Two Cars Algebra Problem

Dear Professor Cram:

One automobile starts out from a town at 8am and travels at an average speed rate of 35mph. Three hours later a second automobile starts out to overtake the first. If the second automobile travels at an average rate of 55mph how long before it overtakes the first?

Holly F., Madison Area Technical College

Thank you for your interest in College-Cram, Holly, and thanks for your question.

Your question is a variant of the classic "A train leaves Chicago going…" question you see on tests like the SAT, but don't let that alarm you. These vary with direction and whether we are solving for how long it takes in time or distance. In this case, you have two vehicles travelling the same direction, with one leaving first and a second having to go faster in order to catch up and we need to know the time it takes.

The basic formulas relate time and velocity (speed as a rate, such as miles per hour) with distance travelled:

    Distance = velocity x time

(For example, at 60 mph for two hours you go 120 miles.)

Now, back to your problem. In order to solve these two simultaneous equations we have to see when the distance travelled by both are the same, so we will need a formula representation for each one, and then set them equal to each other and solve for time. We know the second vehicle left three hours after the first, so we know the time of the second one is three hours less than the time the first one travels.

For the first vehicle, Distance = t hours x 35 mph. For the second vehicle, Distance = t-3 hours x 55 mph. At the point the second overtakes the first, the distance is the same, so let's make them equal:

    t x 35 = (t-3) x 55

We'll want to solve for t-3 (how long the second vehicle took to catch up) in hours.

  • 35t = 55t – 165
  • 165 = 55t – 35t
  • 165 = 20t
  • 8.25 = t
  • 5.25 = t-3

It took 5.25 hours for the second car to catch up.

As a check for your values, see if both cars really travelled the same distance. Car 1 went 8.25 hours at 35 mph, or 288.75 miles. Car 2 went 5.25 hours at 55 mph or 288.75 miles. It worked!

I hope this helps. Let us know if you need anything else.

Good Studying,

Professor Cram