GMAT Math : Relative speeds

Relative speeds is a powerful way of looking at Speed, distance and time GMAT quant questions.

Here is an example:

John, Peter and Alice are driving cars in the same direction. John’s car, the furthest behind, is moving at 100 Kmph. Alice’s car, which is in between Peter and John’s car is moving at 50 Kmph. Peter’s car is moving at 75 Kmph. The distance between John and Alice’s car is 100 Km and Alice and Peters car is 50 Km. John overtakes Alice in time t1 hours and Peter in time t2 hours. What is the value of t2 – t1 ?

A) 3

B) 2

C) 4

D) 6

E) 5

A traditional approach would stipulate forming two linear equations with time, distance and speed inputs and solving the equations to come up with t1, t2 and subsequently t2 – t1. This can be a bit slow and if the numbers are not perfectly round – difficult to calculate.

Let’s solve the above question using relative speeds.

The principle: In case of multiple items moving – in a hypothetical universe, imagine all but 1 object has stopped moving and then picture what its speed would be.

Let’s say, you are moving at 10 Kmph and your friend is moving at 5 Kmph infront of you and in the same direction.

From your perspective, imagine your friend to be stationary then essentially you are moving at (10 – 5) = 5 Kmph, in other words you are catching up with your friend at 5 Kmph.

Coming to the question, if we were to lock Alice position, John would be moving at 100 – 50 Kmph = 50 Kmph. To cover a distance of 100 Km (distance between John and Alice), it would take John – 100/50 = 2 hours

Now, if lock Peter’s position, John would be moving at 100 – 75 Kmph = 25 Kmph. To cover 150 Km (distance between John and Peter), it would take John – 150/25 = 6 hours.

Hence, t2 – t1 = 6 – 2 = 4 hours

The answer is (C). Notice, how we could rapidly move through this question – we did not have to form a single equation to crack it.

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