Problem No. 3. Clock hands

It is 7.30 now. How much time will it pass until the clock hands form the same angle again?

Student tip

Suppose that minute and hour hands move at a constant velocity each.

Solution

At 7.30, the minute hand and the hour hand form an angle of \(45^o\), and they are 7,5 minutes apart. If the hour hand does not move, then after 15 minutes the angle between the minute hand and the hour hand will be \(45^o\) again. But the hour hand also moves with a velocity equal to 5 minutes per hour (60 minutes), so the angle between the minute hand and the hour hand will be the same again, after \(15 + t\) minutes, for some \(t > 0\). 

We are going to determine the value of \(t\) from the fact that for \(15 + t\) minutes, the hour hand should move \(t\) minutes. We have the following proportion

$$5 : 60 = t : (15 + t),$$

which can be solved in the following way

$$5(15+t)=60t$$

$$75+5t=60t$$

$$75=55t,$$

from where

$$t=\frac{75}{55}=\frac{15}{11} \; \text{minutes}.$$

So, the answer is that the clock hands will form the same angle after

$$15 + t = 15 + \frac{15}{11} = 16\frac{4}{11} \; \text{minutes} \approx 16 \; \text{min} \; 22 \; \text{sec}.$$