Bunuel wrote:

Official Solution:

A bus leaves city \(M\) and travels to city \(N\) at a constant speed, at the same time another bus leaves city \(N\) and travels to city \(M\) at the same constant speed. After driving for 2 hours they meet at point \(P\). The following day the buses do the return trip 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

The buses travel at the same constant speed. It would take one bus to travel 4 hours to cover the distance between the cities \(M\) and \(N\) (two buses drove for 2 hours each). We need to find the speed of the bus. If the first bus was delayed by 24 minutes and the second one left 36 minutes earlier, it makes the second bus \(24+36=60\) minutes ahead of the first bus.

The meeting point was 24 miles away on this second day. We know the distance difference between the two meeting points, but we also need to find difference in time those 24 miles were covered. If the second bus drove for 1 hour before the first one departed, each of them had to go for another 1.5 hour to meet (1.5 hour + 1.5 hour + 1 hour). The second bus traveled for 2.5 hours and the first one for 1.5 hour. Therefore the meeting point on the second day was 30 minutes away from that of the previous day.

So the second bus covered 24 miles in 30 minutes, which gives us the speed of the bus, 48 mph. We can calculate the distance as we already know the speed:

\(4*48=192\) miles.

Alternative Explanation

Say the distance between the cities is \(d\) miles.

Since both buses travel at the same constant speed and leave the cities at the same time then they meet at the halfway, so the first meeting point \(P\), is \(\frac{d}{2}\) miles away from \(M\) (and \(N\)).

Next, since the buses meet in 2 hours then the total time to cover \(d\) miles for each bus is 4 hours.

Now, on the second day one bus traveled alone for 1 hour (36min +24min), hence covered \(0.25d\) miles, and \(0.75d\) miles is left to cover.

The buses meet again at the halfway of \(0.75d\), which is 24 miles from \(\frac{d}{2}\):

Hence, \(\frac{d}{2}-24=\frac{0.75d}{2}\), which gives \(d=192\) miles.

Answer: E

i understood explanation , i followed a different approach and resulted in wrong answer but still dont find the reason why its wrong

the only difference on both days is one bus is 1 hour ahead of other

because of this 1 hour they meet 24 miles ahead of the previous meeting point

so why cant we say it took 1 hour for the bus ,which was an hour ahead of other ,to complete the 24 miles

thanks in advance