Clocks
Clock questions for CGPSC CSAT: angle between the hands, overlaps, right angles, mirror images and fast or slow clocks, with exact formulas.
Prerequisites
- Fractions and mixed numbers: \( \frac{240}{11} = 21\frac{9}{11} \) should feel routine.
- Speed as “distance per unit time” — clock hands are just two runners on a circular track (see Basic Numeracy).
- Calendars is the companion topic in General Mental Ability.
The Hook: Two Runners on a Round Track
Look at the clock on an exam-hall wall. The minute hand sprints round the dial once an hour; the hour hand barely moves. At 12:00 they stand together. When will they meet again?
Most people say “1:05”. The real answer is \( 1{:}05\frac{5}{11} \) — a little later, because while the minute hand was catching up, the hour hand kept creeping forward.
The question this page answers: how do you find the angle between the hands at any time, the times when they meet, cross at right angles or point in opposite directions, and what a fast or slow clock really shows?
Why it matters for CGPSC: clock (घड़ी) questions in Prelims Paper II use a handful of exact formulas. Options are usually close fractions like \( 21\frac{9}{11} \) and \( 21\frac{3}{11} \), so the formula — not estimation — decides the mark.
How Fast the Hands Move
The dial is 360°, divided into 60 minute-spaces (6° each) and 12 hour-marks (30° each).
| Hand | Full turn | Speed |
|---|---|---|
| Minute hand | 60 minutes | 6° per minute |
| Hour hand | 12 hours = 720 minutes | 0.5° per minute (30° per hour) |
| Minute hand relative to hour hand | — | 5.5° per minute (55 minute-spaces gained in 60 minutes) |
The Angle Between the Hands
At \( H \) hours and \( M \) minutes:
$$\boxed{\theta = \left|\,30H - \frac{11}{2}M\,\right|}$$If \( \theta \) comes out above 180°, the smaller angle is \( 360^\circ - \theta \).
| Time | Calculation | Angle |
|---|---|---|
| 3:40 | \( \lvert 90 - 220 \rvert \) | 130° |
| 7:20 | \( \lvert 210 - 110 \rvert \) | 100° |
| 2:30 | \( \lvert 60 - 165 \rvert \) | 105° |
| 5:30 | \( \lvert 150 - 165 \rvert \) | 15° |
| 9:15 | \( \lvert 270 - 82.5 \rvert = 187.5 \) → \( 360 - 187.5 \) | 172.5° |
| 10:10 | \( \lvert 300 - 55 \rvert = 245 \) → \( 360 - 245 \) | 115° |
When Do the Hands Meet, Cross or Oppose?
The minute hand gains 5.5° (or \( \frac{11}{12} \) of a minute-space) every minute. Between \( H \) and \( H + 1 \) o’clock, the hands are \( k \) minute-spaces apart at
$$\boxed{M = (5H \pm k)\times\frac{12}{11}\ \text{minutes past } H}$$| Position | \( k \) (minute-spaces) | Example |
|---|---|---|
| Coincide (together) | 0 | Between 4 and 5: \( 20\times\frac{12}{11} = 21\frac{9}{11} \) min past 4 |
| Right angle (90°) | 15 | Between 3 and 4: \( 30\times\frac{12}{11} = 32\frac{8}{11} \) min past 3 (and exactly 3:00) |
| Opposite (180°) | 30 | Between 4 and 5: \( 50\times\frac{12}{11} = 54\frac{6}{11} \) min past 4 |
Take \( 5H + k \) and \( 5H - k \); keep only the answers between 0 and 60.
At what time between 2 and 3 o’clock are the hands at right angles?
\( 5H = 10 \), \( k = 15 \). \( (10 + 15)\times\frac{12}{11} = \frac{300}{11} = 27\frac{3}{11} \). The other value, \( (10 - 15) \), is negative, so there is only one right angle between 2 and 3: at \( 2{:}27\frac{3}{11} \).
How often in a day
The hands coincide every \( \frac{720}{11} = 65\frac{5}{11} \) minutes, not every 60.
| Event | In 12 hours | In 24 hours |
|---|---|---|
| Hands coincide | 11 | 22 |
| Hands opposite (180°) | 11 | 22 |
| Hands in a straight line (together or opposite) | 22 | 44 |
| Hands at right angles | 22 | 44 |
Mirror Images
A clock seen in a vertical mirror shows a time that, added to the real time, makes 12 hours:
$$\boxed{\text{Mirror time} = 11{:}60 - \text{actual time}}$$For times from 12:00 to 12:59, use \( 23{:}60 \) instead.
| Actual time | Mirror shows |
|---|---|
| 3:40 | 8:20 |
| 8:15 | 3:45 |
| 10:35 | 1:25 |
| 12:20 | 11:40 |
| 6:00 or 9:00 | 6:00 or 3:00 |
The rule works both ways: if the mirror shows 4:20, the actual time is \( 11{:}60 - 4{:}20 = 7{:}40 \).
Fast and Slow (Faulty) Clocks
A gaining or losing clock shows the correct time again when its total error reaches 12 hours (720 minutes) on a 12-hour dial:
$$\boxed{\text{Time to show correct time again} = \frac{720}{\text{gain or loss per hour (in minutes)}}\ \text{hours}}$$A clock gaining 5 minutes every hour shows the correct time again after \( \frac{720}{5} = 144 \) hours = 6 days.
A clock is set right at 8 a.m. It gains 10 minutes in every 24 hours. What is the true time when it shows 1 p.m. the next day?
From 8 a.m. to 1 p.m. the next day, the clock shows 29 hours. The clock runs 24 h 10 min (\( 24\frac{1}{6} \) h) for every 24 true hours, so
true time elapsed \( = 29\times\frac{24}{24\frac{1}{6}} = 28.8 \) hours \( = 28 \) h 48 min.
True time \( = \) 8 a.m. \( + \) 28 h 48 min \( = \) 12:48 p.m. the next day.
The hands of a clock coincide every 64 minutes of true time. How much does the clock gain or lose in a day?
In a correct clock they coincide every \( 65\frac{5}{11} \) minutes. This clock completes those \( 65\frac{5}{11} \) clock-minutes in only 64 true minutes, so it gains \( \frac{16}{11} \) minutes every 64 minutes.
Gain per day \( = \frac{16}{11}\times\frac{1440}{64} = \frac{360}{11} = 32\frac{8}{11} \) minutes.
Memory Tricks
- “Thirty H minus eleven-halves M” — the one formula for angles.
- “Six, half, five-and-a-half” — degrees per minute for the minute hand, hour hand and their difference.
- “12 by 11” — every meeting / right-angle / opposite time is a minute-space count multiplied by \( \frac{12}{11} \).
- “11-22-44” — 11 meetings in 12 hours, 22 in a day, and 44 right angles (or straight lines) in a day.
- Mirror: “eleven-sixty minus”.
Don’t Confuse
| This | vs | That |
|---|---|---|
| Hands coincide 22 times a day | Hands at right angles 44 times a day | |
| Hands opposite 22 times a day | Hands in a straight line (together or opposite) 44 times a day | |
| Hands meet every \( 65\frac{5}{11} \) min (correct clock) | Meeting every 64 min → clock gains | |
| Mirror time \( = 11{:}60 - \) time | Angle \( = \lvert 30H - 5.5M \rvert \) |
Common Mistakes
Practice
Level 1: Foundation
- Through how many degrees does the hour hand move in 20 minutes?
- Find the angle between the hands at 6:00 and at 3:00.
- How many times do the hands coincide in a day?
- A clock seen in a mirror shows 9:00. What is the actual time?
Answers
- \( 20\times 0.5 = 10^\circ \).
- 180° and 90°.
- 22 times.
- \( 11{:}60 - 9{:}00 = \) 3:00.
Level 2: Prelims MCQs
Q1. Match List I (time) with List II (angle between the hands):
| List I | List II |
|---|---|
| A. 3:40 | 1. 15° |
| B. 9:15 | 2. 105° |
| C. 2:30 | 3. 130° |
| D. 5:30 | 4. 172.5° |
(a) A-3, B-4, C-1, D-2 (b) A-3, B-4, C-2, D-1 (c) A-4, B-3, C-2, D-1 (d) A-2, B-4, C-3, D-1
Answer: (b) A-3, B-4, C-2, D-1
Q2. At what time between 4 and 5 o’clock will the hands of a clock coincide?
(a) \( 21\frac{9}{11} \) minutes past 4 (b) 20 minutes past 4 (c) \( 22\frac{2}{11} \) minutes past 4 (d) \( 21\frac{3}{11} \) minutes past 4
Answer: (a) 21 9/11 minutes past 4
Q3. The reflection of a wall clock in a plane mirror shows 4:20. What is the actual time?
(a) 8:40 (b) 7:20 (c) 8:20 (d) 7:40
Answer: (d) 7:40
Q4. Consider the following statements about a correctly working clock:
- The hands coincide 22 times in a day.
- The hands are at right angles 48 times in a day.
- The hands are in a straight line (together or opposite) 44 times in a day.
Which of the statements given above is/are correct?
(a) 1 only (b) 1 and 2 only (c) 1 and 3 only (d) 2 and 3 only
Answer: (c) 1 and 3 only
Q5. A clock is set right at 8 a.m. and gains 10 minutes in every 24 hours. What is the true time when it shows 1 p.m. on the following day?
(a) 12:50 p.m. (b) 12:48 p.m. (c) 12:45 p.m. (d) 12:40 p.m.
Answer: (b) 12:48 p.m.
Level 3: Speed Drill — Solve in 60 Seconds
Each item should take under 60 seconds with the formulas on this page. In the reasoning part of Mains Paper IV (GS-II), write the formula before substituting values.
- Angle between the hands at 7:20?
- Angle between the hands at 9:45?
- At what time between 2 and 3 are the hands at right angles?
- At what time between 4 and 5 are the hands opposite each other?
- Actual time 10:35 — what does the mirror show?
- A clock gains 5 minutes every hour. After how many hours will it next show the correct time?
- The hands of a clock coincide every 64 minutes. How much does the clock gain in a day?
Answers with the formula used
- 100° — \( \lvert 210 - 110 \rvert \).
- 22.5° — \( \lvert 270 - 247.5 \rvert \).
- \( 2{:}27\frac{3}{11} \) — \( (10 + 15)\times\frac{12}{11} \).
- \( 4{:}54\frac{6}{11} \) — \( (20 + 30)\times\frac{12}{11} \).
- 1:25 — \( 11{:}60 - 10{:}35 \).
- 144 hours (6 days) — \( \frac{720}{5} \).
- \( 32\frac{8}{11} \) minutes — \( \frac{16}{11}\times\frac{1440}{64} \).
Quick Revision Table
| Idea | Formula / Fact |
|---|---|
| Hand speeds | Minute 6°/min; hour 0.5°/min; relative 5.5°/min |
| Angle | \( \lvert 30H - 5.5M \rvert \); if above 180°, take \( 360 - \theta \) |
| \( k \) spaces apart between H and H+1 | \( (5H \pm k)\times\frac{12}{11} \) min past H |
| Coincide interval | \( 65\frac{5}{11} \) min |
| Per day | 22 coincide, 22 opposite, 44 right angles, 44 straight lines |
| Mirror image | \( 11{:}60 - \) time (\( 23{:}60 \) for 12:xx) |
| Correct again | \( \frac{720}{\text{gain per hour}} \) hours |
| Faster meetings than \( 65\frac{5}{11} \) min | Clock gains |
Key Takeaways
- The angle is \( \lvert 30H - 5.5M \rvert \) — the hour hand moves half a degree every minute, never ignore it.
- Every meeting, right-angle or opposite time is \( (5H \pm k)\times\frac{12}{11} \) minutes past \( H \), with \( k = 0, 15, 30 \).
- In a day: 22 meetings, 22 oppositions, 44 right angles.
- Mirror time \( = 11{:}60 - \) actual time.
- A faulty clock shows the right time again after a total error of 720 minutes; meetings faster than every \( 65\frac{5}{11} \) minutes mean the clock gains.
What’s Next
- General Mental Ability — Quick Revision — series, puzzles, calendars and clocks on one page.
- Calendars — the odd-days method for any date.
- Series and Analogies — the first topic of this chapter.
- Data Interpretation — the next CSAT chapter that rewards exact arithmetic.
- Logical Reasoning — direction sense uses the same angle thinking as clocks.