Question

Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?

• A. \( 64 \frac{6}{11} \)
• B. \( 66 \frac{5}{13} \)
• C. \( 65 \frac{5}{11} \)
• D. \( 66 \frac{6}{13} \)

Hint:

To calculate the time between coinciding hands, use the formula \( T = \frac{12}{11} \times 60 \).

Answer 1

The Correct Option is C

Understanding the Concept:
The hour and minute hands of a clock coincide (overlap) 11 times in a 12-hour period. This happens because the minute hand moves faster and catches up with the hour hand at regular intervals.

Deriving the Time Interval:
The time interval between consecutive coincidences is given by the formula: \[ T = \frac{12}{11} \times 60 \quad \text{(minutes)} \] Simplifying: \[ T = 65 \frac{5}{11} \text{ minutes} \] This means that the hands overlap every 65 minutes and approximately 27 seconds.

Applying to Rohit's Situation:
- Since Rohit observes two coincidences of the hands while at the restaurant, his total time spent is equal to one full cycle of this interval. - Therefore, the total time Rohit stays at the restaurant is \( 65 \frac{5}{11} \) minutes.

Conclusion:
This result is derived based on the relative motion of the hands and their periodic overlap, showing a consistent time interval between coincidences.