Clocks questions with answers and concepts

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Clock problems are a common topic in aptitude tests that involve calculating angles between clock hands, determining times when hands are in specific positions (overlap, opposite, perpendicular), and solving problems related to fast or slow clocks. Understanding the movement rates of hour and minute hands is essential for solving these problems efficiently.

Key formulas and rules

9 rules

Hour hand rate: $\frac{36 0 ^{\circ}}{12 hours} = 3 0^{\circ} / hour = 0. 5^{\circ} / minute$

Minute hand rate: $\frac{36 0 ^{\circ}}{60 minutes} = 6^{\circ} / minute$

Relative speed: $6^{\circ} - 0. 5^{\circ} = 5. 5^{\circ} / min = \frac{11}{2}^{\circ} / min$

Angle between hands at $H$ hours and $M$ minutes: $θ = 30 H - \frac{11}{2} M^{\circ}$

Reflex angle correction: If $θ > 18 0^{\circ}$ , internal angle is $36 0^{\circ} - θ$

Hands coincide (coincident): Every $65 \frac{5}{11} minutes$ (11 times in 12 hours, 22 in 24 hours)

Hands in opposite direction ( $18 0^{\circ}$ ): 11 times in 12 hours (22 in 24 hours)

Hands at right angle ( $9 0^{\circ}$ ): 22 times in 12 hours (44 in 24 hours)

Clock error formula: $True Time = Observed Time \pm Correction$

Key concepts

Understanding Clock Hand Movements

The hour hand completes one full rotation (360 deg) in 12 hours, moving at 30 deg per hour or 0.5 deg per minute. The minute hand completes one full rotation in 60 minutes, moving at 6 deg per minute. The second hand moves at 6 deg per second. The relative speed between minute and hour hands is crucial - the minute hand gains 5.5 deg per minute over the hour hand.

Calculating Angle Between Hands

To find the angle at H hours M minutes, use the formula: |30H - 5.5M|. This gives the angle measured clockwise from the hour hand to the minute hand. If the result exceeds 180 deg, subtract it from 360 deg to get the smaller angle between the two hands. For example, at 3:40, the angle is |30x3 - 5.5x40| = |90 - 220| = 130 deg.

Finding Times When Hands Are in Special Positions

Hands overlap when the angle is 0 deg, occur 11 times in 12 hours (every 65 5/11 minutes). They are opposite (180 deg apart) 11 times in 12 hours. They are perpendicular (90 deg apart) 22 times in 12 hours. To find when hands form a specific angle , use: (30H +/- ) / 5.5 = minutes past H o'clock. The +/- depends on whether the minute hand is ahead (+) or behind (-) the hour hand.

Fast and Slow Clock Problems

A fast clock gains time, a slow clock loses time. If a clock gains/loses x minutes per hour, it shows incorrect time. To find correct time: Calculate total time elapsed, determine total gain/loss, add or subtract from shown time. For example, if a clock gains 5 minutes per hour and shows 4:00 after 6 real hours, the actual time is 3:30 (gained 30 minutes).

Mirror Image of Clock Time

The mirror image of a clock time can be found by subtracting the time from 12:00 (or 11:60 for easier calculation). For time H:M, mirror time = (11 - H):(60 - M). If minutes exceed 60, borrow 1 hour. For example, mirror of 3:20 is 8:40. If the result is 0 hours, use 12. This works because the clock face is symmetric about the vertical 12-6 line.

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Worked examples

Example 1

What is the angle between the hour hand and the minute hand of a clock at 4:40?

Solution

Step 1: Apply the standard clock angle formula.

θ = 30 H - \frac{11}{2} M

Step 2: Substitute $H = 4$ and $M = 40$ .

θ = 30 (4) - \frac{11}{2} (40) = ∣120 - 220∣ = ∣ - 100∣ = 10 0^{\circ}

Answer: 100°.

Example 2

At what exact time between 3:00 and 4:00 will the hour and minute hands of a clock coincide?

Solution

Step 1: Determine initial separation at 3:00.
At 3:00, the minute hand is at $0^{\circ}$ and the hour hand is at $3 \times 3 0^{\circ} = 9 0^{\circ}$ .

Step 2: Calculate relative speed.

v_{rel} = 6^{\circ} / min - 0. 5^{\circ} / min = 5. 5^{\circ} / min = \frac{11}{2}^{\circ} / min

Step 3: Compute catch-up time $M$ .

M = \frac{90}{\frac{11}{2}} = \frac{180}{11} = 16 \frac{4}{11} minutes

Answer: At $3 h 16 \frac{4}{11} m$ (or $16 \frac{4}{11}$ minutes past 3 o'clock).

Example 3

A watch gains 5 minutes per hour. It was synchronized correctly at 9:00 AM. What is the true time when the watch reads 6:30 PM on the same day?

Solution

Step 1: Total elapsed clock time on the fast watch.
From 9:00 AM to 6:30 PM is 9 hours and 30 minutes = 570 minutes.

Step 2: Ratio of true time to fast watch time.
In 60 true minutes, the watch advances 65 minutes:

True Time = 570 \times \frac{60}{65} = 570 \times \frac{12}{13} = \frac{6 , 840}{13} \approx 526.15 minutes = 8 hours 46.15 minutes

Step 3: Advance 9:00 AM by 8 hours 46 minutes.

True Time = 5 : 46 PM

Answer: Approximately 5:46 PM.

Example 4

What is the mirror image time corresponding to 7:25 on a 12-hour analog clock?

Solution

Step 1: Apply the mirror reflection transformation rule.

Reflected Time = 11 : 60 - H : M

Step 2: Substitute $H = 7$ and $M = 25$ .

Hours = 11 - 7 = 4

Minutes = 60 - 25 = 35

Answer: 4:35.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What is the smaller angle between the hands of a clock at 12:05?

12.5 deg
2.5 deg
17.5 deg
27.5 deg

Answer: D. 27.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 12:05 is 27.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What is the smaller angle between the hands of a clock at 1:05?

42.5 deg
12.5 deg
2.5 deg
27.5 deg

Answer: C. 2.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 1:05 is 2.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What is the smaller angle between the hands of a clock at 2:05?

2.5 deg
32.5 deg
17.5 deg
12.5 deg

Answer: B. 32.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 2:05 is 32.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always draw a rough sketch of the clock face to visualize the positions of hour and minute hands - this helps avoid sign errors in calculations.
Remember that the relative speed of 5.5 deg/min comes from (6 deg - 0.5 deg) - the minute hand gains on the hour hand.
For angle problems, if your calculation gives more than 180 deg, subtract from 360 deg to get the smaller angle between the two hands.
When finding specific times (overlap, opposite, perpendicular), note that these occur at regular intervals - hands overlap every 65 5/11 minutes starting from 12:00.
For fast/slow clock problems, use the ratio method: if a clock gains x min per hour, then (60+x) minutes of wrong time = 60 minutes of correct time.
In mirror image problems, remember that left becomes right and vice versa - the sum of the original time and mirror time equals 12:00 (or 11:60 for easier calculation).
For hands perpendicular problems, there are two solutions in each hour (except when the angle condition is at the hour boundary).

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Key concepts

Understanding Clock Hand Movements

Calculating Angle Between Hands

Finding Times When Hands Are in Special Positions

Fast and Slow Clock Problems

Mirror Image of Clock Time

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

What is the angle between the hour hand and the minute hand of a clock at 4:40?

Solution

Step 1: Apply the standard clock angle formula.

θ = 30 H - \frac{11}{2} M

Step 2: Substitute $H = 4$ and $M = 40$ .

θ = 30 (4) - \frac{11}{2} (40) = ∣120 - 220∣ = ∣ - 100∣ = 10 0^{\circ}

Answer: 100°.

Example 2

At what exact time between 3:00 and 4:00 will the hour and minute hands of a clock coincide?

Solution

Step 1: Determine initial separation at 3:00.
At 3:00, the minute hand is at $0^{\circ}$ and the hour hand is at $3 \times 3 0^{\circ} = 9 0^{\circ}$ .

Step 2: Calculate relative speed.

v_{rel} = 6^{\circ} / min - 0. 5^{\circ} / min = 5. 5^{\circ} / min = \frac{11}{2}^{\circ} / min

Step 3: Compute catch-up time $M$ .

M = \frac{90}{\frac{11}{2}} = \frac{180}{11} = 16 \frac{4}{11} minutes

Answer: At $3 h 16 \frac{4}{11} m$ (or $16 \frac{4}{11}$ minutes past 3 o'clock).

Example 3

A watch gains 5 minutes per hour. It was synchronized correctly at 9:00 AM. What is the true time when the watch reads 6:30 PM on the same day?

Solution

Step 1: Total elapsed clock time on the fast watch.
From 9:00 AM to 6:30 PM is 9 hours and 30 minutes = 570 minutes.

Step 2: Ratio of true time to fast watch time.
In 60 true minutes, the watch advances 65 minutes:

True Time = 570 \times \frac{60}{65} = 570 \times \frac{12}{13} = \frac{6 , 840}{13} \approx 526.15 minutes = 8 hours 46.15 minutes

Step 3: Advance 9:00 AM by 8 hours 46 minutes.

True Time = 5 : 46 PM

Answer: Approximately 5:46 PM.

Example 4

What is the mirror image time corresponding to 7:25 on a 12-hour analog clock?

Solution

Step 1: Apply the mirror reflection transformation rule.

Reflected Time = 11 : 60 - H : M

Step 2: Substitute $H = 7$ and $M = 25$ .

Hours = 11 - 7 = 4

Minutes = 60 - 25 = 35

Answer: 4:35.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

What is the smaller angle between the hands of a clock at 12:05?

12.5 deg
2.5 deg
17.5 deg
27.5 deg

Answer: D. 27.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 12:05 is 27.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What is the smaller angle between the hands of a clock at 1:05?

42.5 deg
12.5 deg
2.5 deg
27.5 deg

Answer: C. 2.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 1:05 is 2.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What is the smaller angle between the hands of a clock at 2:05?

2.5 deg
32.5 deg
17.5 deg
12.5 deg

Answer: B. 32.5 deg

Explanation

Using hour angle = 30h + 0.5m and minute angle = 6m, the smaller angle at 2:05 is 32.5 deg.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always draw a rough sketch of the clock face to visualize the positions of hour and minute hands - this helps avoid sign errors in calculations.
Remember that the relative speed of 5.5 deg/min comes from (6 deg - 0.5 deg) - the minute hand gains on the hour hand.
For angle problems, if your calculation gives more than 180 deg, subtract from 360 deg to get the smaller angle between the two hands.
When finding specific times (overlap, opposite, perpendicular), note that these occur at regular intervals - hands overlap every 65 5/11 minutes starting from 12:00.
For fast/slow clock problems, use the ratio method: if a clock gains x min per hour, then (60+x) minutes of wrong time = 60 minutes of correct time.
In mirror image problems, remember that left becomes right and vice versa - the sum of the original time and mirror time equals 12:00 (or 11:60 for easier calculation).
For hands perpendicular problems, there are two solutions in each hour (except when the angle condition is at the hour boundary).

Key formulas and rules

Key concepts

Understanding Clock Hand Movements

Calculating Angle Between Hands

Finding Times When Hands Are in Special Positions

Fast and Slow Clock Problems

Mirror Image of Clock Time

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Understanding Clock Hand Movements

Calculating Angle Between Hands

Finding Times When Hands Are in Special Positions

Fast and Slow Clock Problems

Mirror Image of Clock Time

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.