Clock Problems - Quick Revision
Clock Problems - Quick Revision
Key Points (One-Liners)
- A clock is a 360° circle divided into 60 minute spaces → each minute space = 6°.
- Hour hand speed = 0.5° per minute; minute hand speed = 6° per minute.
- Relative speed of minute hand over hour hand = 5.5° per minute.
- In 60 min the minute hand gains 55 minute spaces over the hour hand.
- Hands coincide once every 65+5/11 min (≈12 times in 12 h).
- Hands are at right angle 22 times in 12 h (44 times in 24 h).
- Hands are in straight line (opposite) 11 times in 12 h.
- Between 2 & 3 o’clock, hands coincide at 2:10+10/11.
- Angle θ between hands = |30H – 5.5M|; if θ > 180° then take 360°–θ.
- When hands swap places, the sum of their original positions = 360°.
- Faulty clock: multiply normal time by (60±x)/60 for ±x min gain/loss per hour.
- 1 minute space = 6° → convert minute spaces to degrees quickly.
- Mirror image time: subtract given time from 11:60 (or 23:60 for 24-h).
- Water image of clock = mirror image except 3 & 9 stay same.
- In 12 h, overtakings (coincidences) = 11, hence interval = 12/11 h.
| Formula/Rule |
Application |
| Angle = |
30H – 5.5M |
| Coincide time = (60/11)×H past H o’clock |
When hands overlap between H & H+1 |
| Straight line (opposite) = (60/11)(2H–1) |
180° apart |
| Right angle = (60/11)(2H±1) |
90° apart (two per hour except 2-3 & 8-9) |
| Minute spaces gained = (5.5)×t |
t = minutes past H o’clock |
| Faulty time ratio = (60±x)/60 |
x = minutes gained(+) or lost(–) per hour |
| Mirror time = 11:60 – given time |
12-h clock |
| Swap places: H+M = 720/11 ≈ 65.45 min |
When hands interchange positions |
| Day gain/loss = x×24×60 min |
x = fraction gained per hour |
| Relative distance = 30H – 5.5M |
Sign shows which hand is ahead |
Memory Tricks
- “5.5 is the key” – relative speed & angle both use 5.5.
- “Coincide → 60/11” – remember 6×10 = 60 & 11 = overtakes per 12 h.
- Mirror: 11:60 minus given – like reversing the clock.
- Straight = 2×Right – 180° formula looks like 2×90° formula.
- Swap sum = 720/11 – 720 is twice 360, easy circle recall.
Common Mistakes
| Mistake |
Correct Approach |
| Forgetting to take θ = 360°–θ when angle >180° |
Always give smaller angle ≤180° |
| Using 30H–5M instead of 30H–5.5M |
Minute hand moves 0.5° per minute extra |
| Counting 12 coincidences in 12 h |
First & last at 12:00 coincide → only 11 unique |
| Mirror image: subtract from 12:00 |
Use 11:60 (or 23:60) to avoid negative |
| Ignoring sign of faulty clock |
Gain → multiply by (60+x)/60; loss → (60–x)/60 |
Last Minute Tips
- Write 5.5 on rough sheet first – every formula needs it.
- Draw tiny 12-h dial; mark 6° & 0.5° for quick visualization.
- Check angle ≤180°; if not, subtract from 360°.
- For “swap” & “mirror”, plug back into angle formula to verify.
- If two options differ by exactly 6°, one is minute-space error – re-calculate.
Quick Practice (5 MCQs)
1. At what time between 4 and 5 o’clock will the hands be 180° apart?
► 4:54+6/11
2. Angle between hands at 3:25 is:
► 47.5°
3. A clock gains 3 min in 48 h. How much will it gain in 72 h?
► 4.5 min
4. Mirror image of 8:40 is:
► 3:20
5. After 12:00, the hands coincide again at:
► 1:05+5/11
BETA
