Problem Solving

A man cycling along the road noticed that every 12 min a bus overtakes him while every 4 min he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the interval between consecutive buses ? anyone knows the answer to this problem ? thanks

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

I don't really know how to solve this. So I am going to just start working on something to see how to get to the answer.

I notice that the buses are moving at speeds relative to the cyclist, and that is all I have to go on. So I am going to go with that.

Let's call the bus speed B and the cyclist speed C.

The buses going his direction are moving at a slower relative speed, B - C.

The ones going the other direction are moving at a faster relative speed, B + C.

We know that the buses are equally spaced. So the buses passing him in his direction are going one space every 12 minutes. 12(B - C) = x

The buses going the other direction are traveling one space every 4 minutes. 4(B + C) = x

I am not sure where I am heading with this but maybe if I just set them equal to each other I'll find a way to get to the answer.

12(B - C) = 4(B + C)

12B - 12C = 4B + 4C

8B = 16C ---> B = 2C

So the buses are going twice as fast as the cyclist. Hmm. How does that help me? I am just going to plug in some numbers and get this thing done.

The easiest numbers I could use are B = 2 and C = 1.

So (B - C) = 1 = 0.5B and (B + C) = 3 = 1.5B

So, thinking back to what I wrote above, I realize that a bus going 0.5B takes 12 minutes to go one space, and a bus going 1.5B takes 4 minutes to go one space.

Sweet. That means that a bus, at speed B, is actually taking 6 minutes to cover a space, and I have my answer.

Choose B.

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.

Thanks, but how/why did you choose 24 units as a distance between consecutive buses?

yass20015 wrote:Thanks, but how/why did you choose 24 units as a distance between consecutive buses?

The distance between consecutive buses can be ANY VALUE.
To make the math easy, I chose a value divisible by the given times (12 minutes and 4 minutes).

To illustrate that we can plug in any value, let the distance between consecutive buses = 12 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 = 12/12 = 1 unit 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 = 12/4 = 3 units per minute.

Adding the two equations, we get:
(b-c) + (b+c) = 1+3
2b = 4
b = 2 units per minute.

Since the rate of each bus is 2 units per minute, and the distance between consecutive buses is 12 units:
The time interval between consecutive buses = d/r = 12/2 = 6 minutes.

Regardless of the distance between consecutive buses, the time interval between consecutive buses is THE SAME:
6 minutes.

The correct answer is B.

though I understand it very less, but it is very good question.

can we solve it as like

relative speed in one direction = B-c

relative speed of oncoming direction = b + c

lets say the distance of two buses is 12meter.

while going at B-c speed for 12 minutes - cyclist see 12/12 = 1 bus

while going at B+c speed for 4 minutes - cyclist see 12/4 = 3 bus

B + c + b-c = 3 +1
2b =4
b=2
c=1

for a distance of 12 meters for every bus sight. bus must start after every 12/2 = 6 mins.

GMATGuruNY 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

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.

Dear GMATGuru,

I failed to put up a simple diagram. Is it possible here?

1- Why did you assume that the bus in one direction has the same speed of the opposite direction? I read the question but could not conclude that.

2- What does 'consecutive buses mean? is it two buses departing from in same direction or a bus depart from one direction and other from opposite direction?

3- why is the distance between buses are constant?

Can you clarify please?

Dear GMATGuru,

I failed to put up a simple diagram. Is it possible here?

Attempting to draw a diagram might complicate rather than simplify the solution.

1- Why did you assume that the bus in one direction has the same speed of the opposite direction? I read the question but could not conclude that.

From the prompt:
all buses...move at a constant speed.
ALL buses = buses moving in the same direction and those moving in the opposite direction.

2- What does 'consecutive buses mean? is it two buses departing from in same direction or a bus depart from one direction and other from opposite direction?

Consecutive buses refers to two successive buses moving in the same direction.

3- why is the distance between buses are constant?

All buses move at a constant speed.
The question stem asks for the time interval between consecutive buses.
The phrase in red implies that the there is a fixed amount of time between the departure of one bus and that of the next bus.
Since the rate of each bus is constant, and the time interval between successive buses is constant, the distance between successive buses is constant.

GMATGuruNY wrote:

3- why is the distance between buses are constant?

All buses move at a constant speed.
The question stem asks for the time interval between consecutive buses.
The phrase in red implies that the there is a fixed amount of time between the departure of one bus and that of the next bus.
Since the rate of each bus is constant, and the time interval between successive buses is constant, the distance between successive buses is constant.

Is the interval time which is 6 minute from the buses of the cyclist direction the same as the buses of the opposite direction?

Mo2men wrote:

GMATGuruNY wrote:

3- why is the distance between buses are constant?

All buses move at a constant speed.
The question stem asks for the time interval between consecutive buses.
The phrase in red implies that the there is a fixed amount of time between the departure of one bus and that of the next bus.
Since the rate of each bus is constant, and the time interval between successive buses is constant, the distance between successive buses is constant.

Is the interval time which is 6 minute from the buses of the cyclist direction the same as the buses of the opposite direction?

Yes.
The time interval is the same in each direction.
If a bus moving in the same direction as the cyclist departs at 12pm, then the next bus moving in this direction will depart 6 minutes later.
If a bus moving in the opposite direction departs at 12pm, then the next bus moving in this direction will depart 6 minutes later.

A follow up to test your understanding!

Suppose I said, "Ha! I've got a much easier way! Why not just use a weighted average? There are three buses coming every 4 minutes from one direction, and one bus coming every 12 minutes from the other! (3*4 + 1*12) / 4 buses = 6, so that's clearly the answer."

Is this valid? Why or why not?