A faulty clock's hands coincide every 64 minutes. How much time does it gain or lose in one true day?

Introduction / Context:
For a perfect clock, the hands coincide every 65 5/11 minutes (720/11 minutes). If a clock shows coincidence every 64 minutes, its rate is not true; this question finds the net daily gain or loss by comparing the clock’s period with the true period.

Given Data / Assumptions:

• True coincidence interval = 720/11 minutes ≈ 65.4545 minutes.
• Faulty clock coincidence interval = 64 minutes.
• Gain/loss assumed uniform over time.

Concept / Approach:
Clock period scales inversely with its rate. If observed period = true period / (1 + x), then 64 = (720/11) / (1 + x). Solve x and convert to daily gain: daily gain = x * 24 hours.

Step-by-Step Solution:
1 + x = (720/11) / 64 = 720 / 704 = 45 / 44.Therefore x = (45/44) - 1 = 1/44.Daily gain = (1/44) * 24 hours = 24*60 / 44 minutes = 1440/44 = 32 8/11 minutes (gain).

Verification / Alternative check:
Since the faulty interval (64) is less than true (≈65.45), the clock runs fast, hence it must gain time — consistent with a positive result.

Why Other Options Are Wrong:
34 2/11 is too large; 32 8/11 loss has the wrong sign; “None” is unnecessary as we have an exact value.

Common Pitfalls:
Subtracting intervals directly instead of using rate scaling; forgetting that period decreases when the clock runs fast.

Final Answer:
32 8/11 minutes gain per day