Clock and Time

PAGE 1 — BASIC CONCEPT OF CLOCK

A clock is a circular instrument used to measure time. In aptitude mathematics, clock problems mainly involve the positions of the hour hand and minute hand.

Three important quantities:

Hour hand: shorter hand showing hours.
Minute hand: longer hand showing minutes.
Second hand: moves once around the clock every minute.

1. CLOCK FACE

A complete clock has 12 equal divisions. Therefore, the angle between two consecutive numbers is:

Angle between two numbers = 360° ÷ 12 = 30°

The complete circular movement of either hand is 360°.

2. SPEED OF CLOCK HANDS

Hand	One complete revolution	Speed
Minute hand	60 minutes	6° per minute
Hour hand	12 hours = 720 minutes	0.5° per minute
Second hand	60 seconds	6° per second

Minute hand = 6° per minute
Hour hand = 0.5° per minute

3. POSITION OF MINUTE HAND

If the minute hand is at m minutes, its angle from 12 is:

Minute-hand angle = 6m°

Example: Find the angle made by the minute hand at 25 minutes.

Angle = 6 × 25 = 150°

4. POSITION OF HOUR HAND

The hour hand does not remain fixed at a number during an hour. It moves continuously as the minutes pass.

Hour-hand angle = 30H + 0.5M

Where H = hour and M = minutes.

Example: Find the position of the hour hand at 4:20.

= 30(4) + 0.5(20)
= 120 + 10
= 130°

TIP: Never use only 30H for the hour hand when minutes are present. Always add the movement caused by the minutes: 0.5M.

PAGE 2 — ANGLE FORMULAS

5. ANGLE BETWEEN HOUR & MINUTE HAND

The angle between the two hands is the difference between their positions.

Angle = |30H + 0.5M − 6M|

Angle = |30H − 5.5M|

If the calculated angle is greater than 180°, the smaller angle is:

Smaller angle = 360° − Calculated angle

Example 1: Find the angle between the hands at 3:00.

Angle = |30(3) − 5.5(0)|
= 90°

Answer: 90°

Example 2: Find the angle at 5:20.

Angle = |30(5) − 5.5(20)|
= |150 − 110|
= 40°

6. IMPORTANT ANGLES AT EXACT HOURS

Time	Angle
1:00	30°
2:00	60°
3:00	90°
4:00	120°
5:00	150°
6:00	180°
7:00	150°
8:00	120°
9:00	90°
10:00	60°
11:00	30°
12:00	0°

7. WHEN DO THE HANDS COINCIDE?

When the hour and minute hands overlap, their angle is 0°.

|30H − 5.5M| = 0
30H = 5.5M
M = 60H ÷ 11

Example: Between 4 and 5, when do the hands coincide?

M = (60 × 4) ÷ 11
= 240 ÷ 11
= 21 9/11 minutes

Therefore, the hands coincide at approximately 4:21:49.

8. WHEN ARE THE HANDS 180° APART?

For opposite hands, the angle is 180°.

|30H − 5.5M| = 180

Example: Between 6 and 7, when are the hands opposite?

|180 − 5.5M| = 180
The valid solution in this interval is:
M = 0

Thus at 6:00, the hands are exactly opposite.

Shortcut: In a 12-hour period, the hour and minute hands coincide 11 times, not 12 times.

PAGE 3 — ADVANCED CLOCK FORMULAS

9. TIME WHEN A GIVEN ANGLE IS FORMED

If the angle between the hands is θ, then:

|30H − 5.5M| = θ

Therefore:

M = (30H ± θ) ÷ 5.5

Since 5.5 = 11/2:

M = 2(30H ± θ) ÷ 11

Example: Between 2 and 3, when will the hands be 60° apart?

|30(2) − 5.5M| = 60
|60 − 5.5M| = 60

One solution is M = 0, giving 2:00.
The other solution:
5.5M = 120
M = 120/5.5
= 21 9/11 minutes.

10. COINCIDENCE OF HANDS

Time between two consecutive coincidences = 65 5/11 minutes

In 12 hours → 11 coincidences
In 24 hours → 22 coincidences

11. RIGHT ANGLE

A right angle means the hands are 90° apart.

|30H − 5.5M| = 90

In 12 hours, the hands form a right angle 22 times. Consequently, in 24 hours they form a right angle 44 times.

12. CLOCK LOSING OR GAINING TIME

Some questions describe a clock that gains or loses a certain number of minutes in a given period.

Gain/Loss per hour = Total Gain/Loss ÷ Number of Hours

Example: A clock gains 5 minutes in 10 hours. How much does it gain in 24 hours?

Gain per hour = 5/10 = 0.5 minute
Gain in 24 hours = 24 × 0.5
= 12 minutes

13. FAST & SLOW CLOCK

Fast clock: shows more time than actual time.
Slow clock: shows less time than actual time.

Error = Shown Time − Actual Time

If a clock gains x minutes in every hour, its displayed time increases faster than real time. If it loses x minutes per hour, its displayed time increases more slowly.

14. MIRROR IMAGE OF A CLOCK

For the mirror image of an analogue clock, use:

Mirror Time = 11:60 − Given Time

Example: Find the mirror image of 4:20.

11:60 − 4:20
= 7:40

Important: The mirror-image shortcut is normally applied to ordinary 12-hour analogue-clock questions.

PAGE 4 — QUICK FORMULA SHEET

Concept	Formula
Clock circle	360°
One division	30°
Minute hand	6M°
Hour hand	30H + 0.5M
Angle between hands	\|30H − 5.5M\|
Smaller angle	360° − larger angle
Coincidence	M = 60H/11
Given angle θ	\|30H − 5.5M\| = θ
Opposite hands	\|30H − 5.5M\| = 180°
Right angle	\|30H − 5.5M\| = 90°
Mirror time	11:60 − Given time

15. SOLVED MIXED EXAMPLES

Example 1: Find the angle between the hands at 7:25.

Angle = |30(7) − 5.5(25)|
= |210 − 137.5|
= 72.5°

Example 2: Find the angle at 10:40.

Angle = |30(10) − 5.5(40)|
= |300 − 220|
= 80°

Example 3: When do the hands coincide between 8 and 9?

M = 60H/11
= 60 × 8 / 11
= 480/11
= 43 7/11 minutes

Therefore, approximately 8:43:38.

Example 4: A clock gains 10 minutes in 20 hours. How much will it gain in 48 hours?

Gain per hour = 10/20 = 0.5 minute
Gain in 48 hours = 48 × 0.5
= 24 minutes

16. PRACTICE QUESTIONS

Find the angle between the hands at 2:30.
Find the angle between the hands at 6:20.
Find the angle between the hands at 9:45.
When do the hands coincide between 3 and 4?
When do the hands coincide between 9 and 10?
Find the mirror image of 5:25.
Find the mirror image of 8:40.
A clock gains 4 minutes in 8 hours. How much will it gain in 24 hours?
At what time between 4 and 5 will the hands be 90° apart?
At what time between 7 and 8 will the hands coincide?

17. ANSWER KEY

1. 105°
2. 70°
3. 22.5°
4. 3:16 4/11 approximately
5. 9:49 1/11 approximately
6. 6:35
7. 3:20
8. 12 minutes
9. 4:05 5/11 and 4:38 2/11 approximately
10. 7:38 2/11 approximately

18. EXAM SHORTCUTS

Shortcut 1: Always remember the basic angle formula:
|30H − 5.5M|

Shortcut 2: Minute hand moves 12 times faster than the hour hand because 6°/minute ÷ 0.5°/minute = 12.

Shortcut 3: For a mirror image, use:
11:60 − Given Time

Common Mistakes: