A fast clock and a slow clock agree again when?

Question

Two clocks with ordinary 12-hour dials are both set to the correct time at noon on the same day. From then on, clock A gains 2 minutes every day and clock B loses 1 minute every day, both steadily.

After how many days will the two clocks next show exactly the same time as each other? (They need not show the correct time.)

Accepted Answer

240 days. The gap. Each day, clock A moves 2 minutes ahead of true time and clock B falls 1 minute behind, so the gap between them grows by 3 minutes per day. When do two 12-hour dials agree? Two clocks show the same time when their readings differ by a whole number of 12-hour cycles, that is, by a multiple of 720 minutes. The first time the gap reaches 720 minutes is after 720 / 3 = 240 days. Check. After 240 days clock A is 480 minutes (8 hours) fast and clock B is 240 minutes (4 hours) slow. Eight hours ahead and four hours behind differ by 12 hours, which on a 12-hour dial looks identical…

A fast clock and a slow clock agree again when?

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