Clock: Angles, Coinciding Hands, Fast and Slow Clocks, Mirror Times
Clock: Angles, Coinciding Hands, Fast and Slow Clocks, Mirror Times
Clock questions use two speeds: the minute hand moves 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° a minute. One formula gives the angle at any time; the same formula, run backwards, gives the times when the hands coincide, are opposite or at right angles.
1. Formula Box
Angle between the hands at H hours M minutes = |30H − 5.5M| (take 360° minus this if it exceeds 180°)
Hands coincide between H and H + 1 at 60H/11 minutes past H
In 12 hours: hands coincide 11 times, are opposite 11 times and at right angles 22 times; in a day these double (22, 22, 44)
Mirror image of a clock time t (vertical mirror) = 11:60 − t, that is 12:00 − t
A clock that gains g minutes per hour shows (60 + g) minutes for every 60 real minutes
| Event | Angle condition |
|---|---|
| Coincide | 0° |
| Right angle | 90° |
| Opposite (straight line) | 180° |
✗ At 3:40 the angle is 40 × 6 − 3 × 30 = 150° | ✓ The hour hand has moved too: |90 − 5.5 × 40| = 130°
✗ Mirror of 3:20 is 9:20 | ✓ 11:60 − 3:20 = 8:40
हिंदी नोट: घड़ी में कोण = |30H − 5.5M|। मिनट की सुई हर मिनट घंटे की सुई से 5.5° आगे बढ़ती है। दर्पण में घड़ी का समय = 11:60 − दिखाया गया समय।
Exam Pointer: These patterns had no verified NTPC shift in our check (clock pages found carried ALP, Group D or other labels) and are tagged syllabus-based. Clock is part of the NTPC reasoning syllabus.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Angle between the hands
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the angle between the hands of a clock at 3:40.
|30 × 3 − 5.5 × 40| = |90 − 220| = 130°.
Answer: 130°
EXAMATLAS LEVEL
Q. Find the angle between the hands at 7:20, and the reflex angle.
|210 − 110| = 100°. The reflex angle is 360° − 100° = 260°.
Answer: 100°; 260°
Pattern 2: Coinciding, opposite and right-angled hands
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. At what time between 4 and 5 o'clock do the hands coincide?
60 × 4/11 = 240/11 = 21 9/11 minutes past 4.
Answer: 21 9/11 minutes past 4
EXAMATLAS LEVEL
Q. At what time between 7 and 8 o'clock are the hands exactly opposite each other?
Set |210 − 5.5M| = 180. 210 − 5.5M = 180 gives M = 30/5.5 = 5 5/11. (The other case, 210 − 5.5M = −180, gives M ≈ 70.9, past 8 o'clock.)
Answer: 5 5/11 minutes past 7
Pattern 3: Fast and slow clocks
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A clock gains 5 minutes every hour. It is set right at 8 AM. What does it show at 2 PM (correct time)?
6 hours pass, so it gains 30 minutes: it shows 2:30 PM.
Answer: 2:30 PM
EXAMATLAS LEVEL
Q. A watch loses 4 minutes every hour. It is set right at 9 AM. When it shows 4 PM the same day, what is the correct time?
The watch shows 56 minutes for every 60 real minutes. It has run 7 hours = 420 minutes on its own dial, which takes 420 × 60/56 = 450 real minutes, 7 hours 30 minutes.
Answer: 4:30 PM
Pattern 4: Mirror image of a clock
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A clock shows 3:20. What time does its mirror image show?
11:60 − 3:20 = 8:40.
Answer: 8:40
EXAMATLAS LEVEL
Q. The mirror image of a clock shows 4:45. What is the actual time? What does the mirror show when the actual time is 12:30?
Actual = 11:60 − 4:45 = 7:15. For 12:30, treat it as 0:30: 11:60 − 0:30 = 11:30.
Answer: 7:15; 11:30
60-Second Revision
- Angle = |30H − 5.5M|.
- Coincide at 60H/11 minutes past H.
- Gaining clock shows (60 + g) per real hour; losing shows (60 − g).
- Mirror time = 11:60 − actual time.