M04-21

Official Solution:

A soccer coach takes \(x\) hours to cycle from his house to his office. When riding at 24 km/h, he arrives 5 minutes late, but at 30 km/h, he arrives 4 minutes early. What is the distance between his house and the office?

A. 18 km
B. 24 km
C. 36 km
D. 40 km
E. 72 km


Denote the distance as \(d\).

Keep in mind that the rates are given in kilometers per hour, while the time is in minutes. Convert the minutes to hours: 5 minutes is \(\frac{1}{12}\) hours, and 4 minutes is \(\frac{1}{15}\) hours.

The equation for the first scenario is \(\frac{d}{24} = x + \frac{1}{12}\). Multiplying by 24, we get: \(d = 24x + 2\).

The equation for the second scenario is \(\frac{d}{30} = x - \frac{1}{15}\). Multiplying by 30, we get: \(d = 30x - 2\).

Subtract the second equation from the first one to obtain: \(6x = 4\).

Solving for \(x\), we find that \(x = \frac{2}{3}\).

Now, substitute this value of \(x\) into either equation to find the distance \(d\). For example, using the first equation: \(d = 24*\frac{2}{3} + 2 = 18\).

Therefore, the distance between the coach's house and his office is 18 kilometers.


Answer: A