Clocks Questions: Easy Tricks to Solve for Competitive Exams

Clocks-related problems are common in competitive exams and assess a candidate’s ability to understand time, angles, and relative motion of clock hands. Mastering this topic requires knowledge of formulas, shortcuts, and problem-solving techniques.

1. Understanding the Basics of Clocks

A clock has two hands:

• Hour Hand (Short Hand) – Moves 30° per hour (360°/12 hours).
• Minute Hand (Long Hand) – Moves 6° per minute (360°/60 minutes).

✅ Important Formula:

• Angle between hands = |(30 × Hour) – (11/2 × Minutes)|
• Relative speed of hands = 5.5° per minute

2. Important Clock Formulas

a) Angle Between Hands

The angle (θ) between the hour and minute hands at any given time is:θ=∣(30H−11/2M)∣

where:

• H = Hour
• M = Minutes

✅ Example:
Find the angle at 3:15. θ=∣(30×3−11/2×15)∣

=∣(90−82.5)∣

=7.5°

b) Overlapping of Hands

The hands of the clock overlap (coincide) every 65.45 minutes (approximately 12 times in 12 hours).

✅ Formula:
Time when hands overlap after H hours: Time=11H×60

✅ Example:
Find when hands overlap after 2:00.= =112×60​=11120​≈10.91 minutes

So, hands overlap at 2:10.91 (around 2:11).

c) When Hands are at Right Angles (90° Apart)

The hands of the clock form a right angle (90°) twice every hour.

✅ Formula:
Time when hands are 90° apart after H hours: Time=11H×60±15

✅ Example:
Find when hands are at 90° apart after 4:00. =114×60±15

=11240±15​

=21.8 min and 38.18 min

So, at 4:21.8 and 4:38.18, hands are at 90°.

d) When Hands are Opposite (180° Apart)

The hands of the clock are opposite (straight line) once every hour.

✅ Formula:
Time when hands are 180° apart after H hours: Time=11H×60±30​

✅ Example:
Find when hands are opposite after 5:00.= =115×60±30

11300±30

=30 min and 32.7 min

So, at 5:30 and 5:32.7, hands are opposite.

e) Gaining or Losing Time

A clock gains or loses time when it does not move at the correct speed.

✅ Formula: Gain/Loss per Day= Total Minutes in Observation Period/Difference×1440

3. Sample Problems

Problem 1: Finding the Angle

Find the angle between the clock hands at 6:40.

✅ Solution: θ=∣(30H−11/2M)∣

=∣(30×6−11/2×40)∣

=∣180−220∣

40°

Problem 2: Finding When Hands Coincide

Find when hands coincide after 7:00.

✅ Solution: Time=11H×60

=117×60

=11/420​≈38.18

So, the hands coincide at 7:38.18.

4. Conclusion

Clock problems are based on the movement of the hands and their angles. By applying the right formulas and understanding the concepts, solving these problems becomes easier and faster. Regular practice will help improve accuracy in exams.