lucalelli88 wrote:

A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

A. 5 minutes

B. 6 minutes

C. 8 minutes

D. 9 minutes

E. 10 minutes

We can let b = speed of the bus and c = speed of the cyclist. Let d = the distance between two consecutive buses (in the same direction). Thus, we want to determine d/b, which is the time interval between two consecutive buses.

When the bus and the cyclist are traveling in the same direction, if we assume a bus overtakes the cyclist at a certain time, then it will be 12 minutes until the next bus overtakes him. Thus, we have:

12(b - c) = d

12b - 12c = d

When the bus and the cyclist are traveling in opposite directions, if we assume the cyclist meets an oncoming bus at a certain time, then it will be 4 minutes until the cyclist meets the next oncoming bus. Thus, we have:

4(b + c) = d

4b + 4c = d

Subtracting 4b + 4c = d from 12b - 12c = d, we have:

8b - 16c = 0

8b = 16c

b = 2c

We see that the bus is twice as fast as the cyclist. Substituting 2c for b in 12(b - c) = d (or in 4(b + c) = d), we have:

12(2c - c) = d

12c = d

Recall that the time interval between two consecutive buses is d/b; thus, we have:

d/b = 12c/2c = 6

Answer: B

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