strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity.\n<ul id=\"eip-id1168469604810\">\n<li>Use variable expressions to represent the number of each type of coin and write them in the table.<\/li>\n<li>Multiply the number times the value to get the total value of each type of coin.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Translate<\/strong> into an equation. Write the equation by adding the total values of all the types of coins.<\/li>\n<li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li>\n<li><strong>Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong>Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<p>You may find it helpful to put all the numbers into the table to make sure they check.<\/p>\n<table id=\"eip-429\" summary=\"...\">\n<thead>\n<tr>\n<th>Type<\/th>\n<th>Number<\/th>\n<th>Value ($)<\/th>\n<th>Total Value<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\"><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Maria has [latex]\\text{\\$2.43}[\/latex] in quarters and pennies in her wallet. She has twice as many pennies as quarters. How many coins of each type does she have?<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nStep 1.&nbsp;<strong>Read<\/strong> the problem carefully.<\/p>\n<ul id=\"fs-id1737044\">\n<li>Determine the types of coins involved. We know that Maria has quarters and pennies.<\/li>\n<li>Create a table to organize the information.\n<ul id=\"fs-id1429247\">\n<li>Label the columns\u201ctype\u201d, \u201cnumber\u201d, \u201cvalue\u201d, \u201ctotal value\u201d.<\/li>\n<li>List the types of coins.<\/li>\n<li>Write in the value of each type of coin.<\/li>\n<li>Write in the total value of all the coins.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<table id=\"fs-id1662723\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Quarters<\/td>\n<td><\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Pennies<\/td>\n<td><\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.43[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 2. <strong>Identify<\/strong> what you are looking for.<br \/>\n[latex]\\text{We are looking for the number of quarters and pennies.}[\/latex]<\/p>\n<p>Step 3. <strong>Name:<\/strong> Represent the number of quarters and pennies using variables.<br \/>\n[latex]\\text{We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.}[\/latex]<br \/>\n[latex]\\text{Let }q\\text{ represent the number of quarters.}[\/latex]<br \/>\n[latex]\\text{Then the number of pennies is }2q[\/latex].<\/p>\n<table id=\"fs-id1876422\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Quarters<\/td>\n<td>[latex]q[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Pennies<\/td>\n<td>[latex]2q[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.43[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Multiply the &#8220;number&#8221; and the &#8220;value&#8221; to get the &#8220;total value&#8221; of each type of coin.<\/p>\n<table id=\"fs-id1749816\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Quarters<\/td>\n<td>[latex]q[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.25q[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Pennies<\/td>\n<td>[latex]2q[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.01\\left(2q\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]2.43[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 4. <strong>Translate.<\/strong> Write the equation by adding the &#8220;total value&#8221; of all the types of coins.<\/p>\n<p>Step 5. <strong>Solve<\/strong> the equation.<\/p>\n<table id=\"eip-id1168468405585\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>Write the equation.<\/td>\n<td>[latex]0.25q+0.01(2q)=2.43[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]0.25q+0.02q=2.43[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]0.27q=2.43[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide by [latex]0.27[\/latex].<\/td>\n<td>[latex]q=9[\/latex] quarters<\/td>\n<\/tr>\n<tr>\n<td>The number of pennies is [latex]2q[\/latex].<\/td>\n<td>[latex]2q[\/latex][latex]2\\cdot\\color{red}{9}[\/latex]<\/p>\n<p>[latex]18[\/latex] pennies<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 6. <strong>Check<\/strong> the answer in the problem.<br \/>\nMaria has [latex]9[\/latex] quarters and [latex]18[\/latex] pennies. Does this make [latex]\\text{\\$2.43}?[\/latex]<br \/>\n[latex]\\begin{array}{cccccc}\\text{9 quarters}\\hfill & & & 9\\left(0.25\\right)\\hfill & =\\hfill & 2.25\\hfill \\\\ \\text{18 pennies}\\hfill & & & 18\\left(0.01\\right)& =\\hfill & \\underset{\\text{_____}}{0.18}\\hfill \\\\ \\text{Total}\\hfill & & & & & \\hfill \\text{\\$2.43}\\quad\\checkmark \\end{array}[\/latex]<\/p>\n<p>Step 7. <strong>Answer<\/strong> the question. Maria has nine quarters and eighteen pennies.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142830\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142830&#38;theme=oea&#38;iframe_resize_id=ohm142830&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example, we&#8217;ll show only the completed table\u2014make sure you understand how to fill it in step by step.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Danny has [latex]\\text{\\$2.14}[\/latex] worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q878656\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q878656\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466711024\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Step 1: <strong>Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Determine the types of coins involved.Create a table.<\/p>\n<\/td>\n<td>Pennies and nickels<\/td>\n<\/tr>\n<tr>\n<td>Write in the value of each type of coin.<\/td>\n<td>Pennies are worth [latex]$0.01[\/latex].Nickels are worth [latex]$0.05[\/latex].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 2: <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the number of pennies and nickels<\/td>\n<\/tr>\n<tr>\n<td>Step 3: <strong>Name.<\/strong> Represent the number of each type of coin using variables.The number of nickels is defined in terms of the number of pennies, so start with pennies.<\/p>\n<\/td>\n<td>Let [latex]p=\\text{number of pennies}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The number of nickels is two more than ten times the number of pennies.<\/td>\n<td>[latex]10p+2=\\text{number of nickels}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Multiply the number and the value to get the total value of each type of coin.<\/p>\n<table id=\"fs-id1748710\" class=\"unnumbered\" summary=\"A table is shown with 4 rows and 4 columns. The top row is a header row. The headings are,\">\n<thead>\n<tr valign=\"top\">\n<th>Type<\/th>\n<th>[latex]\\text{Number}[\/latex]<\/th>\n<th>[latex]\\text{Value (\\$)}[\/latex]<\/th>\n<th>[latex]\\text{Total Value (\\$)}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>pennies<\/td>\n<td>[latex]p[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.01p[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>nickels<\/td>\n<td>[latex]10p+2[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td>[latex]0.05\\left(10p+2\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\"><\/td>\n<td>[latex]\\text{\\$2.14}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 4. <strong>Translate:<\/strong> Write the equation by adding the total value of all the types of coins.<br \/>\nStep 5. <strong>Solve<\/strong> the equation.<\/p>\n<table id=\"eip-id1168048546793\" class=\"unnumbered unstyled\" summary=\"The top line gives the equation 0.01p plus 0.50p plus 0.10 equals 2.14. The next line shows 0.51p plus 0.10 equals 2.14. The next line shows 0.51p equals 2.04. The next line says p equals 4 pennies. Below this, it asks,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.01p+0.50p+0.10=2.14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0.51p+0.10=2.14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0.51p=2.04[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]p=4[\/latex] pennies<\/td>\n<\/tr>\n<tr>\n<td>How many nickels?<\/td>\n<td>[latex]10p+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10(\\color{red}{4})+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]42[\/latex] nickels<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Step 6. <strong>Check.<\/strong> Is the total value of [latex]4[\/latex] pennies and [latex]42[\/latex] nickels equal to [latex]\\text{\\$2.14}?[\/latex]<\/p>\n<p>[latex]\\begin{array}{}\\\\ 4\\left(0.01\\right)+42\\left(0.05\\right)\\stackrel{?}{=}2.14\\\\ 2.14=2.14\\quad\\checkmark \\end{array}[\/latex]<br \/>\nStep 7. <strong>Answer<\/strong> the question. Danny has [latex]4[\/latex] pennies and [latex]42[\/latex] nickels.<\/p>\n<p class=\"p1\"><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142831\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142831&#38;theme=oea&#38;iframe_resize_id=ohm142831&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to solve a word problem that involves finding&nbsp;an amount of coins.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solve a Coin Problem Using an Equation in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vOA7SGQCpr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-4496\">\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 145537, 145542, 142830, 142831. <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>Coins. <strong>Authored by<\/strong>: KMR Photography. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.flickr.com\/photos\/morbokat\/16441865195\/\">https:\/\/www.flickr.com\/photos\/morbokat\/16441865195\/<\/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, 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":25777,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Coins\",\"author\":\"KMR Photography\",\"organization\":\"\",\"url\":\"https:\/\/www.flickr.com\/photos\/morbokat\/16441865195\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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 145537, 145542, 142830, 142831\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4496","chapter","type-chapter","status-web-only","hentry"],"part":3698,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4496","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4496\/revisions"}],"predecessor-version":[{"id":4497,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4496\/revisions\/4497"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/3698"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4496\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4496"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4496"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4496"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}