Clock Problems for Competitive Exams: Formula Guide

Clock problems appear in TCS NQT, Infosys aptitude, and AMCAT quantitative sections, testing four question types: angle between hands, hands coinciding, right-angle positions, and gain-or-loss.

Every formula for these four types follows from two facts about hand speed. This article derives each formula from those facts, then works through examples, including corrections to a widely circulated formula for right angles that produces wrong answers.

The same test paper that includes clock problems also includes coding-and-decoding questions, which test pattern-recognition under a different but related constraint. Clock problems test arithmetic precision; both reward understanding the rule rather than guessing at it.

How Both Hands Move

An analogue clock completes one full rotation (360 degrees) per cycle. The hour hand completes that rotation in 12 hours; the minute hand in 60 minutes.

Hand	Full rotation	Speed
Hour hand	12 hours = 720 minutes	360/720 = 0.5 degrees per minute
Minute hand	60 minutes	360/60 = 6 degrees per minute

The minute hand therefore gains on the hour hand at 6 - 0.5 = 5.5 degrees per minute. This relative speed is the single most important figure in clock problems. Write it down before any timed test.

The Angle Between the Hands: Formula Derivation

At time H hours and M minutes past the hour:

• Hour hand position = 0.5 × (60H + M) = 30H + 0.5M degrees from 12
• Minute hand position = 6M degrees from 12
• Angle between them = |(6M) - (30H + 0.5M)| = |5.5M - 30H|

Factoring: |5.5M - 30H| = |(11/2)M - 30H| = |30H - (11/2)M|

If the result exceeds 180 degrees, subtract from 360 to get the reflex-free angle.

The clock angle problem formalises the same derivation. The formula is standard; the value is in understanding which variables go where.

Worked Examples — Angle

• Q1: Find the angle at 3:15.

• Step 1: H = 3, M = 15

• Step 2: |30 × 3 - (11/2) × 15| = |90 - 82.5| = 7.5 degrees

• Answer: 7.5 degrees

• Q2: Find the angle at 6:40.

• Step 1: H = 6, M = 40

• Step 2: |30 × 6 - (11/2) × 40| = |180 - 220| = 40 degrees

• Answer: 40 degrees

Verification for Q2: at 6:40, the hour hand is at 30 × 6 + 0.5 × 40 = 200 degrees; the minute hand is at 6 × 40 = 240 degrees. Difference = 40 degrees. Confirmed.

When Do the Hands Overlap? The 11-Times Rule

The minute hand gains 5.5 degrees per minute on the hour hand. For them to coincide after H:00, the minute hand must close the 30H-degree gap that exists at the start of the hour.

• Time to coincide after H:00 = 30H / 5.5 = 60H / 11 minutes

For the hands to lap each other (the minute hand gains a full 360 degrees), the time required is 360 / 5.5 = 720/11 = 65 and 5/11 minutes.

In 12 hours (720 minutes), the number of overlaps = 720 / (720/11) = 11 exactly.

Some printed materials say 12 times. That is wrong. The 12th coincidence would fall exactly at 12:00:00, which marks the end of the 12-hour cycle, not an additional meeting within it.

Worked Example — Overlap

• Q: When do the hands next coincide after 2:00?
• Formula: t = 60H / 11 = 60 × 2 / 11 = 120/11 minutes past 2:00
• Convert: 120/11 = 10 and 10/11 minutes = 10 minutes and approximately 55 seconds
• Answer: Hands coincide at approximately 2:10:55

Right Angles and Straight Lines: Corrected Formulas

Right Angles (90 degrees)

Starting from H:00, with the hour hand at 30H degrees:

• At time t minutes, the angle = |5.5t - 30H|
• For 90 degrees: 5.5t - 30H = 90 gives t = (30H + 90) / 5.5 = (60H + 180) / 11
• Or: 30H - 5.5t = 90 gives t = (30H - 90) / 5.5 = (60H - 180) / 11

A formula printed in many practice books states the right-angle time as H × 60/11 plus or minus 15 minutes. The offset 15 is wrong. The correct offset is 180/11 = 16.36 minutes, not 15. This produces meaningfully different answers, as the worked example below shows.

Worked Example — Right Angle at 4:00

• Formula (correct): t = (60 × 4 + 180) / 11 = 420/11 ≈ 38.18 min, and t = (60 × 4 - 180) / 11 = 60/11 ≈ 5.45 min
• Answers: Hands at 90 degrees at 4:05:27 and 4:38:11

The wrong formula gives 240/11 = 21.8 minutes as the first answer. At 4:21.8, the angle is |5.5 × 21.8 - 120| = 0 degrees, which is the coincidence point, not a right angle. The error in the wrong formula is using the coincidence time as the base, then adding or subtracting 15 minutes instead of deriving from the gap formula directly.

Straight Lines (180 degrees)

By the same approach:

• 5.5t - 30H = 180 gives t = (60H + 360) / 11
• 30H - 5.5t = 180 gives t = (60H - 360) / 11

Only solutions where t is between 0 and 60 are valid for the given hour.

Worked Example — Straight Line at 7:00

• Formula: t = (60 × 7 - 360) / 11 = (420 - 360) / 11 = 60/11 ≈ 5.45 min
• Answer: Hands are opposite at approximately 7:05:27
• Verification: Hour hand = 210 + 0.5 × 5.45 = 212.73 degrees; minute hand = 6 × 5.45 = 32.73 degrees; difference = 180 degrees. Confirmed.

The second solution t = (60 × 7 + 360) / 11 = 780/11 ≈ 70.9 minutes falls outside the 7 to 8 hour window and is discarded.

For reference, a cricket ball trajectory calculation uses the same circular-geometry logic (arc length from angular position) that underlies both of these problems.

Fast and Slow Clocks: Gain and Loss

A clock that shows a time later than the actual time is gaining. A clock that shows an earlier time is losing. The formula for daily deviation:

• Gain or loss per day = (deviation in minutes × 1440) / observation period in minutes

Here 1440 = number of minutes in a day. The observation period is the actual elapsed time, not the time shown on the faulty clock.

Worked Example — Daily Gain

• Q: A clock gains 15 minutes in 30 hours. How much does it gain per day?
• Observation period: 30 hours = 1800 minutes
• Deviation: 15 minutes
• Gain per day: (15 × 1440) / 1800 = 21600 / 1800 = 12 minutes per day
• Answer: The clock gains 12 minutes every 24 hours.

A common variant asks for the correct time when a faulty clock reads a specific time. The method: find the true elapsed time from the gain/loss rate, then add or subtract from the last known correct setting.

Mirror Image Problems

A plane mirror reflecting a clock face reverses left and right. In practice this means: a hand shown at angle A degrees clockwise from 12 in the mirror is actually at 360 minus A degrees clockwise from 12 in reality. This subtraction is equivalent to:

• Mirror image formula: Actual time = 11:60 minus (mirror reading), when the mirror reading has non-zero minutes
• Use 12:00 minus (mirror reading) when the mirror shows an exact hour

Worked Example — Mirror Reading 4:25

• Mirror shows: 4:25
• Actual time: 11:60 minus 4:25 = 7:35
• Verification: At 7:35, hour hand = 30 × 7 + 0.5 × 35 = 227.5 degrees; in mirror, 360 - 227.5 = 132.5 degrees, corresponding to roughly the 4-hour-25-minute position. Confirmed.

Worked Example — Mirror Reading 8:10

• Mirror shows: 8:10
• Actual time: 11:60 minus 8:10 = 3:50
• Check: At 3:50, minute hand = 6 × 50 = 300 degrees; mirror shows 360 - 300 = 60 degrees = 10 minutes. Hour hand = 30 × 3 + 0.5 × 50 = 115 degrees; mirror shows 360 - 115 = 245 degrees, corresponding to approximately 8:10. Confirmed.

The IndiaBix clock problems page has a full practice set organised by difficulty, which is useful once you have the formula derivations clear.

The precision that separates a derived formula from a memorised one is the same habit that placement interviewers now test in AI tool-building contexts: knowing why the relative speed is 5.5 degrees per minute, and how the 11-meeting rule follows from it. TinkerLLM at ₹299 applies that same first-principles approach to language models, where you build and verify small tools rather than copying prompts from a tutorial.