Question

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

• A. 32 $\frac{8}{11}$ min
• B. 36 $\frac{5}{11}$ min
• C. 90 min
• D. 96 min
• E.

Hint:

For time-based problems, focus on the discrepancy per cycle, then calculate how it adds up over time.

Answer 1

The Correct Option is A

Since a watch loses time when its hands coincide, we need to calculate how much time is lost per day. The time taken for the hands to coincide is 64 minutes instead of 60. This discrepancy accumulates over time. 
The total loss per day is calculated as 24 hours, or 1440 minutes. 
The difference per minute is $(64 - 60) = 4$ minutes lost every 64 minutes. 
Thus, the total loss per day is calculated as: $$ \frac{4}{64} \times 1440 = 90 \text{ minutes lost per day}. $$ 

Answer 2

Work from the fact that in a perfectly accurate clock, the hands coincide once every \( \frac{720}{11} \) minutes, i.e. \( 65\frac{5}{11} \) minutes, not every 60 minutes.

1. Step 1: Since this watch's hands coincide every 64 real minutes instead of the accurate \( 65\frac{5}{11} \) minutes, the watch completes a full \( 65\frac{5}{11} \)-minute cycle in only 64 minutes of real time, so it is running fast.
2. Step 2: The watch's speed relative to a correct clock is \( \dfrac{65\frac{5}{11}}{64} = \dfrac{720/11}{64} = \dfrac{45}{44} \).
3. Step 3: In 24 hours, i.e. 1440 real minutes, the watch would display \( 1440 \times \dfrac{45}{44} = \dfrac{64800}{44} = 1472\frac{8}{11} \) minutes.
4. Step 4: The difference between what the watch displays and the true 1440 minutes is \( 1472\frac{8}{11} - 1440 = 32\frac{8}{11} \) minutes.

Checking the other options, \( 36\frac{5}{11} \), 90 and 96 minutes do not match this rate-based calculation.

Hence, the correct answer is \( 32\frac{8}{11} \) min.