The minute hand of a clock overtakes the hour hand every 64 minutes of correct time. How much does the clock gain or lose in a day?

A clock's hands meet every 65 5/11 minutes of its own time, faulty or not. If that happens after only 64 real minutes, the clock is racing ahead. Compare how much "clock time" passes against real time, then scale up to a full day of 1,440 real minutes.

Question: Hands meet every 64 real minutes. Gain or loss per day?

1. Correct interval between meetings = 720/11 = 65 5/11 minutes (clock time).
2. This clock shows 65 5/11 minutes while 64 real minutes pass.
3. Gain per 64 real minutes = 65 5/11 - 64 = 1 5/11 = 16/11 minutes.
4. In a day: (1,440 / 64) x 16/11 = 22.5 x 16/11 = 360/11 = 32 8/11 minutes gained.

If the hands met every 66 real minutes instead, the clock would be slow and lose time: (1,440/66) x (66 - 720/11) = (1,440/66) x 6/11 = 11 109/121 minutes a day, roughly 11.9.

const gain = (1440 / 64) * (720 / 11 - 64);
console.log(gain.toFixed(3)); // Output: 32.727

• Concluding the clock is slow because 64 is 'less than' 65 5/11.
• Scaling by 1,440 / 65 5/11 instead of 1,440 / 64; the day is measured in real time.