Problem Solving

A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for 2 hours. The following day the buses do the same trip again at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A. 48
B. 72
C. 96
D. 120
E. 192

IMO : C

the statement "One bus is delayed 24 minutes and the other leaves 36 minutes earlier" makes it 1 hour.
and the statement "If they meet 24 miles from point P" says that the bus has traveled 24 miles in 1 hour.

so since they meet after 2 hours at point P, they must have crossed 48 miles. and since they have the other part of the journey to be made the rest 48 miles should also be added.

So the total distance is 96 miles.

by the way what is the OE?

rahulvsd wrote:A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for 2 hours. The following day the buses do the same trip again at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A. 48
B. 72
C. 96
D. 120
E. 192

here one bus starts after one hr from other bus;

use symmetry to check for the meeting point it will be x/8 from point P, where x is the required distance.

i.e. x/8 = 24
=> x= 192

hence E

rahulvsd wrote:A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for 2 hours. The following day the buses do the same trip again at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A. 48
B. 72
C. 96
D. 120
E. 192

We can plug in the answers, which represent the distance.

On the first day, since each bus travels for 2 hours at the same speed, point P -- the point where the two buses meet -- is 1/2 of the distance.
Since each bus travels 1/2 the distance in 2 hours, the total time needed to cover the entire distance = 4 hours.

On the second day, one bus leaves 1 hour earlier than the other.
Thus, the earlier bus travels 1/4 of the distance on its own.
Then the two buses work together to travel the remaining distance, with each bus traveling 1/2 the remaining distance.

When the correct answer is plugged in, the two buses must meet at a point 24 miles from point P.
Thus, the difference between point P and the distance traveled by the delayed bus must be 24 miles.

Answer choice C: 96 miles
Point P = (1/2)96 = 48 miles.
On the second day, the distance traveled by the earlier bus in the first hour = (1/4)96 = 24.
Remaining distance = 96-24 = 72.
Distance traveled by the delayed bus = (1/2)72 = 36.
Difference between Point P and the distance traveled by the delayed bus = 48-36 = 12.
Since the difference is supposed to be 24 miles -- twice as long -- the correct answer must be double answer choice C.

The correct answer is E.

Answer choice E: 192 miles
Point P = (1/2)192 = 96 miles.
On the second day, the distance traveled by the earlier bus in the first hour = (1/4)192 = 48.
Remaining distance = 192-48 = 144.
Distance traveled by the delayed bus = (1/2)144 = 72.
Difference between Point P and the distance traveled by the delayed bus = 96-72 = 24.
Success!

Hi Mitch,

Can you explain how the delayed bus has to travel 24 miles?

Travel time difference between 2 buses when they travel for the second time = 1 hr

Tb - Ta = 1 ..... (1)

Difference in distance traveled = Db - Da = 24

Db = Speed * Tb
Da = Speed * Ta

Therefore, Db - Da = (Speed * Tb) - (Speed * Ta) = Speed * ( Tb - Ta) = Speed * (1) = 24

Speed = 24 miles/hr

We know that total time = 2 * 2 = 4hrs

Therefore distance between cities = Speed * Time = 24 * 4 = 96 miles

Answer (c)

rahulvsd wrote:Hi Mitch,

Can you explain how the delayed bus has to travel 24 miles?

Please revisit my post above, in which I've fleshed out the reasoning.