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Clocks: Angles, Meeting Hands and Faulty Clocks

Learn how the two hands of a clock move, derive one formula for every angle question, and practise with original questions written for Wisdomhelps readers.

• Beginner to Advanced
• 7 Solved MCQs
• 10-Question Timed Test

Concept: How the Hands Move

A clock face is a full circle of 360° divided into 12 hour marks and 60 minute marks. Every clock question comes from the speed of two hands, so start there.

Step 1: Speed of the minute hand

The minute hand completes 360° in 60 minutes, so it moves 360 ÷ 60 = 6° every minute. One minute mark equals 6°, and 5 minute marks (one hour mark) equal 30°.

Step 2: Speed of the hour hand

The hour hand completes 360° in 12 hours, so it moves 30° every hour. In one minute it moves 30 ÷ 60 = 0.5° (half a degree).

Step 3: Relative speed

The minute hand gains 6 − 0.5 = 5.5° per minute over the hour hand.

Step 4: The angle formula

At H hours and M minutes, the hour hand is at (30H + 0.5M)° and the minute hand is at 6M° from the 12 mark. So:

Angle θ = | 30H − 5.5M |. If the answer is more than 180°, subtract it from 360° to get the smaller angle.

Example: At 4:20, θ = |30 × 4 − 5.5 × 20| = |120 − 110| = 10°.

Step 5: Time for a given angle

Between H and H+1 o'clock, the hands make angle θ when the minute hand is M minutes past H, where

M = 2(30H ± θ)11 minutes past H

For overlap, θ = 0, so M = 60H11. For example, the hands overlap between 2 and 3 at 2 : 101011.

Step 6: Counting facts

• In 12 hours the hands coincide 11 times and point opposite 11 times (22 each in 24 hours).
• They form a right angle 22 times in 12 hours (44 in 24 hours).
• Two successive overlaps are 65511 minutes apart on a correct clock.

Step 7: Mirror and water images

Mirror image of a time T = 11:60 − T. Water image of T = 5:60 − T (add 12 hours if needed). Example: mirror image of 8:15 is 3:45.

Procedure to Solve Clock Problems

1. Identify the type: angle between hands, time for an angle, overlap, image, or fast/slow clock.
2. Write H (hour) and M (minutes) clearly. Convert 24-hour time to 12-hour time first.
3. For angle questions, apply |30H − 5.5M|. If the result is above 180°, take 360° minus it.
4. For time questions, use M = 2(30H ± θ)/11 and check that M is between 0 and 60. Try both signs; both may be valid.
5. For fast or slow clocks, compare how many minutes the faulty clock shows with how many true minutes pass, then use proportion.
6. Re-check by substituting the time back into the angle formula.

Exam tip: Right angle means θ = 90°, straight line means 180°, and overlap means 0°.

7 Solved MCQs

Tap an option to check it, then open the explanation.

Timed Test: 10 Questions

Exam information

Questions	10 (random from a pool of 16)
Time	10 minutes
Marks	1 per correct answer
Negative marking	None
Tools	Mark for review and jump-to-question panel

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