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

The time interval between consecutive buses is equal to how often the buses DEPART from the station: every 5 minutes, every 6 minutes, etc.
All of the buses -- in each direction -- travel at the same uniform speed.
The result is that the distance between consecutive buses is always the same.

Let the distance between consecutive buses = 24 units.
Let b = the rate of each bus and c = the rate of the cyclist.

SAME DIRECTION:
Here, the buses and the cyclist are COMPETING, so we SUBTRACT their rates.
The time needed for the next bus to CATCH UP to the cyclist is 12 minutes.
Thus:
b-c = d/t = 24/12 = 2 units per minute.

OPPOSITE DIRECTIONS:
Here, the buses and the cyclist are WORKING TOGETHER to cover the distance between them, so we ADD their rates.
The time needed for the cyclist and the next oncoming bus to PASS EACH OTHER is 4 minutes.
b+c = d/t = 24/4 = 6 units per minute.

Adding the two equations, we get:
(b-c) + (b+c) = 2+6
2b = 8
b=4 units per minute.

Since the rate of each bus is 4 units per minute and the distance between consecutive buses is 24 units:
The time interval between consecutive buses = d/r = 24/4 = 6 minutes.

The correct answer is B.

Roland2rule wrote:A man cycling along the road noticed that every 12 minutes a bus overtakes him while every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at 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