9.7 Solve a formula for a specific variable

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By the end of this section, you will be able to:
• Use the distance, rate, and time formula
• Solve a formula for a specific variable

Before you get started, take this readiness quiz.

1. Write $35$ miles per gallon as a unit rate.
If you missed this problem, review Ratios and Rates .
2. Solve $6x+24=96.$
If you missed this problem, review Solve Equations with Variables and Constants on Both Sides .
3. Find the simple interest earned after $5$ years on $\text{1,000}$ at an interest rate of $\text{4%}.$
If you missed this problem, review Solve Simple Interest Applications .

Use the distance, rate, and time formula

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!

The math to calculate the distance might look like this:

$\begin{array}{}\\ \text{distance}=\left(\frac{60\phantom{\rule{0.2em}{0ex}}\text{miles}}{1\phantom{\rule{0.2em}{0ex}}\text{hour}}\right)\left(2\phantom{\rule{0.2em}{0ex}}\text{hours}\right)\hfill \\ \text{distance}=120\phantom{\rule{0.2em}{0ex}}\text{miles}\hfill \end{array}$

In general, the formula relating distance, rate, and time is

$\text{distance}\phantom{\rule{0.2em}{0ex}}\text{=}\phantom{\rule{0.2em}{0ex}}\text{rate}·\text{time}$

Distance, rate and time

For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula

$d=rt$

where $d=$ distance, $r=$ rate, and $t=$ time.

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.

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?

Solution

 Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem. $d=?$ $r=12\phantom{\rule{0.2em}{0ex}}\text{mph}$ $t=3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{hours}$ Step 2. Identify what you are looking for. distance traveled Step 3. Name. Choose a variable to represent it. let d = distance Step 4. Translate. Write the appropriate formula for the situation. Substitute in the given information. $d=rt$ $d=12\cdot 3\frac{1}{2}$ Step 5. Solve the equation. $d=42\phantom{\rule{0.2em}{0ex}}\text{miles}$ Step 6. Check: Does 42 miles make sense? Step 7. Answer the question with a complete sentence. Jamal rode 42 miles.

Lindsay drove for $5\frac{1}{2}$ hours at $60$ miles per hour. How much distance did she travel?

330 mi

Trinh walked for $2\frac{1}{3}$ hours at $3$ miles per hour. How far did she walk?

7 mi

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?

Solution

 Step 1. Read the problem. Summarize the information in the problem. $d=520\phantom{\rule{0.2em}{0ex}}\text{miles}$ $r=65\phantom{\rule{0.2em}{0ex}}\text{mph}$ $t=?$ Step 2. Identify what you are looking for. how many hours (time) Step 3. Name: Choose a variable to represent it. let t = time Step 4. Translate. Write the appropriate formula. Substitute in the given information. $d=rt$ $520=65t$ Step 5. Solve the equation. $t=8$ Step 6. Check: Substitute the numbers into the formula and make sure the result is a true statement. $d=rt$ $520\stackrel{?}{=}65\cdot 8$ $520=520>✓$ Step 7. Answer the question with a complete sentence. We know the units of time will be hours because we divided miles by miles per hour. Ray's trip will take 8 hours.

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