A watch has its hands coincide every 64 minutes. How much does the watch gain or lose per day?

Problem restatement
The normal interval between successive coincidences (overlaps) of hands on a correct clock is 720/11 minutes ≈ 65 5/11 minutes. Here, the watch has overlaps every 64 minutes (of true time). Determine the daily gain or loss.

Concept/Approach
If the watch runs at a uniform rate k (indicated minutes per true minute), both hands speed up or slow down by the same factor. The true interval between overlaps becomes (720/11) × (1/k). We are told this true interval equals 64 minutes.

Step-by-Step calculation
(720/11) × (1/k) = 64 1/k = 64 × (11/720) = 704/720 = 88/90 k = 90/88 = 45/44 > 1 (watch runs fast) Daily gain = (k − 1) × 24 × 60 minutes = (1/44) × 1440 = 1440/44 = 32 8/11 minutes

Verification/Alternative
Because the observed interval (64) is less than the standard 65 5/11, the watch must be gaining time. A shorter overlap interval implies a faster indicated time scale.

Common pitfalls

• Assuming a loss because 64 is smaller; in fact, a smaller interval indicates a gain.
• Confusing indicated versus true minutes; always anchor equations in true time.

Final Answer
Gains 32 8/11 minutes/day