# Circular Track Math Problems

Circular track math problems, also known as race track math problems are an interesting class of math word problems. There are generally three types of questions related to circular track math problems:

1) The time to meet at the starting point

2) The time to meet for the 1st time anywhere on the track

3) The number of times they will meet anywhere on the track given fixed amount of time.

There are other variations too such as whether the participants are starting at the same time and point. Whether the participants are heading in the same direction. How many participants there are. Also, calculating the length of the track given the speed and time and meeting points of the participants.

Let’s look at finding the answer to each of these types of questions using the example below.

Two dogs run around a circular track 300 feet long. Fido runs at a steady rate of 15 feet per second. The other dog, Max, runs at a steady rate of 12 feet per second. Assume they start at the same time and point on the track.

#### What is the least number of seconds that will elapse before they are again together at the starting point?

STEP 1: we determine how long each dog will take to complete one lap on the track.

To do this, we simply divide the distance of the track (300 feet) by the speed of each dog (number of feet that can be run by each dog in a given time frame e.g, seconds in this example)

$\frac{track\space distance}{dog\space speed}$ = time for dog to complete one lap

Fido = $\frac{300 feet}{15 feet/second}$ = 20 seconds to complete one lap

Max= $\frac{300 feet}{12 feet/second}$ = 25 seconds to complete one lap

STEP 2: we find the LCM (least common multiple) of 20 and 25

LCM of 20, 25 is 100

ANSWER: It will take 100 seconds for both dogs to meet again at the starting point of the track.

You can easily check this by seeing that Fido will be back at the starting point after 100 seconds b/c he will have completed his 5th lap (20 x 5 = 100). Max will have completed his 4th lap at that same moment (25 x 4 = 100).

#### How long will it take for both dogs to cross paths (anywhere on the track) for the first time after the race begins?

Step 1: Divide the length of the track (same as above) by the relative speed between the two participants.

$\frac{track\space distance}{Fido\space speed - Max\space speed}$

$\frac{300}{15- 12}= \frac{300}{3} = 100\space seconds$

In this case, the answer is the same as in the first question. The dogs meet for the first time after 100 seconds and they happen to meet at the starting point of the track.

#### How many times will the dogs cross paths during a race of a set duration?

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