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<h2>Using the Distance, Rate, and Time Formula</h2>
<p>One formula you’ll 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 travel if you drove at a steady rate of \(60\) miles per hour for \(2\) hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said \(120\) miles, you already know how to use this formula!</p>
<p>The math to calculate the distance might look like this:</p>
<h4>\(\begin{array}{}\text{distance}=\left(\frac{60\phantom{\rule{0.2em}{0ex}}\text{miles}}{1\phantom{\rule{0.2em}{0ex}}\text{hour}}\right)\left(2\phantom{\rule{0.2em}{0ex}}\text{hours}\right)\hfill \\ \text{distance}=120\phantom{\rule{0.2em}{0ex}}\text{miles}\hfill \end{array}\)</h4>
<p>In general, the formula relating distance, rate, and time is</p>
<div data-label="">\(\text{distance}\phantom{\rule{0.2em}{0ex}}\text{=}\phantom{\rule{0.2em}{0ex}}\text{rate}·\text{time}\)</div>
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<h3>Definition: Distance, Rate and Time</h3>
<p>For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula</p>
<div data-label="">\(d=rt\)</div>
<p>where \(d=\) distance, \(r=\) rate, and \(t=\) time.</p>
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<p>Notice that the units we used above for the rate were miles per hour, which we can write as a ratio \(\frac{miles}{hour}.\) Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.</p>
<h3>Optional Video: Solving a Problem using Distance = Rate x Time</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="386" src="https://www.youtube.com/embed/lMO1L_CvH4Y?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="686"></iframe></p>
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<h2>Example</h2>
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<div data-type="example">
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<div>
<p>Jamal rides his bike at a uniform rate of \(12\) miles per hour for \(3\frac{1}{2}\) hours. How much distance has he traveled?</p>
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<h3>Solution</h3>
<table data-label="" style="width: 772px;" summary="The top line says, “Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem.” The word “read” is in bold. Beside this is d equals question mark, then r equals 12 mph, and t equals 3 and a half hours. The next line says, “Step 2. Identify what you are looking for.” The word “identify” is in bold. Beside this is distance traveled. The next line says, “Step 3. Name: Choose a variable to represent it.” The word “name” is in bold. Beside this is let d equal distance. The next line says, “Step 4. Translate: Write the appropriate formula for the situation. Substitute in the given information.” The word “translate” is in bold. Beside this is d equals rt, then d equals 12 mph times 3 and a half hours. The next line says, “Step 5. Solve the equation.” The word “solve” is in bold. Beside this is d equals 42 miles. The next line says, “Step 6. Check.” The word “check” is in bold. Beside this is, “Does 42 miles make sense? Jamal rides 12 miles in 1 hour, 24 miles in 2 hours, 36 miles in 3 hours, 48 miles in 4 hours. 42 miles in 3 and a half hours is reasonable. The last line says, “Step 7. Answer the question with a complete sentence.” The word “answer” is in bold. The answer is given as “Jamal rode 42 miles.">
<tbody>
<tr>
<td style="width: 399px;">Step 1. <strong>Read</strong> the problem.

<div> </div>

You may want to create a mini-chart to summarize the

<div> </div>

information in the problem.</td>
<td style="width: 359px;">\(d=?\)

<div> </div>

\(r=12\phantom{\rule{0.2em}{0ex}}\text{mph}\)

<div> </div>

\(t=3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{hours}\)</td>
</tr>
<tr>
<td style="width: 399px;">Step 2. <strong>Identify</strong> what you are looking for.</td>
<td style="width: 359px;">distance traveled</td>
</tr>
<tr>
<td style="width: 399px;">Step 3. <strong>Name.</strong> Choose a variable to represent it.</td>
<td style="width: 359px;">let <em>d</em> = distance</td>
</tr>
<tr>
<td style="width: 399px;">Step 4. <strong>Translate.</strong>
<div> </div>

Write the appropriate formula for the situation.

<div> </div>

Substitute in the given information.</td>
<td style="width: 359px;">\(d=rt\)

<div> </div>
<div> </div>

\(d=12\cdot 3\frac{1}{2}\)</td>
</tr>
<tr>
<td style="width: 399px;">Step 5. <strong>Solve</strong> the equation.</td>
<td style="width: 359px;">\(d=42\phantom{\rule{0.2em}{0ex}}\text{miles}\)</td>
</tr>
<tr>
<td style="width: 399px;">Step 6. <strong>Check:</strong> Does 42 miles make sense?

<div> </div>
<img alt="." src="https://data.ulearngo.com/assets/e1a8051b-4fe3-4dd7-b9c6-86d3d3511d44"/></td>
<td style="width: 359px;"> </td>
</tr>
<tr>
<td style="width: 399px;">Step 7. <strong>Answer</strong> the question with a complete sentence.</td>
<td style="width: 359px;">Jamal rode 42 miles.</td>
</tr>
</tbody>
</table>
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<h2>Example</h2>
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<div data-type="example">
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<p>Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of \(520\) miles. If he can drive at a steady rate of \(65\) miles per hour, how many hours will the trip take?</p>
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<div>
<h3>Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>Step 1. <strong>Read</strong> the problem.

<div> </div>

Summarize the information in the problem.</td>
<td>\(d=520\phantom{\rule{0.2em}{0ex}}\text{miles}\)

<div> </div>

\(r=65\phantom{\rule{0.2em}{0ex}}\text{mph}\)

<div> </div>

\(t=?\)</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>
<div> </div>

Choose a variable to represent it.</td>
<td>let <em>t</em> = time</td>
</tr>
<tr>
<td>Step 4. <strong>Translate.</strong>
<div> </div>

Write the appropriate formula.

<div> </div>

Substitute in the given information.</td>
<td>\(d=rt\)

<div> </div>

\(520=65t\)</td>
</tr>
<tr>
<td>Step 5. <strong>Solve</strong> the equation.</td>
<td>\(t=8\)</td>
</tr>
<tr>
<td>Step 6. <strong>Check:</strong>
<div> </div>

Substitute the numbers into the formula and make sure

<div> </div>

the result is a true statement.

<div> </div>

\(d=rt\)

<div> </div>

\(520\stackrel{?}{=}65\cdot 8\)

<div> </div>

\(520=520&gt;✓\)</td>
<td> </td>
</tr>
<tr>
<td>Step 7. <strong>Answer</strong> the question with a complete sentence.

<div> </div>

We know the units of time will be hours because

<div> </div>

we divided miles by miles per hour.</td>
<td>Rey's trip will take 8 hours.</td>
</tr>
</tbody>
</table>
</div>
</div>
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</div>
</div>
<h3>Optional Video: Solving a Problem using Distance = Rate x Time</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="386" src="https://www.youtube.com/embed/ArR14Mesu1M?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="686"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

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Using the Distance, Rate, and Time Formula

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Using the Distance, Rate, and Time Formula

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