Clock Problems made clear.
The minute hand moves faster, but the hour hand also moves continuously. Measure both hands from 12 o’clock, then compare their positions.
Key concepts in Clock Problems
Hand speeds
Minute hand: 6° per minute. Hour hand: 0.5° per minute or 30° per hour.
Positions
At H hours M minutes, with H reduced modulo 12: hour angle = 30H+0.5M; minute angle = 6M.
Smaller angle
Compute d = |30H−5.5M|. The smaller angle is min(d,360−d).
Relative motion
Minute hand gains 5.5° per minute. Consecutive overlaps are 360/5.5 = 720/11 minutes apart.
Clock positions at 3:20
| Hand | Movement from 12 | Angle |
|---|---|---|
| Hour | 3×30 + 20×0.5 | 100° |
| Minute | 20×6 | 120° |
| Gap | 120−100 | 20° |
A reliable solving method
- Identify the exact time and use a 12-hour clock value.
- Calculate both hand positions, including the hour hand’s movement.
- Take their absolute difference.
- Choose the smaller or reflex angle according to the question.
Step-by-step solved examples
Find the smaller angle at 3:20.
- Hour hand = 90+10 = 100°.
- Minute hand = 120°.
- Difference = 20°, already the smaller angle.
When do the hands first overlap after 2:00?
- At 2:00 the hour hand is 60° ahead.
- Minute hand closes the gap at 5.5°/minute.
- Time = 60/5.5 = 120/11 minutes, or 10 10/11 minutes after 2.
Find the smaller angle at 8:00.
- Hour hand is at 240°; minute hand is at 0°.
- Direct difference is 240°.
- Smaller angle = 360−240 = 120°.
Clock Problems: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Smaller angle at 6:00?
Show answer and explanation — question 1
Answer D: 180°
The hands point in opposite directions.
Smaller angle at 3:00?
Show answer and explanation — question 2
Answer C: 90°
Three hour divisions equal 90°.
Hour hand movement in 20 minutes?
Show answer and explanation — question 3
Answer B: 10°
20×0.5 = 10°.
Minute hand movement in 15 minutes?
Show answer and explanation — question 4
Answer C: 90°
15×6 = 90°.
Smaller angle at 2:30?
Show answer and explanation — question 5
Answer C: 105°
Hour position 75°, minute position 180°; difference 105°.
Hands overlap how many times in [12:00, next 12:00), a 12-hour interval?
Show answer and explanation — question 6
Answer B: 11
They overlap every 720/11 minutes; the ending endpoint is excluded to avoid double counting.
Frequently asked questions
Why does 5.5 appear in the angle formula?
It is the minute hand’s 6°/minute speed minus the hour hand’s 0.5°/minute.
What about a clock gaining or losing time?
Use the difference between indicated and real elapsed time. Hand-angle formulas alone do not solve a faulty-clock rate question.
Identify the exact time and use a 12-hour clock value. Choose the smaller or reflex angle according to the question.