If a clock's hands coincide every 65 minutes (true time), how much does the clock gain in 24 hours?

Introduction / Context:
True coincidences occur every 720/11 minutes ≈ 65 5/11. If a clock shows coincidence every 65 minutes (shorter than true), it runs fast. We compute the daily gain from the ratio of periods.

Given Data / Assumptions:

• True period = 720/11 minutes.
• Observed period = 65 minutes.

Concept / Approach:
Observed period = true period / (1 + x). Hence 65 = (720/11)/(1 + x). Solve x, then daily gain = x * 24 hours.

Step-by-Step Solution:
1 + x = (720/11) / 65 = 720 / 715 = 144 / 143.x = (144/143) - 1 = 1/143.Daily gain = (1/143) * 24*60 minutes = 1440/143 = 10 10/143 minutes.

Verification / Alternative check:
Because 65 < 65 5/11, the clock must gain; our result is a small positive gain consistent with a slightly fast clock.

Why Other Options Are Wrong:
Other fractional values do not match 1440/143; only 10 10/143 is correct.

Common Pitfalls:
Using difference of periods instead of rate ratio; mixing hours and minutes.

Final Answer:
10 10/143 minutes gain