what is the car's average rate is mph and the ​ bicycle's average rate is mph

Question

Suppose that a cyclist began a 243 mi ride across a state at the western edge of the​ state, at the same time that a car traveling toward it leaves the eastern end of the state. If the bicycle and car met after 4.5 hr and the car traveled 38.4 mph faster than the​ bicycle, find the average rate of each.

Answer 1

Formula you need to know for this problem:

DISTANCE = RATE x TIME

Let’s write a formula for car and a formula for the bike.

1. The time the bike is traveling is the same as the time the car is traveling. They both left at the same time. When they meet, they will stop. This means that both will travel for exactly 4.5 hours.
2. The rate of the bike (miles per hour) is different from the rate of the car. It says that the car travels 38.4 mph faster than the bike. So this means that ...

rate of the bike = r

rate of the car = r + 38.4

1. The distance that the bike travels will be different from the distance that the car travels. The car should travel more miles than the bike. Since we know that the distance between the car and bike is 243 miles and that they are traveling towards each other until they meet ... that means that the distance of the bike and the distance of the car together makes 243 miles.

Distance of bike + distance of car = 243

bike ——-meet here——————————— car

<———-this whole distance is 243 miles———>

NOW WE CAN PUT THIS ALTOGETHER:

(distance of bike) = (rate of bike) x (time of bike)

(distance of bike) = (r) x (4.5)

(distance of bike) = 4.5r

(distance of car) = (rate of car) x (time of car)

(distance of car) = (r + 38.4) x (4.5)

(distance of car) = (4.5)(r) + (4.5)(38.4)

(distance of car) = 4.5r + 172.8

AND SINCE ... distance of bike and distance of car make 243 miles altogether ... WE CAN SAY:

(distance of bike) + (distance of car) = 243

(4.5r) + (4.5r + 172.8) = 243

SOLVE FOR r:

4.5r + 4.5r + 172.8 = 243

9.0r + 172.8 = 243

9.0r = 243 - 172.8

9.0r = 70.2

r = 70.2 ÷ 9.0

r = 7.8

THEREFORE:

rate of bike = r

rate of bike = 7.8 mph

rate of car = r + 38.4

rate of car = 7.8 + 38.4

rate of car = 46.2 mph

ANSWER:

Average rate of bike = 7.8 mph

Average rate of car = 46.2 mph