At what time between 4 and 5 o'clock are the hands of a clock together?

Think of the hands as two runners on a circular track. The minute hand runs 6 degrees a minute and the hour hand 0.5, so the minute hand gains 5.5 degrees every minute. At 4:00 the minute hand is 120 degrees behind. Time to close the gap = gap / relative speed, the same idea as one train catching another.

Question: When between 4 and 5 are the hands together?

1. At 4:00: hour hand at 120 degrees, minute hand at 0. Gap = 120.
2. Relative speed = 6 - 0.5 = 5.5 degrees per minute.
3. Time = 120 / 5.5 = 240 / 11 = 21 9/11 minutes past 4.

In "minute spaces": the minute hand gains 55 minute spaces in 60 minutes. It needs to gain 20 spaces, which takes 20 x 60/55 = 21 9/11 minutes.

Similar: the hands are opposite (180 degrees apart) between 4 and 5 when the minute hand has gained 120 + 180 = 300 degrees: 300/5.5 = 54 6/11 minutes past 4.

console.log((120 / 5.5).toFixed(3)); // Output: 21.818
console.log((300 / 5.5).toFixed(3)); // Output: 54.545

• Answering 4:20 by assuming the hour hand does not move.
• Writing 21 9/11 minutes as 21.9 minutes.