Question

How much time does a watch lose per day if its hands coincide every 64 minutes?

• A. \(30\frac{2}{11}\) minutes
• B. \(31\frac{4}{11}\) minutes
• C. \(32\frac{8}{11}\) minutes
• D. \(33\frac{1}{11}\) minutes

Hint:

Remember that the hands of a correct clock coincide every \[ \frac{720}{11}=65\frac{5}{11} \] minutes. This result is frequently used in clock problems involving gain or loss of time.

Answer 1

The Correct Option is C

Concept: In a correct clock, the minute hand gains \(360^\circ\) over the hour hand in \[ \frac{720}{11}\text{ minutes} \] Therefore, the hands of a correct clock coincide every \[ 65\frac{5}{11}\text{ minutes}. \] If a watch is slow, the interval between two successive coincidences observed on that watch will be different from the interval of a correct clock.

Step 1: Find the loss corresponding to one coincidence interval.
For a correct watch, \[ \text{Interval}=\frac{720}{11}\text{ minutes}. \] For the given watch, \[ \text{Interval}=64\text{ minutes}. \] Hence loss during one coincidence interval is \[ \frac{720}{11}-64 = \frac{720-704}{11} = \frac{16}{11}\text{ minutes}. \]

Step 2: Calculate the loss in one day.
Loss in \(64\) minutes \[ = \frac{16}{11}\text{ minutes}. \] Therefore, loss per minute is \[ \frac{\frac{16}{11}}{64} = \frac{1}{44}\text{ minute}. \] Loss in one day (\(1440\) minutes) \[ = 1440\times\frac{1}{44} = \frac{360}{11} = 32\frac{8}{11}\text{ minutes}. \] Hence, the watch loses \[ \boxed{32\frac{8}{11}\text{ minutes}} \] per day.