Distanct - Speed - Time

[Rohit21]

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?

1. 5 minutes
2. 6 minutes
3. 8 minutes
4. 9 minutes
5. 10 minutes

[aneesh.kg]

Let B the speed of each bus and C the speed of the cyclist.

Let 'd' be the separation between two buses and 't' be the time interval between consecutive buses.
Then d = B*t -- (1)

Relative speed of first bus and cyclist = B - C
d = (B - C)*12 -- (2)

Relative Speed of the oncoming bus and the cyclist = B + C
d = (B + C)*4 --(3)

Using (2) and (3),
B + C = 3*(B - C)
B = 2C

Substituting 'd' from (1) and B = 2C in (2),
B*t = (B - B/2)*12
t = 6 min

[spoiler](B)[/spoiler] is the answer.

[GMATGuruNY]

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?

1. 5 minutes
2. 6 minutes
3. 8 minutes
4. 9 minutes
5. 10 minutes

We are being asked to determine how often the buses DEPART: 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.