Solution: Telling Time Math Puzzle

1. The minute hand is 18 minutes ahead of the hour hand at 3:36.
2. There is no time between 3:00 and 4:00 when the minute hand is 18 minutes behind the hour hand. At 3:00 the minute hand is 15 minutes behind the hour hand.
This interval decreases as the minute hand overtakes the hour hand and is then ahead of it. At 4:00 the minute hand is 40 minutes ahead of the hour hand, or 20 minutes behind it.

Here’s a way to figure out the spacing between the minute hand and the hour hand for any time. Let’s call the minute hand position MP and the “minute position” of the hour hand HP. The time consists of two components: hour and minute, which we will call H and M. So the time is H:M. At 3:45, H = 3 and M = 45.

The position of the minute hand is just the number of minutes. So MP = M.

The hour position is given by

HP = H x 5 + M/12

The hour hand moves 1/12 the distance of the minute hand for any time interval. In an hour, the minute hand makes a full turn around the clock while the hour hand goes 1/12 of the way around.

For example, at 3:30 MP = 30 and HP = 3 x 5 + 30/12 = 15 + 2 1/2 = 17 1/2

The difference between the minute hand position and the hour hand position is

MP – HP = 30 – 17 1/2 = 12 1/2 

That is, the minute hand is 12 1/2 minutes ahead of the hour hand.

At 3:12

MP = 12 and HP = 3 x 5 + 12/12 = 16

MP – HP = 12 – 16 = - 4

The negative indicates that the minute hand is 4 minutes behind the hour hand rather than ahead of it.

Keep in mind that for any time, you can think of the minute hand as being ahead of the hour hand or behind it. At 3:30, the minute hand is 12 1/2 minutes ahead of the hour hand or 47 1/2 minutes behind it. At 3:12, the minute hand is either 4 minutes behind the hour hand or 56 minutes ahead of it.

Normally we use clocks that show 12 hours on the face even though there are 24 hours in a day. There are some clocks that have 24-hour faces.

How would our calculations work on this kind of clock?