Practice Quantitative Aptitude questions on Clocks, covering angle calculations between clock hands, time gained or lost, mirror-image clock problems, relative positions of hands, and finding when clock hands coincide or form specific angles. Takes about 20–25 minutes and is useful for placement tests, TCS NQT, SSC, banking and MBA entrance exams.
Question 1
A watch gains time uniformly. It is 1 minute 30 seconds slow at 6 a.m. on a Monday and 3 minutes 45 seconds fast at 12 noon on the following Saturday. When was it correct?
Explanation
Time from 6 a.m. Monday to 12 noon Saturday = 5 days 6 hours = 126 hours.
Total gain in 126 hours = 1 min 30 s + 3 min 45 s = 5 min 15 s = 5.25 minutes.
The watch gains 1 min 30 s = 1.5 minutes to become correct.
Time needed = 1.5 × 1265.25 = 36 hours = 1 day 12 hours.
So the watch was correct 36 hours after 6 a.m. Monday, that is at 6 p.m. on Tuesday.
Total gain in 126 hours = 1 min 30 s + 3 min 45 s = 5 min 15 s = 5.25 minutes.
The watch gains 1 min 30 s = 1.5 minutes to become correct.
Time needed = 1.5 × 1265.25 = 36 hours = 1 day 12 hours.
So the watch was correct 36 hours after 6 a.m. Monday, that is at 6 p.m. on Tuesday.
Shortcut Method
Gain: 126 h → 5.25 min
1.5 min → 1.5 × 126 ÷ 5.25 = 36 h
6 a.m. Mon + 36 h → ∴ 6 p.m. Tuesday ✓
1.5 min → 1.5 × 126 ÷ 5.25 = 36 h
6 a.m. Mon + 36 h → ∴ 6 p.m. Tuesday ✓
Question 2
Between 2 and 3 o'clock, at what times are the hands of a clock inclined at an angle of 50°?
Explanation
At 2 o'clock the hour hand is 60° ahead of the minute hand, and the minute hand gains 5.5° per minute.
The angle is 50° when the gain is 60 − 50 = 10° (hands 50° apart before overlapping) or 60 + 50 = 110° (hands 50° apart after overlapping).
Gain 10°: time = 10 ÷ 5.5 = 2011 = 1911 min past 2.
Gain 110°: time = 110 ÷ 5.5 = 20 min past 2.
The angle is 50° when the gain is 60 − 50 = 10° (hands 50° apart before overlapping) or 60 + 50 = 110° (hands 50° apart after overlapping).
Gain 10°: time = 10 ÷ 5.5 = 2011 = 1911 min past 2.
Gain 110°: time = 110 ÷ 5.5 = 20 min past 2.
Shortcut Method
Gap at 2:00 = 60°, rate 5.5°/min
gap 50°: gain 10 or 110
10 ÷ 5.5 = 1911; 110 ÷ 5.5 = 20
∴ 1911 and 20 min past 2 ✓
gap 50°: gain 10 or 110
10 ÷ 5.5 = 1911; 110 ÷ 5.5 = 20
∴ 1911 and 20 min past 2 ✓
Question 3
At what time between 8 AM and 9 AM would the two hands of the clock be together?
Explanation
At 8 o'clock the hour hand is 240° ahead of the minute hand.
The minute hand gains 5.5° per minute, so it needs 240 ÷ 5.5 minutes to catch up.
Time = 2405.5 = 48011 = 43711 minutes past 8.
The minute hand gains 5.5° per minute, so it needs 240 ÷ 5.5 minutes to catch up.
Time = 2405.5 = 48011 = 43711 minutes past 8.
Shortcut Method
Gap at 8:00 = 240°, gain = 5.5°/min
240 ÷ 5.5 = 48011
∴ 43711 min past 8 ✓
240 ÷ 5.5 = 48011
∴ 43711 min past 8 ✓
Question 4
The hands of a clock coincide at some time between 10 AM and 11 AM. What is that time?
Explanation
At 10 o'clock the hour hand is 300° ahead of the minute hand.
The minute hand gains 5.5° per minute, so it needs 300 ÷ 5.5 minutes to catch up.
Time = 3005.5 = 60011 = 54611 minutes past 10.
The minute hand gains 5.5° per minute, so it needs 300 ÷ 5.5 minutes to catch up.
Time = 3005.5 = 60011 = 54611 minutes past 10.
Shortcut Method
Gap at 10:00 = 300°, gain = 5.5°/min
300 ÷ 5.5 = 60011
∴ 54611 min past 10 ✓
300 ÷ 5.5 = 60011
∴ 54611 min past 10 ✓
Question 5
Between 10 AM and 4 PM, the hands of a clock coincide how many times?
Explanation
The hands overlap at 60h11 minutes past h.
10:54611 (h = 10), 12:00 (the overlap that follows 11), 1:05511 (h = 1), 2:101011 (h = 2) and 3:16411 (h = 3) all lie between 10:00 and 4:00.
The next overlap is at about 4:21, after 4:00, so the hands overlap 5 times.
10:54611 (h = 10), 12:00 (the overlap that follows 11), 1:05511 (h = 1), 2:101011 (h = 2) and 3:16411 (h = 3) all lie between 10:00 and 4:00.
The next overlap is at about 4:21, after 4:00, so the hands overlap 5 times.
Shortcut Method
Overlaps: 10:54.., 12:00, 1:05.., 2:10.., 3:16..
4:21.. is outside
∴ 5 times ✓
4:21.. is outside
∴ 5 times ✓
Question 6
How many times will the hands of a clock overlap between 1 PM and 5 PM?
Explanation
The hands overlap at 60h11 minutes past h.
1:05511 (h = 1), 2:101011 (h = 2), 3:16411 (h = 3) and 4:21911 (h = 4) all lie between 1:00 and 5:00.
The next overlap is at 5:27311, which is after 5:00.
So the hands overlap 4 times.
1:05511 (h = 1), 2:101011 (h = 2), 3:16411 (h = 3) and 4:21911 (h = 4) all lie between 1:00 and 5:00.
The next overlap is at 5:27311, which is after 5:00.
So the hands overlap 4 times.
Shortcut Method
Overlaps at 1:05.., 2:10.., 3:16.., 4:21..
next = 5:27.. (outside)
∴ 4 times ✓
next = 5:27.. (outside)
∴ 4 times ✓
Question 7
In a week, how many times do the minute hand and the hour hand of a clock coincide?
Explanation
In one day (24 hours) the hands coincide 22 times.
In 7 days = 22 × 7 = 154 times.
In 7 days = 22 × 7 = 154 times.
Shortcut Method
1 day = 22 times
7 days → 22 × 7
∴ 154 times ✓
7 days → 22 × 7
∴ 154 times ✓
Question 8
How many times are the hands of a clock in a straight line (overlapping or opposite) in 5 days?
Explanation
In one day (24 hours) the hands are in a straight line 22 + 22 = 44 times (overlapping 22, opposite 22).
In 5 days = 44 × 5 = 220 times.
In 5 days = 44 × 5 = 220 times.
Shortcut Method
1 day = 44 times
5 days → 44 × 5
∴ 220 times ✓
5 days → 44 × 5
∴ 220 times ✓
Question 9
How many times in 3 days do the two hands of a clock lie along one straight line, pointing in opposite directions?
Explanation
In 12 hours the hands point in opposite directions 11 times, so in one day (24 hours) they do so 22 times.
In 3 days = 22 × 3 = 66 times.
In 3 days = 22 × 3 = 66 times.
Shortcut Method
1 day = 22 times
3 days → 22 × 3
∴ 66 times ✓
3 days → 22 × 3
∴ 66 times ✓
Question 10
In one week, how many times do the hands of a clock stand at right angles?
Explanation
In 12 hours the hands are at right angles 22 times, so in 24 hours they are at right angles 22 × 2 = 44 times.
In one week (7 days) = 44 × 7 = 308 times.
In one week (7 days) = 44 × 7 = 308 times.
Shortcut Method
1 day = 44 times
7 days → 44 × 7
∴ 308 times ✓
7 days → 44 × 7
∴ 308 times ✓
Question 11
At exactly 10:30, the smaller angle between the minute hand and the hour hand of a clock is:
Explanation
The hour hand moves 30° per hour (0.5° per minute) and the minute hand moves 6° per minute.
Hour hand at 10:30 = 10 × 30 + 30 × 0.5 = 315°.
Minute hand at 30 minutes = 30 × 6 = 180°.
Difference = 135°.
Hour hand at 10:30 = 10 × 30 + 30 × 0.5 = 315°.
Minute hand at 30 minutes = 30 × 6 = 180°.
Difference = 135°.
Shortcut Method
Angle = |30H − 5.5M|
|30 × 10 − 5.5 × 30| = |300 − 165| = 135
∴ Angle = 135° ✓
|30 × 10 − 5.5 × 30| = |300 − 165| = 135
∴ Angle = 135° ✓
Question 12
At 7:24, what angle do the hour hand and the minute hand of a clock make with each other?
Explanation
The hour hand moves 30° per hour (0.5° per minute) and the minute hand moves 6° per minute.
Hour hand at 7:24 = 7 × 30 + 24 × 0.5 = 222°.
Minute hand at 24 minutes = 24 × 6 = 144°.
Difference = 78°.
Hour hand at 7:24 = 7 × 30 + 24 × 0.5 = 222°.
Minute hand at 24 minutes = 24 × 6 = 144°.
Difference = 78°.
Shortcut Method
Angle = |30H − 5.5M|
|30 × 7 − 5.5 × 24| = |210 − 132| = 78
∴ Angle = 78° ✓
|30 × 7 − 5.5 × 24| = |210 − 132| = 78
∴ Angle = 78° ✓
Question 13
When will the clock’s hands be in a straight line but pointing in opposite directions between 2 am and 3 am?
Explanation
The hands point in opposite directions when the angle between them is 180 degrees. At 2 o'clock the hour hand is at 60 degrees and the minute hand is at 0.
At t minutes past 2, the minute hand is at 6t degrees and the hour hand at 60 + 0.5t degrees, so 6t - (60 + 0.5t) = 180.
5.5t = 240, so t = 43711 minutes.
At t minutes past 2, the minute hand is at 6t degrees and the hour hand at 60 + 0.5t degrees, so 6t - (60 + 0.5t) = 180.
5.5t = 240, so t = 43711 minutes.
Question 14
What is the angle between the hour and minute hand at 3:40?
Explanation
At 3:40 the hour hand is at (3 × 30) + (40 × 0.5) = 110 degrees, and the minute hand is at 40 × 6 = 240 degrees.
The angle = 240 - 110 = 130 degrees.
The angle = 240 - 110 = 130 degrees.
Question 15
The angle between the minute hand and the hour hand of a clock when the time is 8.30, is
Explanation
At 8:30 the hour hand is at (8 × 30) + (30 × 0.5) = 255 degrees and the minute hand is at 30 × 6 = 180 degrees. The angle between them = 255 - 180 = 75 degrees.
Question 16
A clock is set right at 5 am the clock loses 16 min in 24 hours. What will be the right time when the clock indicates on 3 pm the 8th day?
Explanation
From 5 am on day 1 to 3 pm on the 8th day by the clock is 7 days 10 hours = 178 hours. The clock loses 16 minutes in 24 hours, so 1424 clock minutes correspond to 1440 real minutes, that is 24 clock hours correspond to 24 hours 16 minutes of real time. So 178 clock hours = 178 × 1440 / 1424 = 180 real hours = 7 days 12 hours. So the right time = 5 am + 7 days 12 hours = 5 pm.
Question 17
How much does a watch gain or lose per day, if its hands coincide every 64 minutes?
Explanation
The hands of a correct clock coincide every 65 511 minutes. The watch shows them coincide every 64 minutes, so it loses 65 511 - 64 = 1611 minutes every 64 minutes. In 24 hours (1440 minutes) it loses 1440 / 64 × 1611 = 32 811 minutes.
Question 18
At what time between 9.13 and 10.12 will the hands of a watch be together?
Explanation
The hands are together when the minute hand gains the gap of the hour hand. At 9 o'clock the hour hand is 45 minute-spaces ahead. The minute hand gains 55 minute-spaces in 60 minutes, so it gains 45 spaces in 45 × 60 / 55 = 49 111 minutes. So the hands are together at 49 111 minutes past 9.
Question 19
Two clocks are synchronized at 12 midnight. One clock gains 2 min per hour and the other clock loses 1 min per hour. How many minutes apart will their minute hands be at 11 a.m.?
Explanation
Over 11 hours, the gaining clock displays 11 × 62=682 minutes elapsed and the losing clock displays 11 × 59=649 minutes, a difference of 33 minutes -- not matching any of the listed round numbers.
Question 20
How many times do the hands of a clock coincide in a day?
Explanation
The hands coincide 11 times in every 12-hour period, giving 22 times in a full 24-hour day.
Question 21
Find the number of times the hour hand and the minute hand of a clock are at right angle in a day.
Explanation
In 12 hours, the hands of a clock are at right angles 22 times. Hence, in a day (24 hours), they are at right angles 44 times.
Question 22
Find the angle between the hour hand and the minute hand of a clock when the time is 5.45.
Explanation
Angle traced by the hour hand in 5 hours 45 minutes = 360 × 2312 × 4 = 172.5°. Angle traced by the minute hand in 45 minutes = 270°. Angle between the hands = 270 – 172.5 = 97.5°.
Question 23
An accurate clock is started at noon. By 10 minutes past 3, the hour hand has turned through:
Explanation
The hour hand turns 30 degrees per hour, or 0.5 degrees per minute. From noon to 3:10 is 3 hours 10 minutes = 190 minutes, so it turns 190 × 0.5 = 95 degrees.
Question 24
At what time between 5.30 & 6 will the hands of a watch be at right angle?
Explanation
Between 5:30 and 6 the hands are at a right angle when the minute hand is 90 degrees ahead of the hour hand. At t minutes past 5, the minute hand is at 6t degrees and the hour hand at 150 + 0.5t degrees. So 6t - (150 + 0.5t) = 90, which gives 5.5t = 240 and t = 43 711 minutes.
Question 25
At what time between 3 & 4 will the hands of a watch point in opposite directions?
Explanation
At 3 o'clock the minute hand is 15 minute-spaces behind the hour hand (hour at 15, minute at 0 to 12). For the hands to be opposite, the minute hand must be 30 spaces ahead of the hour hand, so it must gain 15 + 30 = 45 spaces. The minute hand gains 55 spaces in 60 minutes, so it takes 45 × 60 / 55 = 49 111 minutes. The hands are opposite at 49 111 minutes past 3.
