M40-78

Official Solution:

Two cars, each moving at their respective constant speeds, depart simultaneously from points A and B towards each other. After passing each other, the car from point A reaches point B in 25 minutes, while the car from point B reaches point A in 36 minutes. How many minutes did it take for the faster car to cover the entire distance between the two points?

A. 44 minutes
B. 45 minutes
C. 50 minutes
D. 55 minutes
E. 60 minutes


Since the cars departed simultaneously and after meeting, the car from point A reached point B in less time than the car from point B took to reach point A, it's evident that the car from point A is faster.



Assuming they met after \(t\) minutes, then the car from A spent 25 minutes covering the same distance the car from B covered in \(t\) minutes. Since they travel at constant speeds, the ratio of their speeds would be equal to the reciprocal of the ratio of their times: the ratio of speeds of A to that of B = \(\frac{t}{25}\). For instance, if one car took \(t\) minutes to cover 10 kilometers, and another took 25 minutes to cover the same distance, then the ratio of the speed of the first car to that of the second car would be \(\frac{(\frac{10}{t})}{(\frac{10}{25})}=\frac{25}{t}\).

Similarly, since the car from A spent \(t\) minutes covering the same distance the car from B covered in 36 minutes, we'd have the ratio of speeds of A to that of B = \(\frac{36}{t}\).

Equating these two gives: \(\frac{t}{25} = \frac{36}{t}\), which yields \(t^2 = 25 * 36\), and finally, \(t = 5 * 6 = 30\) minutes.

Therefore, the faster car, the one from point A, took \(t + 25 = 30 + 25 = 55\) minutes to cover the entire distance between the two points.


Answer: D