Fast and Slow Clocks: Finding True Time from a Faulty Clock

A clock that gains runs fast: its hands cover more than 60 minutes of dial while 60 true minutes pass. A clock that loses runs slow. Every faulty-clock question is about the ratio between dial time and true time, which stays fixed for a given clock.

From that ratio you can answer the three questions examiners ask: what the clock shows after a given true time, what the true time is when the clock shows a given time, and when the clock will look correct again.

To find what a faulty clock shows, multiply the true time elapsed by the rate. To find the true time from what the clock shows, divide by the rate.

A tempting shortcut is to subtract "one minute per hour" from the shown time. It happens to work here because the numbers were chosen to fit, but in general the number of true hours is what you are solving for, so scaling is the safe method.

A normal clock face repeats every 12 hours. A faulty clock therefore looks correct again once its total error reaches 12 hours, that is 720 minutes, ahead or behind.

If the question involves a 24-hour digital display, the clock must drift 1,440 minutes instead, doubling the answer. Read the question for the kind of clock.

When two clocks drift in opposite directions, their gap grows by the sum of their errors each hour. When both drift the same way, the gap grows by the difference.

The question may instead ask when both clocks will next show the same time. That happens when the gap between them reaches 720 minutes on a 12-hour dial, so divide 720 by the hourly gap.

On a correct clock the hands coincide every 720/11 minutes, about 65 minutes 27 seconds (see the clock angle method for why). If a clock's hands are observed to meet every 65 true minutes, the clock has covered 720/11 dial minutes in only 65 true minutes, so it is running fast.

• ✘ Subtracting the gain instead of scaling by the clock's rate.
• ✘ Using 1,440 minutes for a 12-hour clock face; it repeats after 720.
• ✘ Subtracting the errors of two clocks that drift in opposite directions.
• ✘ Assuming hands that meet every 65 minutes are correct. A correct clock's hands meet every 65 5/11 minutes.

Related: calendar problems by odd days and direction sense.

Why does a fast clock show the correct time after drifting 12 hours?

An ordinary clock face shows only 12 hours. A clock that is exactly 12 hours ahead or behind points to the same numbers as a correct clock, so it looks right even though it is half a day out. A 24-hour digital clock would need to drift a full 24 hours.

Should I convert shown time to true time by adding or by scaling?

Scale it. If the clock runs at 61 dial minutes per 60 true minutes, true time elapsed equals shown time multiplied by 60/61. Simply subtracting the hourly gain works only when you know the exact number of true hours, which is what you are trying to find.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.