Clock problems come up again and again in the Lok Sewa reasoning (IQ) section, for Kharidar, Nayab Subba, Section Officer and bank exams. Questions look tricky (“Find the angle at 4:20”, “How many times do the hands make a right angle in a day?”) but almost all of them are solved with one formula and a few fixed facts. This note gives you the formula, the shortcuts and worked examples, followed by a 20-question quiz.
1. Basic facts about a clock
| Item | Fact |
|---|---|
| Clock face | 360°, divided into 12 hour spaces and 60 minute spaces |
| One hour space (between two numbers) | 30° (= 5 minute spaces) |
| One minute space | 6° |
| Minute hand speed | 6° per minute (360° in 60 minutes) |
| Hour hand speed | 0.5° per minute (30° in 60 minutes) |
| Relative speed of the two hands | 6 − 0.5 = 5.5° per minute |
| Minute hand gains on hour hand | 55 minute spaces in 60 minutes |
2. The one formula: angle between the hands
At H hours and M minutes, the angle between the hour hand and the minute hand is:
θ = | 30H − 5.5M |
If the answer is more than 180°, subtract it from 360° to get the smaller angle. (Use H = 12 as 0 when needed; at 12:30 you get |0 − 165| = 165°.)
| Time | Working | Angle |
|---|---|---|
| 3:00 | |90 − 0| | 90° |
| 3:30 | |90 − 165| | 75° |
| 2:20 | |60 − 110| | 50° |
| 4:20 | |120 − 110| | 10° |
| 5:15 | |150 − 82.5| | 67.5° |
| 9:45 | |270 − 247.5| | 22.5° |
| 8:10 | |240 − 55| | 185° → 360 − 185 = 175° |
3. How many times in a day? (most-asked)
| Position of hands | In 12 hours | In 24 hours (a day) |
|---|---|---|
| Coincide (0°, hands together) | 11 times | 22 times |
| Opposite (180°) | 11 times | 22 times |
| In a straight line (0° or 180°) | 22 times | 44 times |
| At right angle (90°) | 22 times | 44 times |
Why 11 and not 12? Between 11 and 1 o’clock the hands meet only once, at exactly 12:00. Likewise, the hands are opposite only once between 5 and 7 (at 6:00), and they make a right angle only twice from 2 to 4 (the 3:00 position is shared) and from 8 to 10 (9:00 shared).
The hands coincide once every 65 5/11 minutes (= 720/11 minutes), not every 60 minutes.
4. Finding the exact time (between H and H+1 o’clock)
Put the required angle θ into 30H − 5.5M = ±θ and solve for M. This gives three ready-made shortcuts:
| Hands are… | Minutes past H | Example |
|---|---|---|
| Together (0°) | M = (60/11) × H | Between 3 and 4: 180/11 = 16 4/11 min past 3 |
| Opposite (180°) | M = (2/11)(30H ± 180) | Between 4 and 5: (2/11)(300) = 54 6/11 min past 4 |
| At right angle (90°) | M = (2/11)(30H ± 90) | Between 2 and 3: (2/11)(150) = 27 3/11 min past 2 |
Keep only the answers that fall between 0 and 60 minutes. For a right angle you normally get two answers in one hour (for example between 4 and 5: 5 5/11 and 38 2/11 minutes past 4).
5. Mirror image of a clock
- If the time is between 1:00 and 10:59, subtract it from 11:60. Example: 3:40 → 11:60 − 3:40 = 8:20.
- If the time is between 11:00 and 12:59, subtract it from 23:60. Example: 12:20 → 23:60 − 12:20 = 11:40.
- Quick check: the real time and the mirror time always add up to 12:00 (3:40 + 8:20 = 12:00).
6. Fast and slow clocks
- Normally the hands meet every 65 5/11 minutes. If in a clock they meet every less than this (for example every 64 minutes), the clock is fast (gaining); if more, it is slow (losing).
- Example: hands meet every 64 minutes. Gain per 64 minutes = 65 5/11 − 64 = 16/11 min. Gain in a day = 16/11 × (1440 ÷ 64) = 32 8/11 minutes.
- If a clock gains 5 minutes per hour, it shows 1:05 when the correct time is 1:00, so in 12 hours it is 60 minutes ahead.
📌 Key points to remember
- Angle = |30H − 5.5M|; if more than 180°, take 360° minus it.
- Minute hand 6°/min, hour hand 0.5°/min, relative speed 5.5°/min.
- One hour space = 30°; one minute space = 6°.
- Coincide: 22 times a day; opposite: 22 times; right angle: 44 times; straight line: 44 times.
- Hands coincide every 65 5/11 minutes.
- Together between H and H+1: (60/11) × H minutes past H.
- Mirror image: subtract from 11:60 (or 23:60 for 11 and 12 o’clock).
Test yourself: Clock Problems for Lok Sewa
20 questions check what you read about Basic facts about a clock, Finding the exact time (between H and H+1 o’clock) and Fast and slow clocks.