Home › Geometry - angles and clocks

How to find the minute hand from the hour hand's exact position

The rule: multiply the hour-hand mark by 12, then take modulo 60

Number the $60$ small marks around a clock from $0$ at 12 o'clock. The minute hand completes one revolution per hour, while the hour hand completes one revolution per $12$ hours, so the minute hand moves $12$ times as far. If the hour hand is exactly on mark $h$ , the minute hand is on

m \equiv 12 h (mod 60)

The remainder after division by $60$ is the visible minute mark.

Problem

At noon both hands point vertically upward. Example 1. Later, the hour hand points exactly at the small clock mark labeled $19$ minutes. Which minute mark does the minute hand point to? Example 2. Answer the same question when the hour hand points exactly at mark $26$ .

Minute marks also measure the hour hand's angular displacement. The minute hand covers twelve times that angle, with full turns removed.

Answer

Hour hand on mark

19

: minute hand on mark

48

. Hour hand on mark

26

: minute hand on mark

12

Both results were checked from the elapsed time since noon.

Step-by-step solution

Example 1: hour hand on mark 19

The minute hand moves $12$ times the angular distance of the hour hand:

12 \cdot 19 = 228

Remove three complete revolutions, each worth $60$ marks:

228 = 3 \cdot 60 + 48

So the minute hand points at mark $48$ .

Check with elapsed time: the hour hand moves $5$ minute marks per hour, so reaching mark $19$ takes $19/5 = 3.8$ hours. The fractional $0.8$ hour is $0.8 \cdot 60 = 48$ minutes.

Example 2: hour hand on mark 26

12 \cdot 26 = 312

312 = 5 \cdot 60 + 12

So the minute hand points at mark $12$ .

The elapsed-time check gives $26/5 = 5.2$ hours after noon, and $0.2 \cdot 60 = 12$ minutes.

General answer

For any hour-hand mark $h$ , calculate $12 h$ and keep only the remainder after division by $60$ .

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.