Official Solution:Two buses depart simultaneously from cities M and N and travel towards each other at the same constant speed, heading towards cities N and M, respectively. After two hours of travel, they meet at point P and pass each other to continue towards their destination cities. The next day, both buses are scheduled to simultaneously depart back to their original cities at the same constant speed as the day before. However, one of the buses is delayed by 24 minutes while the other departs 36 minutes earlier than originally scheduled. If they meet 24 miles from point P, what is the distance between cities M and N? A. 48
B. 72
C. 96
D. 120
E. 192
Let the distance between cities M and N be \(d\) miles.
Since both buses travel at the same constant speed and leave their respective cities at the same time, they will meet each other at the halfway point. Therefore, the first meeting point P is located at a distance of \(\frac{d}{2}\) miles from both cities M and N.
Next, since the buses meet each other after 2 hours of travel, the time taken by each bus to cover the entire distance is 4 hours.
On the second day, one of the buses departs 36 minutes earlier than scheduled, while the other bus is delayed by 24 minutes. As a result, the first bus travels alone for 1 hour and covers a distance of \(\frac{d}{4}\) miles. Therefore, when the second bus begins to move, the distance between the two buses is \(d - \frac{d}{4} = \frac{3}{4}*d\) miles.
Since both buses travel at the same constant speed, they will meet each other at the halfway point of the remaining distance of \(\frac{3}{4}*d\) miles, hence \(\frac{3}{8}*d\) miles from the city that the delayed bus departed from.
Given that this meeting point is 24 miles from point P, we have the equation \(\frac{d}{2} - 24 = \frac{3}{8}*d\).
Solving this equation gives us \(d = 192\) miles.
Answer: E
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