Question

THE FAULTY WATCH

One day I found a strange thing happening to my watch—the minute hand and the hour hand were coming together every sixty-five minutes. I decided to have it seen to. Was my watch gaining or losing time, and how much per hour?

Answer 1

your watch according to me was gaining time at approx. 25 sec per hour

Answer 2

Given,

The minute hand and hour hand come together every sixty five minutes

To Find,

Whether the watch is gaining or losing time and how much per hour.

Solution,

Consider for one rotation in a clock, a total of 360° are covered.

Thus for the minute hand it needs an hour, that is 60 minutes to complete one rotation, that is 360°.

In this on hour, the hour hand moves by one part, that is to the next digit in the clock.

⇒ $\frac{360}{12} = 30$ (since clock has 12 equal parts)

Thus when the minute hand completes one rotation, hour hand moves by 30°

To cover this 30°, the minute hand requires five minutes as follows:

360° → 60 min

30° → ?

On cross multiplication,

$30(\frac{60}{360}) =5$

Thus for every 65 minutes, minute and hour hands meet in a clock.

Henceforth, there is no loss or gain of time and the watch is working properly.