Clocks
CSAT · Logical Reasoning
Introduction
Clock questions involve the angle between the hands, times when the hands coincide or are at right angles, and clocks that gain or lose time. All follow from the relative speed of the two hands.
Key concepts
Hand speeds
- Minute hand: 360° in 60 min = 6° per minute.
- Hour hand: 360° in 12 h = 0.5° per minute (30° per hour).
- The minute hand gains 5.5° per minute on the hour hand (55 minute-spaces per hour).
Angle at a given time
θ = |30H − 5.5M|. If θ > 180°, the smaller angle is 360° − θ. At 3:40: |90 − 220| = 130°.
Coincidence and straight lines
The hands coincide 11 times in 12 hours (22 times a day) – not 12, because between 11 and 1 they coincide only once, at 12:00. Between H and H + 1 o'clock they coincide at 60H/11 minutes past H.
They are at right angles 22 times in 12 hours (44 a day), and in a straight line (opposite or together) 22 times in 12 hours.
Faulty clocks
A clock that gains g minutes per hour shows the correct time plus g × hours elapsed. To find when it shows a certain time, set up the ratio of its speed to a correct clock.
Mirror images
The mirror time of H:M is 11:60 − H:M (for times between 1:00 and 11:00).
- θ = |30H − 5.5M|
- Coincide at 60H/11 min past H
- Gain 5.5° per minute
- Mirror time = 11:60 − time
Solved examples
Example 1
A clock gains 4 minutes every hour. It is set right at 9:00 a.m. What time will it show when the correct time is 8 hours later?
(a) 17:36 (b) 17:04 (c) 17:32 plus 0 h → 17:32 (d) 17:28
- In 8 hours it gains 4 × 8 = 32 minutes.
- Correct time 17:00 + 32 min = 17:32 (24-hour clock).
Answer: (c) 17:32 plus 0 h → 17:32
Example 2
At what time between 9 and 10 o'clock are the hands of a clock together?
(a) 51 9/11 min past 9 (b) 45 min past 9 (c) 49 1/11 min past 9 (49 1/11 min) (d) 54 1/11 min past 9 (49 1/11 min)
- The minute hand gains 5.5° per minute on the hour hand.
- At 9:00 it is 270° behind; time to close the gap = 270 ÷ 5.5 = 60×9/11 = 49 1/11 minutes.
Answer: (c) 49 1/11 min past 9 (49 1/11 min)
Example 3
At what time between 4 and 5 o'clock are the hands of a clock together?
(a) 24 6/11 min past 4 (b) 21 9/11 min past 4 (21 9/11 min) (c) 23 9/11 min past 4 (21 9/11 min) (d) 20 min past 4
- The minute hand gains 5.5° per minute on the hour hand.
- At 4:00 it is 120° behind; time to close the gap = 120 ÷ 5.5 = 60×4/11 = 21 9/11 minutes.
Answer: (b) 21 9/11 min past 4 (21 9/11 min)
Example 4
A clock gains 3 minutes every hour. It is set right at 8:00 a.m. What time will it show when the correct time is 6 hours later?
(a) 14:18 plus 0 h → 14:18 (b) 14:42 (c) 14:03 (d) 14:21
- In 6 hours it gains 3 × 6 = 18 minutes.
- Correct time 14:00 + 18 min = 14:18 (24-hour clock).
Answer: (a) 14:18 plus 0 h → 14:18
Example 5
A clock gains 3 minutes every hour. It is set right at 6:00 a.m. What time will it show when the correct time is 12 hours later?
(a) 18:24 (b) 18:36 plus 0 h → 18:36 (c) 18:03 (d) 18:39
- In 12 hours it gains 3 × 12 = 36 minutes.
- Correct time 18:00 + 36 min = 18:36 (24-hour clock).
Answer: (b) 18:36 plus 0 h → 18:36
Example 6
At what time between 1 and 2 o'clock are the hands of a clock together?
(a) 5 5/11 min past 1 (5 5/11 min) (b) 6 5/11 min past 1 (5 5/11 min) (c) 8 2/11 min past 1 (d) 5 min past 1
- The minute hand gains 5.5° per minute on the hour hand.
- At 1:00 it is 30° behind; time to close the gap = 30 ÷ 5.5 = 60×1/11 = 5 5/11 minutes.
Answer: (a) 5 5/11 min past 1 (5 5/11 min)
Example 7
When are the hands of a clock first at right angles after 6 o'clock?
(a) 17 min past 6 (b) 49 1/11 min past 6 (c) 45 min past 6 (d) 16 4/11 min past 6
- At 6:00 the hour hand is 180° ahead.
- For the first right angle the minute hand must gain 90° (so it is 90° behind the hour hand).
- Time = 90 ÷ 5.5 = 16 4/11 minutes.
Answer: (d) 16 4/11 min past 6
Example 8
A clock gains 4 minutes every hour. It is set right at 6:00 a.m. What time will it show when the correct time is 24 hours later?
(a) 05:24 (b) 6:36 plus 1 h → 07:36 (c) 06:40 (d) 06:04
- In 24 hours it gains 4 × 24 = 96 minutes.
- Correct time 06:00 + 96 min = 07:36 (24-hour clock).
Answer: (b) 6:36 plus 1 h → 07:36
- For right angles: solve 30H − 5.5M = ±90.
- Use fractions of 11 for coincidence times: 5 5/11, 10 10/11, 16 4/11 …
- Check the answer by computing the angle at the option time.
- Ignoring the hour hand's movement within the hour (3:30 is not 90°, it is 75°).
- Saying the hands coincide 24 times a day (it is 22).
- Giving the reflex angle instead of the smaller one.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. A clock gains 2 minutes every hour. It is set right at 9:00 a.m. What time will it show when the correct time is 6 hours later?
(a) 15:14 (b) 15:48 (c) 15:12 plus 0 h → 15:12 (d) 15:02
Q2. At what time between 10 and 11 o'clock are the hands of a clock together?
(a) 54 6/11 min past 10 (54 6/11 min) (b) 50 min past 10 (c) 57 3/11 min past 10 (d) 59 6/11 min past 10 (54 6/11 min)
Q3. What is the angle between the hour and minute hands of a clock at 11:20?
(a) 140° (b) 210° (c) 220° (d) 147 1/2°
Q4. What is the angle between the hour and minute hands of a clock at 6:55?
(a) 150° (b) 122 1/2° (c) 237 1/2° (d) 130°
Q5. What is the angle between the hour and minute hands of a clock at 10:55?
(a) 10° (b) 2 1/2° (c) 357 1/2° (d) 30°
Q6. When are the hands of a clock first at right angles after 9 o'clock?
(a) 32 8/11 min past 9 (b) 65 5/11 min past 9 (c) 33 min past 9 (d) 60 min past 9
Q7. At what time between 5 and 6 o'clock are the hands of a clock together?
(a) 25 min past 5 (b) 27 3/11 min past 5 (27 3/11 min) (c) 30 3/11 min past 5 (27 3/11 min) (d) 30 min past 5
Q8. When are the hands of a clock first at right angles after 7 o'clock?
(a) 22 min past 7 (b) 54 6/11 min past 7 (c) 50 min past 7 (d) 21 9/11 min past 7
Q9. When are the hands of a clock first at right angles after 2 o'clock?
(a) 32 min past 2 (b) 25 min past 2 (c) 28 min past 2 (d) 27 3/11 min past 2
Q10. What is the angle between the hour and minute hands of a clock at 5:15?
(a) 292 1/2° (b) 60° (c) 67 1/2° (d) 75°
Q11. What is the angle between the hour and minute hands of a clock at 12:15?
(a) 277 1/2° (b) 90° (c) 82 1/2° (d) 90 1/2°
Q12. At what time between 3 and 4 o'clock are the hands of a clock together?
(a) 15 min past 3 (b) 19 1/11 min past 3 (c) 18 4/11 min past 3 (16 4/11 min) (d) 16 4/11 min past 3 (16 4/11 min)
Q1 – (c) 15:12 plus 0 h → 15:12. In 6 hours it gains 2 × 6 = 12 minutes. Correct time 15:00 + 12 min = 15:12 (24-hour clock).
Q2 – (a) 54 6/11 min past 10 (54 6/11 min). The minute hand gains 5.5° per minute on the hour hand. At 10:00 it is 300° behind; time to close the gap = 300 ÷ 5.5 = 60×10/11 = 54 6/11 minutes.
Q3 – (a) 140°. Angle = |30H − 5.5M| (take 360° minus this if it exceeds 180°). = |30 × 11 − 5.5 × 20| = |330 − 110| = 220°. Smaller angle = 140°.
Q4 – (b) 122 1/2°. Angle = |30H − 5.5M| (take 360° minus this if it exceeds 180°). = |30 × 6 − 5.5 × 55| = |180 − 302 1/2| = 122 1/2°. Smaller angle = 122 1/2°.
Q5 – (b) 2 1/2°. Angle = |30H − 5.5M| (take 360° minus this if it exceeds 180°). = |30 × 10 − 5.5 × 55| = |300 − 302 1/2| = 2 1/2°. Smaller angle = 2 1/2°.
Q6 – (a) 32 8/11 min past 9. At 9:00 the hour hand is 270° ahead. For the first right angle the minute hand must gain 180° (so it is 90° behind the hour hand). Time = 180 ÷ 5.5 = 32 8/11 minutes.
Q7 – (b) 27 3/11 min past 5 (27 3/11 min). The minute hand gains 5.5° per minute on the hour hand. At 5:00 it is 150° behind; time to close the gap = 150 ÷ 5.5 = 60×5/11 = 27 3/11 minutes.
Q8 – (d) 21 9/11 min past 7. At 7:00 the hour hand is 210° ahead. For the first right angle the minute hand must gain 120° (so it is 90° behind the hour hand). Time = 120 ÷ 5.5 = 21 9/11 minutes.
Q9 – (d) 27 3/11 min past 2. At 2:00 the hour hand is 60° ahead. For the first right angle the minute hand must gain 150° (so it is 90° ahead of the hour hand). Time = 150 ÷ 5.5 = 27 3/11 minutes.
Q10 – (c) 67 1/2°. Angle = |30H − 5.5M| (take 360° minus this if it exceeds 180°). = |30 × 5 − 5.5 × 15| = |150 − 82 1/2| = 67 1/2°. Smaller angle = 67 1/2°.
Q11 – (c) 82 1/2°. Angle = |30H − 5.5M| (take 360° minus this if it exceeds 180°). = |30 × 0 − 5.5 × 15| = |0 − 82 1/2| = 82 1/2°. Smaller angle = 82 1/2°.
Q12 – (d) 16 4/11 min past 3 (16 4/11 min). The minute hand gains 5.5° per minute on the hour hand. At 3:00 it is 90° behind; time to close the gap = 90 ÷ 5.5 = 60×3/11 = 16 4/11 minutes.
- 6° and 0.5° per minute
- θ = |30H − 5.5M|
- 22 coincidences a day
- Coincide 60H/11 min past H
