{"id":8948,"date":"2017-05-02T14:45:55","date_gmt":"2017-05-02T14:45:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=8948"},"modified":"2018-01-10T01:16:07","modified_gmt":"2018-01-10T01:16:07","slug":"using-the-distance-rate-and-time-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/using-the-distance-rate-and-time-formula\/","title":{"raw":"Using the Distance, Rate, and Time Formula","rendered":"Using the Distance, Rate, and Time Formula"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the problem-solving method to solve problems using the distance, rate, and time formula<\/li>\r\n<\/ul>\r\n<\/div>\r\nOne formula you\u2019ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you traveled if you drove at a steady rate of [latex]60[\/latex] miles per hour for [latex]2[\/latex] hours? (This might happen if you use your car\u2019s cruise control while driving on the Interstate.) If you said [latex]120[\/latex] miles, you already know how to use this formula!\r\n\r\nThe math to calculate the distance might look like this:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\text{distance}=\\left(\\Large\\frac{60\\text{ miles}}{1\\text{ hour}}\\normalsize\\right)\\left(2\\text{ hours}\\right)\\hfill \\\\ \\text{distance}=120\\text{ miles}\\hfill \\end{array}[\/latex]<\/p>\r\nIn general, the formula relating distance, rate, and time is\r\n<p style=\"text-align: center;\">[latex]\\text{distance}\\text{=}\\text{rate}\\cdot \\text{time}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Distance, Rate, and Time<\/h3>\r\nFor an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula\r\n<p style=\"text-align: center;\">[latex]d=rt[\/latex]<\/p>\r\n<p style=\"text-align: center;\">where [latex]d=[\/latex] distance, [latex]r=[\/latex] rate, and [latex]t=[\/latex] time.<\/p>\r\n\r\n<\/div>\r\nNotice that the units we used above for the rate were miles per hour, which we can write as a ratio [latex]\\Large\\frac{miles}{hour}[\/latex]. Then when we multiplied by the time, in hours, the common units \"hour\" divided out. The answer was in miles.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nJamal rides his bike at a uniform rate of [latex]12[\/latex] miles per hour for [latex]3\\Large\\frac{1}{2}[\/latex] hours. How much distance has he traveled?\r\n\r\nSolution:\r\n<table id=\"eip-id1168468716988\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.\r\n\r\nYou may want to create a mini-chart to summarize the\r\ninformation in the problem.<\/td>\r\n<td>[latex]d=?[\/latex]\r\n\r\n[latex]r=12\\text{mph}[\/latex]\r\n\r\n[latex]t=3\r\n\r\n\\Large\\frac{1}{2}\\normalsize\\text{hours}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>distance traveled<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>d<\/em> = distance<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula for the situation.\r\n\r\nSubstitute in the given information.<\/td>\r\n<td>[latex]d=rt[\/latex]\r\n\r\n[latex]d=12\\cdot 3\\Large\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]d=42\\text{ miles}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Does 42 miles make sense?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224214\/CNX_BMath_Figure_09_07_009_img-01.png\" alt=\".\" \/><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\r\n<td>Jamal rode 42 miles.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom400\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145553&amp;theme=oea&amp;iframe_resize_id=mom400\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145550&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we provide another example of how to solve for distance given rate and time.\r\n\r\nhttps:\/\/youtu.be\/lMO1L_CvH4Y\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of [latex]520[\/latex] miles. If he can drive at a steady rate of [latex]65[\/latex] miles per hour, how many hours will the trip take?\r\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190834\"]<\/p>\r\nSolution:\r\n<table id=\"eip-id1168466303184\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.\r\n\r\nSummarize the information in the problem.<\/td>\r\n<td>[latex]d=520[\/latex] miles\r\n\r\n[latex]r=65[\/latex] mph\r\n\r\n[latex]t=?[\/latex]<\/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 hours (time)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name:<\/strong>\r\n\r\nChoose a variable to represent it.<\/td>\r\n<td>let [latex]t[\/latex]\u00a0= time<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula.\r\n\r\nSubstitute in the given information.<\/td>\r\n<td>[latex]d=rt[\/latex]\r\n\r\n[latex]520=65t[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]t=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\nSubstitute the numbers into the formula and make sure\r\nthe result is a true statement.\r\n\r\n[latex]d=rt[\/latex]\r\n\r\n[latex]520\\stackrel{?}{=}65\\cdot 8[\/latex]\r\n\r\n[latex]520=520\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.\r\n\r\nWe know the units of time will be hours because\r\nwe divided miles by miles per hour.<\/td>\r\n<td>Rey's trip will take [latex]8[\/latex] hours.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom500\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145619&amp;theme=oea&amp;iframe_resize_id=mom500\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom50\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145620&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we show another example of how to find rate given distance and time.\r\n\r\nhttps:\/\/youtu.be\/3rYh32ErDaE","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the problem-solving method to solve problems using the distance, rate, and time formula<\/li>\n<\/ul>\n<\/div>\n<p>One formula you\u2019ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you traveled if you drove at a steady rate of [latex]60[\/latex] miles per hour for [latex]2[\/latex] hours? (This might happen if you use your car\u2019s cruise control while driving on the Interstate.) If you said [latex]120[\/latex] miles, you already know how to use this formula!<\/p>\n<p>The math to calculate the distance might look like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\text{distance}=\\left(\\Large\\frac{60\\text{ miles}}{1\\text{ hour}}\\normalsize\\right)\\left(2\\text{ hours}\\right)\\hfill \\\\ \\text{distance}=120\\text{ miles}\\hfill \\end{array}[\/latex]<\/p>\n<p>In general, the formula relating distance, rate, and time is<\/p>\n<p style=\"text-align: center;\">[latex]\\text{distance}\\text{=}\\text{rate}\\cdot \\text{time}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Distance, Rate, and Time<\/h3>\n<p>For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula<\/p>\n<p style=\"text-align: center;\">[latex]d=rt[\/latex]<\/p>\n<p style=\"text-align: center;\">where [latex]d=[\/latex] distance, [latex]r=[\/latex] rate, and [latex]t=[\/latex] time.<\/p>\n<\/div>\n<p>Notice that the units we used above for the rate were miles per hour, which we can write as a ratio [latex]\\Large\\frac{miles}{hour}[\/latex]. Then when we multiplied by the time, in hours, the common units &#8220;hour&#8221; divided out. The answer was in miles.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Jamal rides his bike at a uniform rate of [latex]12[\/latex] miles per hour for [latex]3\\Large\\frac{1}{2}[\/latex] hours. How much distance has he traveled?<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168468716988\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/p>\n<p>You may want to create a mini-chart to summarize the<br \/>\ninformation in the problem.<\/td>\n<td>[latex]d=?[\/latex]<\/p>\n<p>[latex]r=12\\text{mph}[\/latex]<\/p>\n<p>[latex]t=3    \\Large\\frac{1}{2}\\normalsize\\text{hours}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>distance traveled<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>d<\/em> = distance<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula for the situation.<\/p>\n<p>Substitute in the given information.<\/td>\n<td>[latex]d=rt[\/latex]<\/p>\n<p>[latex]d=12\\cdot 3\\Large\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]d=42\\text{ miles}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Does 42 miles make sense?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224214\/CNX_BMath_Figure_09_07_009_img-01.png\" alt=\".\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.<\/td>\n<td>Jamal rode 42 miles.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom400\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145553&amp;theme=oea&amp;iframe_resize_id=mom400\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145550&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we provide another example of how to solve for distance given rate and time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Solve a Problem using Distance = Rate x Time\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lMO1L_CvH4Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of [latex]520[\/latex] miles. If he can drive at a steady rate of [latex]65[\/latex] miles per hour, how many hours will the trip take?<\/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:<\/p>\n<table id=\"eip-id1168466303184\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/p>\n<p>Summarize the information in the problem.<\/td>\n<td>[latex]d=520[\/latex] miles<\/p>\n<p>[latex]r=65[\/latex] mph<\/p>\n<p>[latex]t=?[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>how many hours (time)<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name:<\/strong><\/p>\n<p>Choose a variable to represent it.<\/td>\n<td>let [latex]t[\/latex]\u00a0= time<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute in the given information.<\/td>\n<td>[latex]d=rt[\/latex]<\/p>\n<p>[latex]520=65t[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]t=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p>Substitute the numbers into the formula and make sure<br \/>\nthe result is a true statement.<\/p>\n<p>[latex]d=rt[\/latex]<\/p>\n<p>[latex]520\\stackrel{?}{=}65\\cdot 8[\/latex]<\/p>\n<p>[latex]520=520\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.<\/p>\n<p>We know the units of time will be hours because<br \/>\nwe divided miles by miles per hour.<\/td>\n<td>Rey&#8217;s trip will take [latex]8[\/latex] hours.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"mom500\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145619&amp;theme=oea&amp;iframe_resize_id=mom500\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom50\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145620&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to find rate given distance and time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Find the Rate Given Distance and Time\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3rYh32ErDaE?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-8948\">\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 145550, 145553,145619,145620. <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><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Find the Rate Given Distance and Time. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3rYh32ErDaE\">https:\/\/youtu.be\/3rYh32ErDaE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Example: Solve a Problem using Distance = Rate x Time. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lMO1L_CvH4Y\">https:\/\/youtu.be\/lMO1L_CvH4Y<\/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 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145553,145619,145620\",\"author\":\"Lumen 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