In a faulty clock, the minute hand overtakes the hour hand every 64 minutes of correct time. How much time does the clock gain in a day?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(32\frac{8}{11}\) minutes
The minute hand gains \(6^\circ-0.5^\circ=5.5^\circ\) a minute on the hour hand, so in a correct clock it overtakes the hour hand once every \(\displaystyle\frac{360}{5.5}=\frac{720}{11}=65\frac{5}{11}\) minutes of the clock’s own time. Here this happens after only 64 real minutes, so the clock runs fast.
In 64 real minutes the clock shows \(65\frac{5}{11}\) minutes
Gain in 64 minutes \(\displaystyle =65\frac{5}{11}-64=1\frac{5}{11}=\frac{16}{11}\) minutes
A day has \(24\times60=1440\) minutes, which is \(\displaystyle\frac{1440}{64}=22.5\) such periods
Gain in a day \(\displaystyle =22.5\times\frac{16}{11}=\frac{360}{11}=32\frac{8}{11}\) minutes