# Equations and Problem Solving

## Distance-Rate-Time Problems

An object that moves at a constant rate is said to be in uniform motion. The formula *d* = *rt* gives the relationship between distance *d*, rate *r*, and time *t*. Uniform motion problems may involve objects going the same direction, opposite directions, or round trips.

In the diagram below, the two vehicles are traveling the same direction at different rates. The distances the vehicles travel are the same.

Since the distances are equal, the products of rate and time for the two cars are equal. For the vehicles shown, 40 5 = 50 4.

A table can also help you understand relationships in distance-rate-time problems.

#### Same-Direction Travel

EngineeringA train leaves a train station at 1 p.m. It travels at an average rate of 60 mi/h. A high-speed train leaves the same station an hour later. It travels at an average rate of 96 mi/h. The second train follows the same route as the first train on a track parallel to the first. In how many hours will the second train catch up with the first train?

For uniform motion problems that involve a round trip, it is important to remember that the distance going is equal to the distance returning.

Since the distances are equal, the products of rate and time for traveling in both directions are equal. That is, 20 3 = 30 2.

##### Reading Math

The total travel time is for a round trip. If it takes *x* out of a 2-hour round trip to get to the store, then 2 – *x* is the time it will take for the drive home.

#### Round-Trip Travel

Noya drives into the city to buy a software program at a computer store. Because of traffic conditions, she averages only 15 mi/h. On her drive home she averages 35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the computer store?

For uniform motion problems involving two objects moving in opposite directions, you can write equations using the fact that the sum of their distances is the total distance.

#### Opposite-Direction Travel

Jane and Peter leave their home traveling in opposite directions on a straight road. Peter drives 15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find Peter's rate and Jane's rate.

Exercises Click here to view the Exercises for this lesson.