{"id":520,"date":"2023-02-01T00:04:27","date_gmt":"2023-02-01T00:04:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/using-the-distance-rate-and-time-formula\/"},"modified":"2023-02-01T00:04:27","modified_gmt":"2023-02-01T00:04:27","slug":"using-the-distance-rate-and-time-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/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":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Use the problem-solving method to solve problems using the distance, rate, and time formula<\/li><\/ul><\/div>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!\n\nThe math to calculate the distance might look like this:\n\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]\n\nIn general, the formula relating distance, rate, and time is\n\n<\/p><p style=\"text-align: center\">[latex]\\text{distance}\\text{=}\\text{rate}\\cdot \\text{time}[\/latex]\n\n<\/p><div class=\"textbox shaded\"><h3>Distance, Rate, and Time<\/h3>For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula\n\n<p style=\"text-align: center\">[latex]d=rt[\/latex]\n\n<\/p><p style=\"text-align: center\">where [latex]d=[\/latex] distance, [latex]r=[\/latex] rate, and [latex]t=[\/latex] time.\n\n<\/p><\/div>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 \"hour\" divided out. The answer was in miles.\n\n<div class=\"textbox exercises\"><h3>example<\/h3>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?\n\nSolution:\n\n<table id=\"eip-id1168468716988\" class=\"unnumbered unstyled\" summary=\"The top line says, \"><tbody><tr><td>Step 1. <strong>Read<\/strong> the problem.You may want to create a mini-chart to summarize the\ninformation in the problem.\n\n<\/td><td>[latex]d=?[\/latex][latex]r=12\\text{mph}[\/latex]\n\n[latex]t=3\n\n\\Large\\frac{1}{2}\\normalsize\\text{hours}[\/latex]\n\n<\/td><\/tr><tr><td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td><td>distance traveled<\/td><\/tr><tr><td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td><td>let <em>d<\/em> = distance<\/td><\/tr><tr><td>Step 4. <strong>Translate.<\/strong>Write the appropriate formula for the situation.\n\nSubstitute in the given information.\n\n<\/td><td>[latex]d=rt[\/latex][latex]d=12\\cdot 3\\Large\\frac{1}{2}[\/latex]\n\n<\/td><\/tr><tr><td>Step 5. <strong>Solve<\/strong> the equation.<\/td><td>[latex]d=42\\text{ miles}[\/latex]<\/td><\/tr><tr><td>Step 6. <strong>Check:<\/strong> Does 42 miles make sense?<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=\".\">\n\n<\/td><td><\/td><\/tr><tr><td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.<\/td><td>Jamal rode 42 miles.<\/td><\/tr><\/tbody><\/table><\/div>&nbsp;\n\n<div class=\"textbox key-takeaways\"><h3>try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145553&amp;theme=oea&amp;iframe_resize_id=mom400[\/embed]\n\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145550&amp;theme=oea&amp;iframe_resize_id=mom40[\/embed]\n\n\n\n<\/div>In the following video we provide another example of how to solve for distance given rate and time.\n\nhttps:\/\/youtu.be\/lMO1L_CvH4Y\n\n<div class=\"textbox exercises\"><h3>Example<\/h3>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?\n\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\n\n<\/p><p class=\"p1\">[hidden-answer a=\"190834\"]\n\nSolution:\n\n<\/p><table id=\"eip-id1168466303184\" class=\"unnumbered unstyled\" summary=\".\"><tbody><tr><td>Step 1. <strong>Read<\/strong> the problem.Summarize the information in the problem.\n\n<\/td><td>[latex]d=520[\/latex] miles[latex]r=65[\/latex] mph\n\n[latex]t=?[\/latex]\n\n<\/td><\/tr><tr><td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td><td>how many hours (time)<\/td><\/tr><tr><td>Step 3. <strong>Name:<\/strong>Choose a variable to represent it.\n\n<\/td><td>let [latex]t[\/latex]&nbsp;= time<\/td><\/tr><tr><td>Step 4. <strong>Translate.<\/strong>Write the appropriate formula.\n\nSubstitute in the given information.\n\n<\/td><td>[latex]d=rt[\/latex][latex]520=65t[\/latex]\n\n<\/td><\/tr><tr><td>Step 5. <strong>Solve<\/strong> the equation.<\/td><td>[latex]t=8[\/latex]<\/td><\/tr><tr><td>Step 6. <strong>Check:<\/strong>Substitute the numbers into the formula and make sure\nthe result is a true statement.\n\n[latex]d=rt[\/latex]\n\n[latex]520\\stackrel{?}{=}65\\cdot 8[\/latex]\n\n[latex]520=520\\quad\\checkmark [\/latex]\n\n<\/td><td><\/td><\/tr><tr><td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.We know the units of time will be hours because\nwe divided miles by miles per hour.\n\n<\/td><td>Rey's trip will take [latex]8[\/latex] hours.<\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div>&nbsp;\n\n<div class=\"textbox key-takeaways\"><h3>try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145619&amp;theme=oea&amp;iframe_resize_id=mom500[\/embed]\n\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145620&amp;theme=oea&amp;iframe_resize_id=mom50[\/embed]\n\n\n\n<\/div>In the following video we show another example of how to find rate given distance and time.\n\nhttps:\/\/youtu.be\/3rYh32ErDaE\n\n\n","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.You may want to create a mini-chart to summarize the<br \/>\ninformation in the problem.<\/p>\n<\/td>\n<td>[latex]d=?[\/latex][latex]r=12\\text{mph}[\/latex]<\/p>\n<p>[latex]t=3  \\Large\\frac{1}{2}\\normalsize\\text{hours}[\/latex]<\/p>\n<\/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>Write the appropriate formula for the situation.<\/p>\n<p>Substitute in the given information.<\/p>\n<\/td>\n<td>[latex]d=rt[\/latex][latex]d=12\\cdot 3\\Large\\frac{1}{2}[\/latex]<\/p>\n<\/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?<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=\".\" \/><\/p>\n<\/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=\"ohm145553\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145553&#38;theme=oea&#38;iframe_resize_id=ohm145553&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145550\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145550&#38;theme=oea&#38;iframe_resize_id=ohm145550&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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.Summarize the information in the problem.<\/p>\n<\/td>\n<td>[latex]d=520[\/latex] miles[latex]r=65[\/latex] mph<\/p>\n<p>[latex]t=?[\/latex]<\/p>\n<\/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>Choose a variable to represent it.<\/p>\n<\/td>\n<td>let [latex]t[\/latex]&nbsp;= time<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong>Write the appropriate formula.<\/p>\n<p>Substitute in the given information.<\/p>\n<\/td>\n<td>[latex]d=rt[\/latex][latex]520=65t[\/latex]<\/p>\n<\/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>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]<\/p>\n<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question with a complete sentence.We know the units of time will be hours because<br \/>\nwe divided miles by miles per hour.<\/p>\n<\/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=\"ohm145619\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145619&#38;theme=oea&#38;iframe_resize_id=ohm145619&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145620\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145620&#38;theme=oea&#38;iframe_resize_id=ohm145620&#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 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-520\">\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 <\/section>","protected":false},"author":538461,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Find the Rate Given Distance and Time\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/3rYh32ErDaE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example: Solve a Problem using Distance = Rate x Time\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/lMO1L_CvH4Y\",\"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 145550, 145553,145619,145620\",\"author\":\"Lumen 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