Distance-Rate-Time and other Problems

Distance-Rate-Time and other Products

In this lesson, we will investigate the relationship between the distance traveled, the rate or speed or travel, and the time that it takes to travel that distance at that rate. We will also look at a few other related products.

Distance = (Rate)(Time)

The equation that relates distance, rate, and time is

d = rt

Where d is the distance traveled, r is the rate, and t is the time. On the CAHSEE exam, you will be given two of these and will be asked to use the above equation to find the third.

Example 1

It took Markus half an hour to drive home from work. He averaged 34 miles per hour. How far does Markus live from his work?

Solution

We are given that it takes 1/2 an hour for the trip. This is a time:

t = 1/2

We are given that he averages 34 miles per hour. This is a rate:

r = 34

We are asked how few he has traveled. This is a distance. We use the d=rt equation:

d = rt

= (34)(1/2)

= 17

Markus lives 17 miles from work.

Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 1

The current along the beach is moving towards the south at 1.5 miles per hour. If a piece of debris is placed into the water, how far will the current take it in 6 hours?

Example 2

Elena always rides her bicycle at a speed of 15 miles per hour. On Sunday, she goes on a 24 mile bike ride. How many hours does this ride take?

Solution

The speed of 15 miles per hour is a rate. The key words that tell us that this is a rate are "speed" and "miles per hour". We can write:

r = 15

Next, 24 miles is a distance. We have:

d = 24

Now use the d=rt equation to get

24 = 15t

To solve this, divide both sides by 15 to get

t = 24/15

Both are divisible by 3, so this fraction reduces to

t = 8/5 = 1.6

Elena's ride takes 1.6 hours.

Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 2

Roberto will be driving from Los Angeles to Las Vegas tomorrow. He can average 60 miles per hour for the trip. It is 282 miles from Los Angeles to Las Vegas. How long will it take Roberto to drive from Los Angeles to Las Vegas?

Example 3

Juan has just completed a 5 kilometer race in 20 minutes. What was his average speed in kilometers per hour?

Solution

First, 5 kilometers is a distance, so

d = 5

Next, we are given 20 minutes and we are asked to present the speed in kilometers per hour rather than kilometers per minute. We need to convert 20 minutes to hours. Since there are 60 minutes per hour, we divide by 60 to find out how many hours it took

t = 20 minutes / 60 minutes per hour

= 1/3 hours

Now we can use the d=rt equation:

5 = (r)(1/3)

If we multiply both sides by 3, we get

(3)(5) = (r)(1/3)(3)

r = 15

Juan's average speed was 15 kilometers per hour.

Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 3

It takes 2 hours for a turtle to move a distance of 60 meters. How fast in kilometers per hour is the turtle moving?

Problems Involving Work

Related to the d=rt equation are problems that involve work. We will look at a few such examples.

Example 4

If one person paints the outside of a house, then it will take that person 56 hours to complete the job. If a team of 4 people each work 7 hours per day, how many days will it take the team to paint the outside of the house?

Solution

The strategy that we will take is to first find out how many person-hours each day the house is being painted. Since there are 4 people on the job and each works 7 hours in a day, there are

(4)(7) = 28

person-hours each day. We can now divide the number of hours to complete the job by the number of person-hours to find the total number of days:

Days = (56 hours) / (28 person-hours per day)

= 2 days

It will take 2 days for the team to finish painting the house.

Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 4

It would take one person 72 hours to harvest the orange grove alone. If a three person team is hired for 8 hours per day per person, how many days will it take the team to harvest the orange grove?

Now for one more related example.

Example 5

A DVD burner can burn 12 DVDs per hour. A factory that has 150 DVD burners is in operation for 10 hours each day. How many DVDs will the factor burn per day?

Solution

Since one burner burns 12 DVDs per hour and there are 150 DVDs, we can multiply to find how many DVDs are burned in the factory:

Total DVDs per hour = (12)(150)

= 1800 DVDs per hour

Since the factory is in operation for 10 hours each day, we just multiply by 10 to get the total number of DVDs burned per day:

DVDs per day = (1800 DVDs per hour)(10 hours per day)

= 18,000 DVDs per day

Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 5

Maria can complete 32 arithmetic problems per minute. If she works a arithmetic problems for 5 minutes without stopping, how many arithmetic problems will she be able to complete?