# Q. How many times the hour hand and the minute hand coincide in a clock between 10:00 a.m. and 2:00 p.m. (same day)?

a) 3 times

b) 4 times

c) 5 times

d) 6 times

Correct answer: a) 3 times

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

## Clock Hands Coincidence between 10:00 a.m. and 2:00 p.m.

Step 1: Understand the movement of clock hands

– Hour hand: Makes a complete 360° rotation in 12 hours (30° per hour)

– Minute hand: Makes a complete 360° rotation in 1 hour (6° per minute)

Step 2: Calculate the relative speed of the minute hand to the hour hand

– Relative speed = Speed of minute hand – Speed of hour hand

– In one minute: Minute hand moves 6°, Hour hand moves 0.5°

– Relative speed = 6° – 0.5° = 5.5° per minute

Step 3: Calculate the time interval between coincidences

– For coincidence, the minute hand needs to “catch up” to the hour hand by 360°

– Time for one coincidence = 360° ÷ 5.5° per minute = 65.45 minutes

Step 4: Count coincidences between 10:00 a.m. and 2:00 p.m.

– Total time period = 4 hours = 240 minutes

– Number of coincidences = 240 minutes ÷ 65.45 minutes ≈ 3.67

Step 5: Interpret the result

– Since we can only have whole number coincidences, this means 3 complete coincidences occur within the given time frame.

Therefore, the correct answer is a) 3 times.