Clock Problems Questions for Placement Exams - Complete Question Bank

Last Updated: March 2026

Introduction to Clock Problems

Clock Problems are a fascinating and important topic in quantitative aptitude that test your understanding of angles, relative speed, and time calculations. These problems appear frequently in placement exams and competitive examinations due to their practical nature and the logical thinking they require.

Why are Clock Problems Important?

Regular Appearance: 2-4 questions typically appear in every exam
Conceptual Clarity: Tests understanding of relative motion and angles
Quick Solving: Most problems can be solved in 30-60 seconds with formulas
High Scoring: Formula-based approach ensures consistent accuracy
Real-World Application: Time and angle calculations are practical skills

Types of Clock Problems

Angle Between Hands: Finding the angle between hour and minute hands
Coincidence of Hands: When do the hands overlap
Opposite Hands: When are hands 180° apart
Right Angle: When are hands perpendicular (90°)
Fast/Slow Clocks: Clocks gaining or losing time
Time Calculation: Finding time based on hand positions
Mirror Time: Finding actual time from mirror image

Important Formulas and Concepts

Basic Clock Facts

- Clock face: 360° circle
- 12 hours marked on dial
- Each hour mark = 360°/12 = 30° apart
- Each minute mark = 360°/60 = 6° apart
- Hour hand completes 360° in 12 hours
- Minute hand completes 360° in 60 minutes

Speed of Hands

Minute Hand:
- Speed = 360° per 60 minutes = 6° per minute
- Speed = 6°/min

Hour Hand:
- Speed = 360° per 12 hours = 30° per hour
- Speed = 30°/60 min = 0.5° per minute
- Speed = 0.5°/min

Relative Speed (Minute - Hour):
- Relative Speed = 6° - 0.5° = 5.5° per minute
- Relative Speed = 5.5°/min or 11/2 °/min

Angle Formula

Angle between hands at H hours M minutes:

θ = |30H - 5.5M|

Or equivalently:
θ = |30H - (11/2)M|

If θ > 180°, take 360° - θ for the smaller angle.

Special Positions

Hands Coincide (0° angle):
- Occurs 11 times in 12 hours
- Interval between coincidences = 12/11 hours = 65 5/11 minutes
- Times: 12:00, ~1:05, ~2:10, ~3:16, ~4:21, ~5:27, ~6:32, ~7:38, ~8:43, ~9:49, ~10:54

Hands Opposite (180° angle):
- Occurs 11 times in 12 hours
- Interval = 12/11 hours
- First after 12:00 is at ~12:32

Hands at Right Angle (90°):
- Occurs 22 times in 12 hours
- Interval = 12/22 hours = 32 8/11 minutes between consecutive right angles

Fast and Slow Clocks

If a clock gains/loses x minutes per hour:
- In 1 hour of correct time, it shows (60 ± x) minutes
- The ratio of faulty to correct time = (60 ± x) : 60

To find correct time when faulty clock shows a time:
Correct Time = Faulty Time × (60 / (60 ± x))

Mirror Image of Clock

Mirror Time Formula:
If actual time is H:M, mirror shows:
- For H = 11 or 12: (11-H):(60-M) with adjustments
- General: (11-H):(60-M) if M ≠ 0, or (12-H):0 if M = 0

Simpler method:
Mirror Time = 12:00 - Actual Time

Example: Actual = 3:40, Mirror = 12:00 - 3:40 = 8:20

Practice Questions

Set 1: Angle Between Hands

Question 1 (Easy): What is the angle between the hour hand and minute hand at 3:00?

Solution: Using formula: θ = |30H - 5.5M| At 3:00: H = 3, M = 0 θ = |30×3 - 5.5×0| = |90 - 0| = 90°

Question 2 (Easy): Find the angle between the clock hands at 6:30.

Solution: H = 6, M = 30 θ = |30×6 - 5.5×30| = |180 - 165| = 15°

Question 3 (Medium): What is the angle between the hands at 4:40?

Solution: H = 4, M = 40 θ = |30×4 - 5.5×40| = |120 - 220| = |-100| = 100°

Since 100° < 180°, this is the smaller angle.

Question 4 (Medium): At what time between 2 and 3 o'clock will the hands be at 30° to each other?

Solution: We need θ = 30° |30×2 - 5.5M| = 30 |60 - 5.5M| = 30

Two cases: Case 1: 60 - 5.5M = 30 5.5M = 30 M = 30/5.5 = 60/11 = 5 5/11 minutes

Case 2: 60 - 5.5M = -30 5.5M = 90 M = 90/5.5 = 180/11 = 16 4/11 minutes

Question 5 (Hard): Find the reflex angle between the hands at 10:25.

Solution: H = 10, M = 25 θ = |30×10 - 5.5×25| = |300 - 137.5| = 162.5°

Reflex angle = 360° - 162.5° = 197.5°

Set 2: Coincidence of Hands

Question 6 (Easy): At what time between 1 and 2 o'clock do the hands coincide?

Solution: For coincidence, angle = 0° |30×1 - 5.5M| = 0 30 - 5.5M = 0 5.5M = 30 M = 30/5.5 = 60/11 = 5 5/11 minutes

Question 7 (Medium): How many times do the hands of a clock coincide in a day?

Solution: Hands coincide every 12/11 hours. In 12 hours, they coincide 11 times (not 12, because the 12th coincidence is at the start of the next cycle). In 24 hours, they coincide 22 times.

Question 8 (Medium): At what time between 4 and 5 o'clock will the hands point in opposite directions?

Solution: For opposite, angle = 180° |30×4 - 5.5M| = 180 |120 - 5.5M| = 180

Case 1: 120 - 5.5M = 180 -5.5M = 60 M = -60/5.5 (negative, not valid)

Case 2: 120 - 5.5M = -180 -5.5M = -300 M = 300/5.5 = 600/11 = 54 6/11 minutes

Question 9 (Hard): The hands of a clock are together at 12:00. When will they next form a straight line (180°)?

Solution: From coincidence to opposite is half the cycle. Interval between coincidences = 12/11 hours Interval to opposite = (12/11)/2 = 6/11 hours = 360/11 minutes = 32 8/11 minutes

Question 10 (Hard): How many times in a day are the hands of a clock in a straight line (either coinciding or opposite)?

Solution: Coincidence: 22 times per day Opposite: 11 times in 12 hours × 2 = 22 times per day

But wait, let's verify:

Coincide: 22 times
Opposite: 22 times
Total straight line positions = 44 times

Actually, in each 12-hour period, hands are straight 11 times coinciding + 11 times opposite = 22 times. In 24 hours: 44 times.

Set 3: Right Angle Positions

Question 11 (Easy): At what time between 3 and 4 o'clock will the hands be at right angles?

Solution: For right angle, θ = 90° |30×3 - 5.5M| = 90 |90 - 5.5M| = 90

Case 1: 90 - 5.5M = 90 5.5M = 0 M = 0 → 3:00 (valid, but at start of hour)

Case 2: 90 - 5.5M = -90 5.5M = 180 M = 180/5.5 = 360/11 = 32 8/11 minutes

Question 12 (Medium): How many times are the hands at right angles in 12 hours?

Solution: Hands are at right angles 22 times in 12 hours. They form 90° angle twice in each hour (approximately), but in the 12-hour cycle, there are 22 occurrences.

More precisely: In 12 hours, the hands are perpendicular 22 times.

Question 13 (Medium): Find the times between 5 and 6 o'clock when the hands are perpendicular.

Solution: |30×5 - 5.5M| = 90 |150 - 5.5M| = 90

Case 1: 150 - 5.5M = 90 5.5M = 60 M = 60/5.5 = 120/11 = 10 10/11 minutes

Case 2: 150 - 5.5M = -90 5.5M = 240 M = 240/5.5 = 480/11 = 43 7/11 minutes

Question 14 (Hard): A clock strikes 4, and the hands are at right angles. How many minutes will pass before they are at right angles again?

Solution: At 4:00, hands are at 120° (not 90°). So this is between 4 and 5.

First right angle after 4:00: From Q13, at 4:43 7/11

Wait, the question says "strikes 4" meaning 4:00. But at 4:00, angle is 120°, not 90°.

Let me re-read: "A clock strikes 4" - this could mean during the 4th hour (between 4 and 5).

Actually, let's assume the question means at some time during the 4 o'clock hour, hands are perpendicular, and we need time until next perpendicular.

Interval between consecutive right angles = 12/22 hours = 32 8/11 minutes.

Question 15 (Hard): How many times in a day do the hands form an angle of 30°?

Solution: In each 12-hour period, for each angle θ (except 0° and 180°), the hands form that angle 22 times.

For 30°: 22 times in 12 hours. In 24 hours: 44 times.

Actually, verification: For any angle θ where 0 < θ < 180, hands form that angle 22 times in 12 hours (once going in each direction).

Set 4: Fast and Slow Clocks

Question 16 (Easy): A watch gains 5 minutes per hour. If it is set correctly at 12 noon, what time will it show at 6:00 PM (actual time)?

Solution: Actual time elapsed = 6 hours Watch gains 5 minutes per hour. Total gain = 6 × 5 = 30 minutes

Watch shows: 6:00 + 0:30 = 6:30 PM

Question 17 (Medium): A clock loses 3 minutes every hour. If it shows 4:00 PM when the correct time is 5:00 PM, when was it set correctly?

Solution: Clock loses 3 minutes per hour. For every 60 minutes actual, clock shows 57 minutes.

Actual time is 5:00 PM, clock shows 4:00 PM. Difference = 1 hour = clock is 60 minutes slow.

Time to lose 60 minutes at 3 min/hour = 60/3 = 20 hours.

Clock was set correctly 20 hours ago. 5:00 PM - 20 hours = 9:00 PM (previous day)

Question 18 (Medium): A watch which gains uniformly is 5 minutes slow at 8:00 AM on Sunday and is 5 minutes fast at 8:00 PM on the following Sunday. When was it correct?

Solution: Total time = 7 days = 168 hours Total gain = 5 min (slow to correct) + 5 min (correct to fast) = 10 minutes

Rate of gain = 10/168 minutes per hour

To go from 5 min slow to correct: needs to gain 5 minutes. Time = 5 / (10/168) = 5 × 168/10 = 84 hours = 3.5 days

8:00 AM Sunday + 3.5 days = 8:00 PM Wednesday

Question 19 (Hard): A clock gains 10 minutes every 3 hours. If it is set right at 6:00 AM, what is the true time when the clock shows 3:00 PM?

Solution: Clock gains 10 min per 3 hours = 10/3 min per hour.

From 6:00 AM to 3:00 PM on clock = 9 hours shown on clock.

Ratio: For every 60 + 10/3 = 190/3 minutes on clock, actual time = 60 minutes.

Clock shows 9 hours = 540 minutes. Actual time = 540 × (60) / (190/3) = 540 × 180 / 190 = 97200/190 = 511.58 minutes ≈ 8.53 hours

Actual time = 6:00 AM + 8 hours 32 minutes = 2:32 PM

Question 20 (Hard): Two clocks start together. One gains 2 minutes per hour, the other loses 3 minutes per hour. After how many hours will they show the same time again?

Solution: Clock A gains 2 min/hour → shows 62 min for every 60 actual Clock B loses 3 min/hour → shows 57 min for every 60 actual

Difference in rates = 2 + 3 = 5 minutes per hour

For them to show same time, difference must be 12 hours = 720 minutes (full cycle).

Time = 720 / 5 = 144 hours = 6 days

Set 5: Time Calculations

Question 21 (Easy): At what time between 7 and 8 o'clock will the minute hand be 5 minutes ahead of the hour hand?

Solution: Minute hand gains 5.5 minutes (in minute marks) per minute on hour hand. At 7:00, minute hand is 35 minutes behind hour hand.

We want minute hand 5 minutes ahead. Total gain needed = 35 + 5 = 40 minutes.

Time = 40 / 5.5 = 80/11 = 7 3/11 minutes past 7.

Question 22 (Medium): How many minutes is the minute hand ahead of the hour hand at 5:24?

Solution: Position of hour hand: 5 + 24/60 = 5.4 hours = 5 hours 24 minutes mark Position of minute hand: 24 minutes mark

In minute marks (each number = 5 minutes): Hour hand at: 5.4 × 5 = 27 minute marks Minute hand at: 24 minute marks

Minute hand is actually 3 minute marks BEHIND, not ahead.

Wait, the question assumes minute hand is ahead. Let me recalculate differently.

At 5:24: Hour hand: 5 × 5 + 24 × (1/12) = 25 + 2 = 27 minutes (on the minute scale) Minute hand: 24 minutes

Difference = 27 - 24 = 3 minutes (hour hand ahead)

If the question states minute hand is ahead, the time might be different.

Alternatively, perhaps the question means something else.

Actually, re-reading: "How many minutes is the minute hand ahead" - if it's behind, answer is negative or re-interpret.

At 5:24, hour hand is 3 minutes ahead, so minute hand is 3 minutes behind.

Question 23 (Medium): The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much does the clock gain or lose per day?

Solution: Normal interval between overtakes = 65 5/11 minutes = 720/11 minutes.

Given interval = 65 minutes = 715/11 minutes.

The clock is faster since overtakes happen sooner.

In 65 minutes of correct time, the clock completes one full relative cycle.

Gain per 65 minutes = 65 5/11 - 65 = 5/11 minutes.

Gain per day (1440 minutes): Gain = (5/11) × (1440/65) = (5/11) × (288/13) = 1440/143 minutes ≈ 10.07 minutes

Question 24 (Hard): At what time between 8 and 9 o'clock will the two hands be equidistant from 8 on the clock face but not together?

Solution: At 8:00, hour hand is at 8, minute hand is at 12. We want both hands equidistant from 8 mark.

Let minute hand be at M minutes. Position: M/5 (in hour marks from 12) Distance from 8: |M/5 - 8| or considering wraparound.

Hour hand position: 8 + M/60 Distance from 8: M/60

For equidistant: M/60 = |M/5 - 8| (considering minute hand is before 8)

Case: Minute hand before 8 (M < 40): M/60 = 8 - M/5 M/60 + M/5 = 8 M/60 + 12M/60 = 8 13M/60 = 8 M = 480/13 = 36 12/13 minutes

Question 25 (Hard): A clock is set right at 8:00 AM. The clock gains 10 minutes in 24 hours. What will be the true time when the clock indicates 1:00 PM on the following day?

Solution: Clock shows: From 8:00 AM to 1:00 PM next day = 29 hours

Clock gains 10 minutes in 24 hours. So for every 24 hours actual, clock shows 24 + 10/60 = 24.1667 hours.

Ratio: Clock time / Actual time = (24 + 1/6) / 24 = 145/144

If clock shows 29 hours, actual time = 29 × 144/145 = 28.8 hours = 28 hours 48 minutes.

8:00 AM + 28 hours 48 minutes = 12:48 PM (following day)

Set 6: Mirror Time

Question 26 (Easy): What is the mirror image of 3:00 on a clock?

Solution: Mirror time = 12:00 - 3:00 = 9:00

Or use formula: (11-H):(60-M) for H < 11 (11-3):(60-0) = 8:60 = 9:00

Question 27 (Medium): A clock shows a time in a mirror as 4:20. What is the actual time?

Solution: Actual time = 12:00 - 4:20 = 7:40

Or: (11-4):(60-20) = 7:40

Question 28 (Medium): The mirror image of a clock shows 9:45. What is the actual time?

Solution: Actual = 12:00 - 9:45 = 2:15

Or: (11-9):(60-45) = 2:15

Question 29 (Hard): If the time in a mirror appears to be 12:30, what is the actual time?

Solution: Using formula: For 12:30, special case. Mirror of 12:30 = 11:30 (symmetric about 6-12 line)

Or calculation: 12:00 - 12:30 doesn't work directly.

(11-12):(60-30) = (-1):30 → Not valid.

For times involving 12, convert to 0 or use: (12-H-1):(60-M) when M > 0

(11-0):(60-30) = 11:30 for H=12 case? Actually use 0 for 12.

Actually: Mirror of 12:30: Visualize: At 12:30, hour hand is halfway between 12 and 1, minute hand at 6. Mirror image: hour hand halfway between 12 and 11 (i.e., at 11:30 position), minute hand at 6. So mirror shows what looks like 11:30, meaning actual is 11:30? No wait...

If mirror shows 12:30, actual is: 12:00 - 12:30 would be negative. Actual = 11:30

Question 30 (Hard): A man looks at a clock through a mirror and sees the time as 6:15. He accidentally looks at the clock directly later and realizes he is 30 minutes late for a meeting scheduled at the mirror time. What time is it actually when he looks directly?

Solution: Mirror shows 6:15 Actual time when he looked = 12:00 - 6:15 = 5:45

Meeting was scheduled at mirror time = 6:15 actual? No...

"Meeting scheduled at the mirror time" - this means the time he thought it was (6:15) was the scheduled time.

He is 30 minutes late, so actual time now = 6:15 + 30 = 6:45

Wait, but he looked at mirror and saw 6:15 when actual was different.

Let me re-read: "sees the time as 6:15" (in mirror), "meeting scheduled at the mirror time" (meaning at the time shown in mirror = 6:15), "30 minutes late".

So he arrives when actual time is such that he's 30 minutes past 6:15. Actual arrival time = 6:45.

But when he looked in mirror and saw 6:15, what was actual time? Mirror 6:15 → Actual = 12:00 - 6:15 = 5:45

So at 5:45 actual, mirror showed 6:15. He is 30 minutes late for 6:15 meeting, so he arrives at 6:45.

Time elapsed = 6:45 - 5:45 = 1 hour.

Companies and Exams That Frequently Ask Clock Problems

Campus Placement Exams

TCS NQT: 2-3 clock questions
Infosys: 2-4 questions (angles, fast/slow)
Wipro: 2-3 questions (basic angles)
Cognizant: 3-4 questions (mixed)
HCL: 2-3 questions
Accenture: 3-5 questions (comprehensive)
IBM: 2-4 questions
Capgemini: 2-3 questions

Government Exams

IBPS PO/Clerk: 2-4 questions
SBI PO/Clerk: 2-3 questions
SSC CGL: 2-4 questions
SSC CHSL: 1-3 questions
Railway Exams: 2-4 questions

Preparation Tips for Clock Problems

1. Memorize the Speed Values

Minute hand: 6° per minute
Hour hand: 0.5° per minute
Relative speed: 5.5° per minute

2. Master the Angle Formula

θ = |30H - 5.5M|
Use this for all angle calculations
Take the smaller angle if result > 180°

3. Understand Relative Motion

Think of it as a race where minute hand chases hour hand
Relative speed determines when they meet
5.5° per minute is key

4. Practice Common Times

Memorize coincidence times: ~1:05, ~2:10, ~3:16, etc.
Right angles occur approximately every 32 minutes
Opposite hands at ~12:32, ~1:38, etc.

5. Fast/Slow Clock Formula

Understand the ratio approach
If clock gains x min per hour, ratio = (60+x):60
Apply carefully to find correct/faulty time

6. Mirror Time Shortcut

Mirror time = 12:00 - Actual time
Or use (11-H):(60-M) formula
Visualize the clock for verification

7. Check Your Answers

Verify angles make sense (0° to 180°)
Ensure times are within valid ranges
Double-check AM/PM distinctions

Frequently Asked Questions (FAQ)

Q1: Why do hands coincide 11 times in 12 hours, not 12?

A: The hands coincide every 65 5/11 minutes, not every hour. In 12 hours, they coincide at 12:00, ~1:05, ~2:10, etc., but the 11th coincidence is just before 1:00 (actually at ~12:00 next cycle). The 12th would be at the next 12:00, which starts a new 12-hour cycle.

Q2: How do I remember the angle formula?

A: Hour hand moves 30° per hour and 0.5° per minute. Minute hand moves 6° per minute. The difference gives |30H + 0.5M - 6M| = |30H - 5.5M|.

Q3: What's the fastest way to solve coincidence problems?

A: Use the fact that coincidences happen every 12/11 hours. For nth coincidence after 12:00, time = n × 12/11 hours past 12.

Q4: How do I handle fast/slow clock problems?

A: Find the ratio of faulty time to correct time. If clock gains 5 min/hour, it shows 65 min for every 60 actual. Use this ratio to convert between shown and actual times.

Q5: Why does the mirror time formula work?

A: The mirror image reflects the clock across the 12-6 line. The sum of actual and mirror positions equals 12 hours (or the clock face becomes symmetric).

Quick Reference: Key Formulas

Concept	Formula
Angle at H:M	\|30H - 5.5M\|
Hands coincide every	12/11 hours = 65 5/11 minutes
Hands opposite every	12/11 hours (alternate with coincide)
Hands at 90° every	12/22 hours = 32 8/11 minutes
Mirror time	12:00 - Actual Time
Fast clock ratio	(60 + gain) : 60
Slow clock ratio	(60 - loss) : 60

Master clock problems through formula application and practice!

Clock Problems Questions Placement