Between 5 and 6, a lady looked at her watch and, mistaking the hour hand for…

On a clock face the minute hand sweeps 12 times as fast as the hour hand: for every 1 minute-space the hour hand advances, the minute hand advances 12 minute-spaces. When a person swaps the two hands while reading the time, the true time and the misread time correspond to two different readings built from the very same hand positions, and the stated gap between them can be turned into a linear equation in the hour hand's offset.

1. Let x be the number of minute-spaces the hour hand has moved past the 5 mark, so the true time lies between 5:00 and 6:00.

2. Since the minute hand moves 12 times as fast as the hour hand, when the hour hand is x spaces past 5 the minute hand has moved 12x spaces past 12 — so the true time is 5 hours and 12x minutes, i.e. (300 + 12x) minutes past 12.

3. The hour hand's own position is (25 + x) minute-spaces past 12 (since 5 o'clock itself is 25 spaces past 12). Misreading this hour-hand position as the minute hand, and the minute hand's position as an hour-hand reading near the '4' mark, the lady perceived the time as 4 hours and (25 + x) minutes, i.e. (265 + x) minutes past 12.

4. Her perceived time was 57 minutes earlier than the true time, so (300 + 12x) - (265 + x) = 57.

5. Simplifying: 35 + 11x = 57, so 11x = 22, giving x = 2.

6. Substituting back, the true time = 300 + 12(2) = 324 minutes past 12 = 5 hours 24 minutes, i.e. 5:24.

Check: at 5:24 the minute hand is at 24 spaces and the hour hand is at 25 + 2 = 27 spaces. Reading 27 (the hour hand's true position) as minutes, and 24 (the minute hand's true position) as an hour-hand reading near the '4' mark, gives a perceived time of 4:27, i.e. 267 minutes past 12. The true time, 324 minutes past 12, is exactly 324 - 267 = 57 minutes later, matching the given condition and confirming 5:24 as the correct time.