{"id":16113,"date":"2019-10-01T17:41:09","date_gmt":"2019-10-01T17:41:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-solve-distance-rate-and-time-problems\/"},"modified":"2020-10-22T09:11:46","modified_gmt":"2020-10-22T09:11:46","slug":"read-solve-distance-rate-and-time-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-solve-distance-rate-and-time-problems\/","title":{"raw":"7.4.a - Problems Involving Formulas I","rendered":"7.4.a &#8211; Problems Involving Formulas I"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve a formula for a specific variable<\/li>\r\n \t<li>Use the distance, rate, and time formula<\/li>\r\n \t<li>Apply for the steps for solving word problems to interest rate problems<\/li>\r\n<\/ul>\r\n<\/div>\r\nThere\u00a0is often a well-known formula or relationship that applies to a word problem. For example, if you were to plan a road trip, you would\u00a0want to know how long it would take you to reach your destination.\u00a0[latex]d=rt[\/latex] is a well-known relationship that associates distance traveled, the rate at which you travel, and how long the travel takes.\r\n<h2>Distance, Rate, and Time<\/h2>\r\nIf you know two of the quantities in the relationship [latex]d=rt[\/latex], you can easily find the third using methods for solving linear equations. For example, if you\u00a0know that you\u00a0will be traveling on a road with a speed limit of\u00a0[latex]30\\frac{\\text{ miles }}{\\text{ hour }}[\/latex] for [latex]2[\/latex] hours, you can\u00a0find the distance you would travel by\u00a0multiplying rate times time or [latex]\\left(30\\frac{\\text{ miles }}{\\text{ hour }}\\right)\\left(2\\text{ hours }\\right)=60\\text{ miles }[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]184997[\/ohm_question]\r\n\r\n<\/div>\r\nWe can generalize this idea depending on what information we are given and what we are looking for. For example, if we need to find time, we could solve the\u00a0[latex]d=rt[\/latex] equation for [latex]t[\/latex] using division:\r\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\frac{d}{r}=t[\/latex]<\/p>\r\nLikewise, if we want to find rate, we can isolate [latex]r[\/latex] using division:\r\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\frac{d}{t}=r[\/latex]<\/p>\r\nIn the following examples, you will see how this formula is applied to answer questions about ultra marathon running.\r\n\r\n[caption id=\"attachment_3540\" align=\"alignright\" width=\"266\"]<img class=\"size-medium wp-image-3540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/05184602\/Screen-Shot-2016-05-05-at-11.45.10-AM-266x300.png\" alt=\"Ann Trason\" width=\"266\" height=\"300\" \/> Ann Trason[\/caption]\r\n\r\nUltra marathon running (defined as anything longer than\u00a0[latex]26.2[\/latex] miles) is becoming very popular among women even though it remains a male-dominated niche sport. Ann Trason has broken twenty world records in her career. One such record was the American River\u00a0[latex]50[\/latex]-mile Endurance Run, which\u00a0begins in Sacramento, California, and ends in Auburn, California.[footnote]\"Ann Trason.\" Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/Ann_Trason.[\/footnote] In 1993, Trason finished the run with a time of [latex]6:09:08[\/latex]<strong>.<\/strong> \u00a0The men's record for the same course was set in 1994\u00a0by Tom Johnson, who finished the course with a time of \u00a0[latex]5:33:21[\/latex].[footnote]\u00a0\"American River [latex]50[\/latex] Mile Endurance Run.\" Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run.[\/footnote]\r\n\r\nIn the next examples, we will use the [latex]d=rt[\/latex] formula to answer the following questions about the two runners.\r\n<ol>\r\n \t<li>What was each runner's rate for their record-setting runs?<\/li>\r\n \t<li>By the time Johnson had finished, how many more miles did Trason have to run?<\/li>\r\n \t<li>How much further could Johnson have run if he had run as long as Trason?<\/li>\r\n \t<li>What was each runner's time for running one mile?<\/li>\r\n<\/ol>\r\nTo make answering the questions easier, we will round the two runners' times to\u00a0[latex]6[\/latex] hours and [latex]5.5[\/latex] hours.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat was each runner's rate for their record-setting runs? Round to two decimal places.\r\n[reveal-answer q=\"55589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"55589\"]\r\n\r\n<strong>Read and Understand:<\/strong>\u00a0We\u00a0are looking for rate and we know distance and time, so we can use the idea:\u00a0[latex]d=rt[\/latex], [latex]\\frac{d}{t}=r[\/latex]. Let's solve one runner's rate using the original formula, and one using the rearranged formula.\r\n\r\n<strong>Define and Translate:<\/strong> Because there are two runners, making a table to organize this information helps. Note how we keep units to help us keep track of how all the terms are related to each other.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]r[\/latex]<\/td>\r\n<td>[latex]6[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]r[\/latex]<\/td>\r\n<td>\u00a0[latex]5.5[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Write and Solve:<\/strong>\r\n\r\nTrason's rate:\r\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]50\\text{ miles }=r\\left(6\\text{ hours}\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]50=6r[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]r\\approx 8.33[\/latex]<\/p>\r\nJohnson's rate:\r\n<p style=\"text-align: center\">[latex]r=\\frac{d}{t}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]r=\\frac{50\\text{ miles}}{5.5\\text{ hours}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]r=\\frac{50}{5.5}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]r\\approx 9.10[\/latex]<\/p>\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nWe can fill in our table with this information.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]6[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]9.1[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]5.5[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow that we know each runner's rate, we can answer the second question.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nBy the time Johnson had finished, how many more miles did Trason have to run?\r\n[reveal-answer q=\"747303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"747303\"]\r\n\r\nHere is the table we created for reference:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]6[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>\u00a0[latex]5.5[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Read and Understand:<\/strong>\u00a0We are looking for how many miles Trason still had on the trail when Johnson had finished after [latex]5.5[\/latex] hours. This is a distance, and we know rate and time.\r\n\r\n<strong>Define and Translate:<\/strong>\u00a0We can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]d[\/latex], and the time Trason had run is [latex]5.5[\/latex] hours.\r\n\r\n<strong>Write and Solve:<\/strong>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}d=rt\\\\\\\\d=8.33\\frac{\\text{ miles }}{\\text{ hour }}\\left(5.5\\text{ hours}\\right)\\\\\\\\d=45.82\\text{ miles}\\end{array}[\/latex]<\/p>\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nHave we answered the question? We were asked to find how many more miles she had to run after [latex]5.5[\/latex] hours. \u00a0What we have found is how long she had run after [latex]5.5[\/latex] hours. We need to subtract [latex]d=45.82\\text{ miles }[\/latex] from the total distance of the course.\r\n<p style=\"text-align: center\">[latex]50\\text{ miles }-45.82\\text{ miles }=4.18\\text{ miles }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe third question is similar to the second. Now that we know each runner's rate, we can answer questions about individual distances or times.\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nHow much further could Johnson have run if he had run for the same amount of time as Trason?\r\n\r\n[reveal-answer q=\"757303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757303\"]\r\n\r\nHere is the table we created for reference:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]6[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]5.5[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Read and Understand:<\/strong>\u00a0The word further implies we are looking for a distance.\r\n\r\n<strong>Define and Translate:<\/strong>\u00a0We can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]d[\/latex], the\u00a0time is [latex]6[\/latex] hours, and Johnson's rate is [latex]9.1\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]\r\n\r\n<strong>Write and Solve:<\/strong>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}d=rt\\\\\\\\d=9.1\\frac{\\text{ miles }}{\\text{ hour }}\\left(6\\text{ hours }\\right)\\\\\\\\d=54.6\\text{ miles }\\end{array}[\/latex].<\/p>\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nHave we answered the question? We were asked to find how many more miles\u00a0Johnson would have run if he had run at his rate of\u00a0[latex]9.1\\frac{\\text{ miles }}{\\text{ hour }}[\/latex] for\u00a0[latex]6[\/latex] hours.\r\n\r\nJohnson would have run [latex]54.6[\/latex] miles, so that's [latex]4.6[\/latex] more miles than he ran during the race.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will tackle the last question, where we are asked to find a time for each runner.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhat was each runner's time for running one mile?\r\n\r\n[reveal-answer q=\"757309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757309\"]\r\n\r\nHere is the table we created for reference:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]6[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]50[\/latex] miles<\/td>\r\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]5.5[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Read and Understand:<\/strong>\u00a0we are looking for time, and this time our distance has changed from \u00a0[latex]50[\/latex] miles to [latex]1[\/latex] mile, so we can use\r\n<p style=\"text-align: center\">[latex]d=rt\\\\\\frac{d}{r}=t[\/latex]<\/p>\r\n<strong>Define and Translate:<\/strong> we can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]t[\/latex], the distance is [latex]1[\/latex] mile, and we know each runner's rate. It may help to create a new table:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Runner<\/th>\r\n<th>Distance =<\/th>\r\n<th>(Rate )<\/th>\r\n<th>(Time)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Trason<\/td>\r\n<td>[latex]1[\/latex] mile<\/td>\r\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]t[\/latex] hours<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Johnson<\/td>\r\n<td>[latex]1[\/latex] mile<\/td>\r\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\r\n<td>[latex]t[\/latex] hours<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Write and Solve:<\/strong>\r\n\r\nTrason:\r\n\r\nWe will need to divide to isolate time.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}d=rt\\\\\\\\1\\text{ mile }=8.33\\frac{\\text{ miles }}{\\text{ hour }}\\left(t\\text{ hours }\\right)\\\\\\\\\\frac{1\\text{ mile }}{\\frac{8.33\\text{ miles }}{\\text{ hour }}}=t\\text{ hours }\\\\\\\\0.12\\text{ hours }=t\\end{array}[\/latex].<\/p>\r\n[latex]0.12[\/latex] \u00a0hours is about \u00a0[latex]7.2[\/latex] minutes, so Trason's time for running one mile was about [latex]7.2[\/latex] minutes. WOW! She did that for \u00a0[latex]6[\/latex] hours!\r\n\r\nJohnson:\r\n\r\nWe will need to divide to isolate time.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}d=rt\\\\\\\\1\\text{ mile }=9.1\\frac{\\text{ miles }}{\\text{ hour }}\\left(t\\text{ hours }\\right)\\\\\\\\\\frac{1\\text{ mile }}{\\frac{9.1\\text{ miles }}{\\text{ hour }}}=t\\text{ hours }\\\\\\\\0.11\\text{ hours }=t\\end{array}[\/latex].<\/p>\r\n[latex]0.11[\/latex] hours is about [latex]6.6[\/latex] minutes, so Johnson's time for running one mile was about \u00a0[latex]6.6[\/latex] minutes. WOW! He did that for [latex]5.5[\/latex] hours!\r\n\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nHave we answered the question? We were asked to find how long it took each runner to run one mile given the rate at which they ran the whole \u00a0[latex]50[\/latex]-mile course. \u00a0Yes, we answered our question.\r\n\r\nTrason's mile time was [latex]7.2\\frac{\\text{minutes}}{\\text{mile}}[\/latex] and Johnsons' mile time was\u00a0[latex]6.6\\frac{\\text{minutes}}{\\text{mile}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of answering many rate questions given distance and time.\r\n\r\nhttps:\/\/youtu.be\/3WLp5mY1FhU\r\n<h2>Simple Interest<\/h2>\r\nIn order to entice customers to invest their money, many banks will offer interest-bearing accounts. The accounts work like this: a customer deposits a certain amount of money (called the Principal, or [latex]P[\/latex]), which then grows slowly according to the interest rate ([latex]r[\/latex], measured in percent) and the length of time ([latex]t[\/latex], usually measured in months) that the money stays in the account. The amount earned over time is called the interest ([latex]I[\/latex]), which is then given to the customer.\r\n<div class=\"textbox shaded\"><img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"76\" height=\"67\" \/>Beware! Interest rates are commonly given as yearly rates, but can also be monthly, quarterly, bimonthly, or even some custom amount of time. It is important that the units\u00a0of time and the units of the interest rate match. You will see why this matters in a later example.<\/div>\r\nThe simplest way to calculate interest earned on an account is through the formula [latex]\\displaystyle I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]\r\n\r\nIf we know any of the three amounts related to this equation, we can find the fourth. For example, if we want to find the time it will take to accrue a specific amount of interest, we can solve for [latex]t[\/latex] using division:\r\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\\\\\ \\frac{I}{{P}\\,\\cdot \\,r}=\\frac{P\\cdot\\,r\\,\\cdot \\,t}{\\,P\\,\\cdot \\,r}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{t}=\\frac{I}{\\,r\\,\\cdot \\,t}\\end{array}[\/latex]<\/p>\r\nBelow is a table showing the result of solving for each individual variable in the formula.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Solve For<\/th>\r\n<th>Result<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]I[\/latex]<\/td>\r\n<td>[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]P[\/latex]<\/td>\r\n<td>[latex]{P}=\\frac{I}{{r}\\,\\cdot \\,t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]r[\/latex]<\/td>\r\n<td>[latex]{r}=\\frac{I}{{P}\\,\\cdot \\,t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]t[\/latex]<\/td>\r\n<td>\u00a0[latex]{t}=\\frac{I}{{P}\\,\\cdot \\,r}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the next examples, we will show how to substitute given values into the simple interest formula, and decipher which variable to solve for.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIf a customer deposits a principal of\u00a0[latex]$2000[\/latex] at a monthly rate of \u00a0[latex]0.7\\%[\/latex], what is the total amount that she has after [latex]24[\/latex] months?\r\n\r\n[reveal-answer q=\"57640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"57640\"]\r\n\r\nSubstitute in the values given for the Principal, Rate, and Time.\r\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{l}I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\I=2000\\cdot 0.7\\%\\cdot 24\\end{array}[\/latex]<\/p>\r\nRewrite \u00a0[latex]0.7\\%[\/latex] as the decimal \u00a0[latex]0.007[\/latex], then multiply.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}I=2000\\cdot 0.007\\cdot 24\\\\I=336\\end{array}[\/latex]<\/p>\r\nAdd the interest and the original principal amount to get the total amount in her account.\r\n<p style=\"text-align: center\">[latex] \\displaystyle 2000+336=2336[\/latex]<\/p>\r\nShe has \u00a0[latex]$2336[\/latex] after \u00a0[latex]24[\/latex] months.[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another example of finding an account balance after a given amount of time, principal invested, and a rate.\r\n\r\nhttps:\/\/youtu.be\/XkGgEEMR_00\r\n\r\nIn the following example, you will see why it is important to make sure the units of the interest rate match the units of time when using the simple interest formula.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAlex invests\u00a0[latex]$600[\/latex] at\u00a0[latex]3.5\\%[\/latex] monthly interest for\u00a0[latex]3[\/latex] years. What amount of interest has Alex earned?\r\n\r\n[reveal-answer q=\"97640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"97640\"]\r\n\r\n<strong>Read and Understand:<\/strong> The question asks\u00a0for an amount, so we can substitute what we are given into the simple interest formula\u00a0[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]\r\n\r\n<strong>Define and Translate:<\/strong>\u00a0we know [latex]P[\/latex], [latex]r[\/latex], and [latex]t[\/latex] so we can use substitution. [latex]r[\/latex]\u00a0 =[latex]0.035[\/latex], [latex]P[\/latex] = \u00a0[latex]$600[\/latex], and [latex]t[\/latex] =[latex]3[\/latex] years. We have to be careful, because [latex]r[\/latex] is in months, and [latex]t[\/latex] is in years. \u00a0We need to change [latex]t[\/latex] into months, because we can't change the rate\u2014it is set by the bank.\r\n<p style=\"text-align: center\">[latex]{t}=3\\text{ years }\\cdot12\\frac{\\text{ months }}{\\text {year }}=36\\text{ months }[\/latex]<\/p>\r\n<strong>Write and Solve:<\/strong>\r\n\r\nSubstitute the given values into the formula.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l} I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\\\\\I=600\\,\\cdot \\,0.035\\,\\cdot \\,36\\\\\\\\{I}=756\\end{array}[\/latex]<\/p>\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nWe were asked what amount Alex earned, which is the amount provided by the formula. In the previous example, we were asked the total amount in the account, which included the principal and interest earned.\r\n\r\nAlex has earned [latex]$756[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of how to find the amount of interest earned after an investment has been sitting for a given monthly interest.\r\n\r\nhttps:\/\/youtu.be\/mRV5ljj32Rg\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAfter \u00a0[latex]10[\/latex] years, Jodi's account balance has earned \u00a0[latex]$1080[\/latex] in interest. The rate on the account is \u00a0[latex]0.09\\%[\/latex] monthly. What was the original amount she invested in the account?\r\n\r\n[reveal-answer q=\"97641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"97641\"]\r\n\r\n<strong>Read and Understand:<\/strong> The question asks\u00a0for the original amount invested, the principal. We are given a length of time in years, and an interest rate in months, and the amount of interest earned.\r\n\r\n<strong>Define and Translate:<\/strong>\u00a0we know [latex]I[\/latex] = \u00a0[latex]$1080[\/latex], [latex]r[\/latex] =[latex]0.009[\/latex], and [latex]t[\/latex] =[latex]10[\/latex] years, so we can use [latex]{P}=\\frac{I}{{r}\\,\\cdot \\,t}[\/latex]\r\n\r\nWe also need to make sure the units on the interest rate and the length of time match, and they do not. We need to change time into months again.\r\n<p style=\"text-align: center\">[latex]{t}=10\\text{ years }\\cdot12\\frac{\\text{ months }}{\\text{ year }}=120\\text{ months }[\/latex]<\/p>\r\n<strong>Write and Solve:<\/strong>\r\n\r\nSubstitute the given values into the formula\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}{P}=\\frac{I}{{R}\\,\\cdot \\,T}\\\\\\\\{P}=\\frac{1080}{{0.009}\\,\\cdot \\,120}\\\\\\\\{P}=\\frac{1080}{1.08}=1000\\end{array}[\/latex]<\/p>\r\n<strong>Check and Interpret:<\/strong>\r\n\r\nWe were asked to find the principal given the amount of interest earned on an\u00a0account. \u00a0If we substitute [latex]P[\/latex] =[latex]$1000[\/latex] into the formula\u00a0[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex] we get\r\n<p style=\"text-align: center\">[latex]I=1000\\,\\cdot \\,0.009\\,\\cdot \\,120\\\\I=1080[\/latex]<\/p>\r\nOur solution checks out. Jodi invested \u00a0[latex]$1000[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe last video shows another example of finding the principal amount invested based on simple interest.\r\nhttps:\/\/youtu.be\/vbMqN6lVoOM\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question]196954[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next section, we will apply our problem-solving method to problems involving dimensions of geometric shapes.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve a formula for a specific variable<\/li>\n<li>Use the distance, rate, and time formula<\/li>\n<li>Apply for the steps for solving word problems to interest rate problems<\/li>\n<\/ul>\n<\/div>\n<p>There\u00a0is often a well-known formula or relationship that applies to a word problem. For example, if you were to plan a road trip, you would\u00a0want to know how long it would take you to reach your destination.\u00a0[latex]d=rt[\/latex] is a well-known relationship that associates distance traveled, the rate at which you travel, and how long the travel takes.<\/p>\n<h2>Distance, Rate, and Time<\/h2>\n<p>If you know two of the quantities in the relationship [latex]d=rt[\/latex], you can easily find the third using methods for solving linear equations. For example, if you\u00a0know that you\u00a0will be traveling on a road with a speed limit of\u00a0[latex]30\\frac{\\text{ miles }}{\\text{ hour }}[\/latex] for [latex]2[\/latex] hours, you can\u00a0find the distance you would travel by\u00a0multiplying rate times time or [latex]\\left(30\\frac{\\text{ miles }}{\\text{ hour }}\\right)\\left(2\\text{ hours }\\right)=60\\text{ miles }[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm184997\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=184997&theme=oea&iframe_resize_id=ohm184997&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We can generalize this idea depending on what information we are given and what we are looking for. For example, if we need to find time, we could solve the\u00a0[latex]d=rt[\/latex] equation for [latex]t[\/latex] using division:<\/p>\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\frac{d}{r}=t[\/latex]<\/p>\n<p>Likewise, if we want to find rate, we can isolate [latex]r[\/latex] using division:<\/p>\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\frac{d}{t}=r[\/latex]<\/p>\n<p>In the following examples, you will see how this formula is applied to answer questions about ultra marathon running.<\/p>\n<div id=\"attachment_3540\" style=\"width: 276px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3540\" class=\"size-medium wp-image-3540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/05184602\/Screen-Shot-2016-05-05-at-11.45.10-AM-266x300.png\" alt=\"Ann Trason\" width=\"266\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-3540\" class=\"wp-caption-text\">Ann Trason<\/p>\n<\/div>\n<p>Ultra marathon running (defined as anything longer than\u00a0[latex]26.2[\/latex] miles) is becoming very popular among women even though it remains a male-dominated niche sport. Ann Trason has broken twenty world records in her career. One such record was the American River\u00a0[latex]50[\/latex]-mile Endurance Run, which\u00a0begins in Sacramento, California, and ends in Auburn, California.<a class=\"footnote\" title=\"&quot;Ann Trason.&quot; Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/Ann_Trason.\" id=\"return-footnote-16113-1\" href=\"#footnote-16113-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> In 1993, Trason finished the run with a time of [latex]6:09:08[\/latex]<strong>.<\/strong> \u00a0The men&#8217;s record for the same course was set in 1994\u00a0by Tom Johnson, who finished the course with a time of \u00a0[latex]5:33:21[\/latex].<a class=\"footnote\" title=\"\u00a0&quot;American River [latex]50[\/latex] Mile Endurance Run.&quot; Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run.\" id=\"return-footnote-16113-2\" href=\"#footnote-16113-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>In the next examples, we will use the [latex]d=rt[\/latex] formula to answer the following questions about the two runners.<\/p>\n<ol>\n<li>What was each runner&#8217;s rate for their record-setting runs?<\/li>\n<li>By the time Johnson had finished, how many more miles did Trason have to run?<\/li>\n<li>How much further could Johnson have run if he had run as long as Trason?<\/li>\n<li>What was each runner&#8217;s time for running one mile?<\/li>\n<\/ol>\n<p>To make answering the questions easier, we will round the two runners&#8217; times to\u00a0[latex]6[\/latex] hours and [latex]5.5[\/latex] hours.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What was each runner&#8217;s rate for their record-setting runs? Round to two decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q55589\">Show Solution<\/span><\/p>\n<div id=\"q55589\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:<\/strong>\u00a0We\u00a0are looking for rate and we know distance and time, so we can use the idea:\u00a0[latex]d=rt[\/latex], [latex]\\frac{d}{t}=r[\/latex]. Let&#8217;s solve one runner&#8217;s rate using the original formula, and one using the rearranged formula.<\/p>\n<p><strong>Define and Translate:<\/strong> Because there are two runners, making a table to organize this information helps. Note how we keep units to help us keep track of how all the terms are related to each other.<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]r[\/latex]<\/td>\n<td>[latex]6[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]r[\/latex]<\/td>\n<td>\u00a0[latex]5.5[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p>Trason&#8217;s rate:<\/p>\n<p style=\"text-align: center\">[latex]d=rt[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]50\\text{ miles }=r\\left(6\\text{ hours}\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]50=6r[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]r\\approx 8.33[\/latex]<\/p>\n<p>Johnson&#8217;s rate:<\/p>\n<p style=\"text-align: center\">[latex]r=\\frac{d}{t}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]r=\\frac{50\\text{ miles}}{5.5\\text{ hours}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]r=\\frac{50}{5.5}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]r\\approx 9.10[\/latex]<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>We can fill in our table with this information.<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]6[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]9.1[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]5.5[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now that we know each runner&#8217;s rate, we can answer the second question.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>By the time Johnson had finished, how many more miles did Trason have to run?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q747303\">Show Solution<\/span><\/p>\n<div id=\"q747303\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here is the table we created for reference:<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]6[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>\u00a0[latex]5.5[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Read and Understand:<\/strong>\u00a0We are looking for how many miles Trason still had on the trail when Johnson had finished after [latex]5.5[\/latex] hours. This is a distance, and we know rate and time.<\/p>\n<p><strong>Define and Translate:<\/strong>\u00a0We can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]d[\/latex], and the time Trason had run is [latex]5.5[\/latex] hours.<\/p>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}d=rt\\\\\\\\d=8.33\\frac{\\text{ miles }}{\\text{ hour }}\\left(5.5\\text{ hours}\\right)\\\\\\\\d=45.82\\text{ miles}\\end{array}[\/latex]<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>Have we answered the question? We were asked to find how many more miles she had to run after [latex]5.5[\/latex] hours. \u00a0What we have found is how long she had run after [latex]5.5[\/latex] hours. We need to subtract [latex]d=45.82\\text{ miles }[\/latex] from the total distance of the course.<\/p>\n<p style=\"text-align: center\">[latex]50\\text{ miles }-45.82\\text{ miles }=4.18\\text{ miles }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The third question is similar to the second. Now that we know each runner&#8217;s rate, we can answer questions about individual distances or times.<\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>How much further could Johnson have run if he had run for the same amount of time as Trason?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q757303\">Show Solution<\/span><\/p>\n<div id=\"q757303\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here is the table we created for reference:<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]6[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]5.5[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Read and Understand:<\/strong>\u00a0The word further implies we are looking for a distance.<\/p>\n<p><strong>Define and Translate:<\/strong>\u00a0We can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]d[\/latex], the\u00a0time is [latex]6[\/latex] hours, and Johnson&#8217;s rate is [latex]9.1\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/p>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}d=rt\\\\\\\\d=9.1\\frac{\\text{ miles }}{\\text{ hour }}\\left(6\\text{ hours }\\right)\\\\\\\\d=54.6\\text{ miles }\\end{array}[\/latex].<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>Have we answered the question? We were asked to find how many more miles\u00a0Johnson would have run if he had run at his rate of\u00a0[latex]9.1\\frac{\\text{ miles }}{\\text{ hour }}[\/latex] for\u00a0[latex]6[\/latex] hours.<\/p>\n<p>Johnson would have run [latex]54.6[\/latex] miles, so that&#8217;s [latex]4.6[\/latex] more miles than he ran during the race.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will tackle the last question, where we are asked to find a time for each runner.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>What was each runner&#8217;s time for running one mile?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q757309\">Show Solution<\/span><\/p>\n<div id=\"q757309\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here is the table we created for reference:<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]6[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]50[\/latex] miles<\/td>\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]5.5[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Read and Understand:<\/strong>\u00a0we are looking for time, and this time our distance has changed from \u00a0[latex]50[\/latex] miles to [latex]1[\/latex] mile, so we can use<\/p>\n<p style=\"text-align: center\">[latex]d=rt\\\\\\frac{d}{r}=t[\/latex]<\/p>\n<p><strong>Define and Translate:<\/strong> we can use the formula\u00a0[latex]d=rt[\/latex] again. This time the unknown is [latex]t[\/latex], the distance is [latex]1[\/latex] mile, and we know each runner&#8217;s rate. It may help to create a new table:<\/p>\n<table>\n<thead>\n<tr>\n<th>Runner<\/th>\n<th>Distance =<\/th>\n<th>(Rate )<\/th>\n<th>(Time)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Trason<\/td>\n<td>[latex]1[\/latex] mile<\/td>\n<td>[latex]8.33[\/latex] [latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]t[\/latex] hours<\/td>\n<\/tr>\n<tr>\n<td>Johnson<\/td>\n<td>[latex]1[\/latex] mile<\/td>\n<td>[latex]9.1[\/latex]\u00a0[latex]\\frac{\\text{ miles }}{\\text{ hour }}[\/latex]<\/td>\n<td>[latex]t[\/latex] hours<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p>Trason:<\/p>\n<p>We will need to divide to isolate time.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}d=rt\\\\\\\\1\\text{ mile }=8.33\\frac{\\text{ miles }}{\\text{ hour }}\\left(t\\text{ hours }\\right)\\\\\\\\\\frac{1\\text{ mile }}{\\frac{8.33\\text{ miles }}{\\text{ hour }}}=t\\text{ hours }\\\\\\\\0.12\\text{ hours }=t\\end{array}[\/latex].<\/p>\n<p>[latex]0.12[\/latex] \u00a0hours is about \u00a0[latex]7.2[\/latex] minutes, so Trason&#8217;s time for running one mile was about [latex]7.2[\/latex] minutes. WOW! She did that for \u00a0[latex]6[\/latex] hours!<\/p>\n<p>Johnson:<\/p>\n<p>We will need to divide to isolate time.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}d=rt\\\\\\\\1\\text{ mile }=9.1\\frac{\\text{ miles }}{\\text{ hour }}\\left(t\\text{ hours }\\right)\\\\\\\\\\frac{1\\text{ mile }}{\\frac{9.1\\text{ miles }}{\\text{ hour }}}=t\\text{ hours }\\\\\\\\0.11\\text{ hours }=t\\end{array}[\/latex].<\/p>\n<p>[latex]0.11[\/latex] hours is about [latex]6.6[\/latex] minutes, so Johnson&#8217;s time for running one mile was about \u00a0[latex]6.6[\/latex] minutes. WOW! He did that for [latex]5.5[\/latex] hours!<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>Have we answered the question? We were asked to find how long it took each runner to run one mile given the rate at which they ran the whole \u00a0[latex]50[\/latex]-mile course. \u00a0Yes, we answered our question.<\/p>\n<p>Trason&#8217;s mile time was [latex]7.2\\frac{\\text{minutes}}{\\text{mile}}[\/latex] and Johnsons&#8217; mile time was\u00a0[latex]6.6\\frac{\\text{minutes}}{\\text{mile}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of answering many rate questions given distance and time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Problem Solving Using Distance, Rate, Time (Running)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3WLp5mY1FhU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simple Interest<\/h2>\n<p>In order to entice customers to invest their money, many banks will offer interest-bearing accounts. The accounts work like this: a customer deposits a certain amount of money (called the Principal, or [latex]P[\/latex]), which then grows slowly according to the interest rate ([latex]r[\/latex], measured in percent) and the length of time ([latex]t[\/latex], usually measured in months) that the money stays in the account. The amount earned over time is called the interest ([latex]I[\/latex]), which is then given to the customer.<\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"76\" height=\"67\" \/>Beware! Interest rates are commonly given as yearly rates, but can also be monthly, quarterly, bimonthly, or even some custom amount of time. It is important that the units\u00a0of time and the units of the interest rate match. You will see why this matters in a later example.<\/div>\n<p>The simplest way to calculate interest earned on an account is through the formula [latex]\\displaystyle I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]<\/p>\n<p>If we know any of the three amounts related to this equation, we can find the fourth. For example, if we want to find the time it will take to accrue a specific amount of interest, we can solve for [latex]t[\/latex] using division:<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\\\\\ \\frac{I}{{P}\\,\\cdot \\,r}=\\frac{P\\cdot\\,r\\,\\cdot \\,t}{\\,P\\,\\cdot \\,r}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{t}=\\frac{I}{\\,r\\,\\cdot \\,t}\\end{array}[\/latex]<\/p>\n<p>Below is a table showing the result of solving for each individual variable in the formula.<\/p>\n<table>\n<thead>\n<tr>\n<th>Solve For<\/th>\n<th>Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]I[\/latex]<\/td>\n<td>[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]P[\/latex]<\/td>\n<td>[latex]{P}=\\frac{I}{{r}\\,\\cdot \\,t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r[\/latex]<\/td>\n<td>[latex]{r}=\\frac{I}{{P}\\,\\cdot \\,t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]t[\/latex]<\/td>\n<td>\u00a0[latex]{t}=\\frac{I}{{P}\\,\\cdot \\,r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the next examples, we will show how to substitute given values into the simple interest formula, and decipher which variable to solve for.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>If a customer deposits a principal of\u00a0[latex]$2000[\/latex] at a monthly rate of \u00a0[latex]0.7\\%[\/latex], what is the total amount that she has after [latex]24[\/latex] months?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q57640\">Show Solution<\/span><\/p>\n<div id=\"q57640\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute in the values given for the Principal, Rate, and Time.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{l}I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\I=2000\\cdot 0.7\\%\\cdot 24\\end{array}[\/latex]<\/p>\n<p>Rewrite \u00a0[latex]0.7\\%[\/latex] as the decimal \u00a0[latex]0.007[\/latex], then multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}I=2000\\cdot 0.007\\cdot 24\\\\I=336\\end{array}[\/latex]<\/p>\n<p>Add the interest and the original principal amount to get the total amount in her account.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 2000+336=2336[\/latex]<\/p>\n<p>She has \u00a0[latex]$2336[\/latex] after \u00a0[latex]24[\/latex] months.<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video shows another example of finding an account balance after a given amount of time, principal invested, and a rate.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simple Interest - Determine Account Balance (Monthly Interest)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XkGgEEMR_00?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following example, you will see why it is important to make sure the units of the interest rate match the units of time when using the simple interest formula.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Alex invests\u00a0[latex]$600[\/latex] at\u00a0[latex]3.5\\%[\/latex] monthly interest for\u00a0[latex]3[\/latex] years. What amount of interest has Alex earned?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q97640\">Show Solution<\/span><\/p>\n<div id=\"q97640\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:<\/strong> The question asks\u00a0for an amount, so we can substitute what we are given into the simple interest formula\u00a0[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex]<\/p>\n<p><strong>Define and Translate:<\/strong>\u00a0we know [latex]P[\/latex], [latex]r[\/latex], and [latex]t[\/latex] so we can use substitution. [latex]r[\/latex]\u00a0 =[latex]0.035[\/latex], [latex]P[\/latex] = \u00a0[latex]$600[\/latex], and [latex]t[\/latex] =[latex]3[\/latex] years. We have to be careful, because [latex]r[\/latex] is in months, and [latex]t[\/latex] is in years. \u00a0We need to change [latex]t[\/latex] into months, because we can&#8217;t change the rate\u2014it is set by the bank.<\/p>\n<p style=\"text-align: center\">[latex]{t}=3\\text{ years }\\cdot12\\frac{\\text{ months }}{\\text {year }}=36\\text{ months }[\/latex]<\/p>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p>Substitute the given values into the formula.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l} I=P\\,\\cdot \\,r\\,\\cdot \\,t\\\\\\\\I=600\\,\\cdot \\,0.035\\,\\cdot \\,36\\\\\\\\{I}=756\\end{array}[\/latex]<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>We were asked what amount Alex earned, which is the amount provided by the formula. In the previous example, we were asked the total amount in the account, which included the principal and interest earned.<\/p>\n<p>Alex has earned [latex]$756[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of how to find the amount of interest earned after an investment has been sitting for a given monthly interest.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simple Interest - Determine Interest Balance (Monthly Interest)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/mRV5ljj32Rg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>After \u00a0[latex]10[\/latex] years, Jodi&#8217;s account balance has earned \u00a0[latex]$1080[\/latex] in interest. The rate on the account is \u00a0[latex]0.09\\%[\/latex] monthly. What was the original amount she invested in the account?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q97641\">Show Solution<\/span><\/p>\n<div id=\"q97641\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Read and Understand:<\/strong> The question asks\u00a0for the original amount invested, the principal. We are given a length of time in years, and an interest rate in months, and the amount of interest earned.<\/p>\n<p><strong>Define and Translate:<\/strong>\u00a0we know [latex]I[\/latex] = \u00a0[latex]$1080[\/latex], [latex]r[\/latex] =[latex]0.009[\/latex], and [latex]t[\/latex] =[latex]10[\/latex] years, so we can use [latex]{P}=\\frac{I}{{r}\\,\\cdot \\,t}[\/latex]<\/p>\n<p>We also need to make sure the units on the interest rate and the length of time match, and they do not. We need to change time into months again.<\/p>\n<p style=\"text-align: center\">[latex]{t}=10\\text{ years }\\cdot12\\frac{\\text{ months }}{\\text{ year }}=120\\text{ months }[\/latex]<\/p>\n<p><strong>Write and Solve:<\/strong><\/p>\n<p>Substitute the given values into the formula<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}{P}=\\frac{I}{{R}\\,\\cdot \\,T}\\\\\\\\{P}=\\frac{1080}{{0.009}\\,\\cdot \\,120}\\\\\\\\{P}=\\frac{1080}{1.08}=1000\\end{array}[\/latex]<\/p>\n<p><strong>Check and Interpret:<\/strong><\/p>\n<p>We were asked to find the principal given the amount of interest earned on an\u00a0account. \u00a0If we substitute [latex]P[\/latex] =[latex]$1000[\/latex] into the formula\u00a0[latex]I=P\\,\\cdot \\,r\\,\\cdot \\,t[\/latex] we get<\/p>\n<p style=\"text-align: center\">[latex]I=1000\\,\\cdot \\,0.009\\,\\cdot \\,120\\\\I=1080[\/latex]<\/p>\n<p>Our solution checks out. Jodi invested \u00a0[latex]$1000[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The last video shows another example of finding the principal amount invested based on simple interest.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Simple Interest - Determine Principal Balance (Monthly Interest)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vbMqN6lVoOM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm196954\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=196954&theme=oea&iframe_resize_id=ohm196954&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next section, we will apply our problem-solving method to problems involving dimensions of geometric shapes.<\/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-16113\">\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>Image: Ann Trason Trail Running. <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><li>Rates. <strong>Provided 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><li>Problem Solving Using Distance, Rate, Time (Running). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3WLp5mY1FhU\">https:\/\/youtu.be\/3WLp5mY1FhU<\/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>Simple Interest - Determine Account Balance (Monthly Interest). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/XkGgEEMR_00\">https:\/\/youtu.be\/XkGgEEMR_00<\/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>Simple Interest - Determine Interest Balance (Monthly Interest). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mRV5ljj32Rg\">https:\/\/youtu.be\/mRV5ljj32Rg<\/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>Simple Interest - Determine Principal Balance (Monthly Interest). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vbMqN6lVoOM\">https:\/\/youtu.be\/vbMqN6lVoOM<\/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>Ann Trason. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/en.wikipedia.org\/wiki\/Ann_Trason\">https:\/\/en.wikipedia.org\/wiki\/Ann_Trason<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>American River 50 Mile Endurance Run. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run\">https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-16113-1\">\"Ann Trason.\" Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/Ann_Trason. <a href=\"#return-footnote-16113-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-16113-2\">\u00a0\"American River [latex]50[\/latex] Mile Endurance Run.\" Wikipedia. Accessed May 05, 2016. https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run. <a href=\"#return-footnote-16113-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":169554,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Image: Ann Trason Trail Running\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ann Trason\",\"author\":\"\",\"organization\":\"Wikipedia\",\"url\":\"https:\/\/en.wikipedia.org\/wiki\/Ann_Trason\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"American River 50 Mile Endurance Run\",\"author\":\"\",\"organization\":\"Wikipedia\",\"url\":\"https:\/\/en.wikipedia.org\/wiki\/American_River_50_Mile_Endurance_Run\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Rates\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Problem Solving Using Distance, Rate, Time (Running)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/3WLp5mY1FhU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simple Interest - 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