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Distance, Rate, and Time

Distance, Rate, and Time

Author: Sophia Tutorial
Description:

Solve for either distance, rate, or time in a given situation.

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Tutorial
what's covered
  1. Relationship between Distance Rate and Time
  2. Solving an Equation for Distance
  3. Solving an Equation for Rate
  4. Solving an Equation for Time

1. Relationship between Distance, Rate, and Time

An application of linear equations can be found in distance problems. When solving distance problems we will use the relationship d = rt or distance equals rate (speed) times time. There are a couple of ways we can represent this relationship:

d equals r times t

r equals d over t

t equals d over r


2. Solving an Equation for Distance

The first example on how to use the distance-rate-time relationship is solving for distance. Jason ran for 45 minutes, or 0.75 hours, at a speed of 7 miles per hour. How far did he run?

We want to know how far he ran so that going to be the distance. We can see that the rate is going to be 7 miles per hour and the time is going to be 0.75 hours. We can use the following equation:

d equals r times t

Using this formula, we are going to multiply the rate, which is 7 miles per hour, or 7 miles in 1 hour, by the time, which is 0.75 hours, or 0.75 hours over 1.

d equals fraction numerator 7 space m i l e s over denominator 1 space h o u r end fraction times fraction numerator 0.75 space h o u r s over denominator 1 end fraction

Remember from doing unit conversions, when we want to figure out our units, the goal is to get units to cancel. We can see that hours in the denominator and hours in the numerator are going to cancel out. So the answer is going to be in terms of miles, which is good since we are trying to find distance.

d equals fraction numerator 7 space m i l e s over denominator 1 space up diagonal strike h o u r end strike end fraction times fraction numerator 0.75 space up diagonal strike h o u r s end strike over denominator 1 end fraction

We just need to multiply and simplify the fractions. Multiplying 7 times 0.75 equals 5.25 and 1 times 1 is just 1.

d equals fraction numerator left parenthesis 7 times 0.75 right parenthesis space m i l e s over denominator 1 end fraction equals fraction numerator 5.25 space m i l e s over denominator 1 end fraction equals 5.25 space m i l e s

Running for 45 minutes, or 0.75 hours, at a speed of 7 miles per hour, Jason ran a total distance of 5.25 miles.


3. Solving an Equation for Rate

In this second example, Lee biked to his friend's house in 2 hours. He knows that his friend's house is 42 miles from where he started and he wants to know how fast did he bike.

Finding how fast Lee bikes is going to be a rate. We know that the distance is 42 miles and the time is 2 hours. So we can use this formula for rate being equal to distance over time.

r equals d over t

We can start by substituting the value for distance, 42 miles, and the value for time, 2 hours.

r equals fraction numerator 42 space m i l e s over denominator 2 space h o u r s end fraction

We can simplify these two numbers. 42 divided by 2 is going to equal 21. None of the units cancel out, so the units of the answer are going to be in terms of miles per hour.

r equals fraction numerator 42 space m i l e s over denominator 2 space h o u r s end fraction equals 21 fraction numerator space m i l e s over denominator h o u r s end fraction

Biking 42 miles in 2 hours, Lee's rate is 21 miles per hour.


4. Solving an Equation for Time

In this third example. Shira wants to rollerblade to the corner store. She knows she can rollerblade at 12 miles per hour and that the store is 4 miles away. How long will it take for her to get there?

We want to find how long it will take, which means we want to find the time. We know the distance and the rate so we can use this formula for time, which again time is equal to the distance over the rate divided by the rate.

t equals d over r

The distance is going to be 4 miles and the rate is going to be 12 miles per hour, or 12 miles in 1 hour.

t equals fraction numerator 4 space m i l e s over denominator 12 space begin display style fraction numerator m i l e s over denominator h o u r end fraction end style end fraction

Dividing by 12 is the same as multiplying by 1/12, so we can rewrite this as 4 miles times 1 hour/12 miles, which still describes the speed. We can put 4 miles over 1 and see how the units cancel.

t equals fraction numerator 4 space up diagonal strike m i l e s end strike over denominator 1 end fraction times fraction numerator 1 space h o u r over denominator 12 space up diagonal strike m i l e s end strike end fraction equals fraction numerator 4 space h o u r s over denominator 12 end fraction equals 1 third h o u r s

So the miles in the numerator of the first fraction will cancel with the miles in the denominator of the second fraction. Now we are left with hours as the unit, which is good because we are trying to find time. 4 times 1 is 4. 1 times 12 is 12, and this reduces to 1/3. It will take Shira 1/3 hours. If we want this fraction in minutes, we can just multiply it by the conversion factor 1 hour = 60 minutes.

fraction numerator 1 space up diagonal strike h o u r end strike over denominator space space 3 space space space space space space space space space end fraction times fraction numerator 60 space m i n u t e s over denominator 1 space up diagonal strike h o u r end strike end fraction equals fraction numerator 60 space m i n u t e s over denominator 3 end fraction equals 20 space m i n u t e s

It will take Shira 20 minutes to rollerblade 4 miles at a rate of 12 miles per hour.

summary
In the relationship between distance, rate, and time, the variables D equals distance, R equals rate, and T equals time. The formula that relates these variables is distance is equal to the rate times the time. We can also say that rate is equal to the distance divided by the time, A third equation is time is equal to the distance divided by the rate. We can use any of these formulas when solving for a distance, rate, or time.

Formulas to Know
Distance, Rate, Time

d i s tan c e equals r a t e times t i m e