Clock problems look unique, but they’re secretly just relative speed problems in disguise. Two hands (hour and minute) are moving in the same direction on a circular track (the clock face). The minute hand is faster and keeps lapping the hour hand. Once we see it this way, everything falls into place.
Speed of the Clock Hands
Let’s break this down. The minute hand completes one full circle (360°) in 60 minutes, so it moves at 6° per minute. The hour hand completes one full circle in 12 hours = 720 minutes, so it moves at 0.5° per minute. The minute hand gains on the hour hand at 5.5° per minute — this is the relative speed.
The Clock Angle Diagram
Each number on the clock represents 30° (since 360°/12 = 30°). So at 3 o’clock, the angle is 3 × 30° = 90°. At 6 o’clock, it’s 180°. This is the foundation.
The Master Formula
For any time H hours and M minutes, the angle between the hands is:
Angle = |30H - 5.5M|
If the result is more than 180°, subtract from 360° (since we want the smaller angle).
In simple language: 30H gives us where the hour hand would be if it were exactly on the hour. Then 5.5M adjusts for the fact that the minute hand is at M × 6° and the hour hand has moved an extra M × 0.5° since the hour.
Why does this work? At H:M, the hour hand is at (30H + 0.5M)° and the minute hand is at 6M°. The difference is |30H + 0.5M - 6M| = |30H - 5.5M|.
Example 1: Finding the angle
Find the angle between clock hands at 3:20.
Angle = |30(3) - 5.5(20)| = |90 - 110| = |-20| = 20°
Example 2: Another angle calculation
Find the angle at 7:45.
Angle = |30(7) - 5.5(45)| = |210 - 247.5| = |-37.5| = 37.5°
Since 37.5° < 180°, the answer is 37.5°.
Example 3: Angle greater than 180
Find the angle at 9:10.
Angle = |30(9) - 5.5(10)| = |270 - 55| = 215°
Since 215° > 180°, the smaller angle = 360 - 215 = 145°
“At What Time” Problems
These flip the formula around — we know the angle and need to find the time.
Example 4: When are hands at 90°?
At what time between 4 and 5 will the hands be at a right angle?
We need |30(4) - 5.5M| = 90.
Case 1: 120 - 5.5M = 90 → 5.5M = 30 → M = 30/5.5 = 60/11 = 5 and 5/11 minutes past 4.
Case 2: 120 - 5.5M = -90 → 5.5M = 210 → M = 210/5.5 = 420/11 = 38 and 2/11 minutes past 4.
So the hands are at right angles at 4:05 5/11 and 4:38 2/11.
Example 5: When do hands coincide?
At what time between 7 and 8 do the hands overlap?
We need |30(7) - 5.5M| = 0.
210 - 5.5M = 0 → M = 210/5.5 = 420/11 = 38 and 2/11 minutes past 7.
So the hands coincide at 7:38 2/11 (approximately 7:38:11).
Important Counts
How many times do the hands form certain angles in 12 hours?
Why 11 and not 12? Because between 11 and 1, the hands coincide only once (at 12:00), not twice. Same for the straight line — between 5 and 7, the hands form 180° only once (at 6:00), not twice.
In 24 hours: Double everything. Coincidences = 22 times. Right angles = 44 times.
Interval between consecutive coincidences: 12 hours / 11 = 12/11 hours = 1 hour 5 minutes 27.27 seconds ≈ 65 5/11 minutes.
Faulty Clock Problems
A faulty clock runs too fast or too slow. We need to figure out what the actual time is when the faulty clock shows a certain time, or vice versa.
Key concept: If a clock gains 5 minutes per hour, then in a “true” hour, it shows 65 minutes. If it loses 5 minutes per hour, it shows 55 minutes.
Example 6: Fast clock
A clock gains 5 minutes every hour. If it was set correctly at 12 noon, what is the true time when it shows 6:00 PM?
The clock shows 6 hours = 360 minutes have passed (on the clock).
In every real 60 minutes, the clock shows 65 minutes.
Real time elapsed = 360 × (60/65) = 360 × 12/13 = 4320/13 ≈ 332.3 minutes ≈ 5 hours 32 minutes 18 seconds.
True time ≈ 5:32 PM (when the clock shows 6:00 PM).
Example 7: Slow clock
A clock loses 3 minutes every hour. It is set right at 8 AM. What does it show when the actual time is 8 PM?
Real time elapsed = 12 hours = 720 minutes.
In every 60 real minutes, the clock shows 57 minutes.
Clock shows: 720 × (57/60) = 720 × 19/20 = 684 minutes = 11 hours 24 minutes.
Clock shows 7:24 PM when actual time is 8 PM.
Mirror/Reflection Problems
When we see a clock in a mirror, the time appears reversed. To find the actual time from a mirror image:
Actual time = 11:60 - Mirror time (or equivalently, 12:00 - mirror time)
If mirror shows 2:25, actual time = 11:60 - 2:25 = 9:35. If mirror shows 8:40, actual time = 11:60 - 8:40 = 3:20.
This works because a mirror reversal on a clock is like subtracting from 12:00.
Common Exam Variations
- Find the angle at a given time.
- Find the time when hands form a specific angle.
- How many times do hands coincide / form right angles in X hours?
- Faulty clock: find true time or clock time.
- Mirror reflection: find actual time from mirror image.
- “Between X and Y o’clock, at what time are the hands at angle θ?”
Practice Problems
Problem 1: Find the angle between the clock hands at 5:30.
Problem 2: At what time between 3 and 4 o’clock do the minute and hour hands coincide?
Problem 3: A clock gains 4 minutes every hour. It is set correctly at 10 AM. What is the true time when the clock shows 4 PM?
Answers
Problem 1: Angle = |30(5) - 5.5(30)| = |150 - 165| = 15°. The answer is 15°.
Problem 2: We need |30(3) - 5.5M| = 0 → 90 = 5.5M → M = 90/5.5 = 180/11 = 16 and 4/11 minutes. The hands coincide at 3:16 4/11 (approximately 3:16:22).
Problem 3: The clock shows 6 hours = 360 minutes. In every real 60 minutes, the clock advances 64 minutes. Real time = 360 × 60/64 = 360 × 15/16 = 337.5 minutes = 5 hours 37.5 minutes. True time = 3:37:30 PM (about 3:37 PM).