A correct clock has its hands coincide every 65 5/11 minutes (720/11 minutes). If a watch has its hands coincide every 64 minutes (true time), how much does this watch gain or lose per day?

Introduction / Context:
The interval between successive coincidences of the hands depends on their relative angular speed. A correct clock’s hands coincide every 720/11 minutes (≈ 65.4545 min). If a watch shows coincidences every 64 true minutes, its hands are moving faster relative to each other than they should, meaning the watch is running fast overall.

Given Data / Assumptions:

• Correct interval T_correct = 720/11 min.
• Observed interval T_watch = 64 min (true minutes).
• Relative speed is proportional to the watch’s rate r compared with correct: r = T_correct / T_watch.

Concept / Approach:
If coincidences occur more frequently than 720/11 min, the relative speed is higher by factor r = (720/11) / 64 = 45/44. Thus the watch runs (45/44 − 1) of the true rate fast. Multiply this fractional gain by the minutes in a day to get daily gain.

Step-by-Step Solution:

Verification / Alternative check:
360/11 ≈ 32.727 min/day fast. Over 11 days, the watch would be ahead by 360 minutes (= 6 hours), consistent with the proportional reasoning.

Why Other Options Are Wrong:
Loss values contradict the increased coincidence frequency; 96 or 90 minutes are far larger than the derived precise gain; 328/11 does not follow from the computed rate.

Common Pitfalls:
Using 64 as “watch minutes” instead of true minutes changes the model and yields a different (loss) result; misapplying relative speed as 6 − 0.5 without scaling.

Final Answer:
Gain 360/11 min per day.