# 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

**Explanation : **

## Hour Hand & Minute Hand

**1) First, let’s understand how the hands move:**

Hour hand makes a complete 360° rotation in 12 hours, so it moves at 360°/12 = 30° per hour or 0.5° per minute

Minute hand makes a complete 360° rotation in 1 hour, so it moves at 360°/60 = 6° per minute

**2) For hands to coincide, they must point at the same angle. Let’s use the formula:**

Let ‘t’ be minutes after start time

Minute hand angle = 6t degrees

Hour hand angle = (initial hour angle + 0.5t) degrees

When they coincide: 6t = 30h + 0.5t

Where h is hours passed from start of clock (0 to 12)

**3) From 10:00 AM to 2:00 PM is 4 hours**

At 10:00, hour hand is at 300°

Coincidences happen when: 6t = 300 + 0.5t

5.5t = 300

t = 54.545… minutes after 10:00

**4) Similarly, we can find all coincidences:**

First: Around 10:54 AM

Second: Around 12:00 noon

Third: Around 1:05 PM

Therefore, in the given 4-hour period from 10:00 AM to 2:00 PM, the hour and minute hands coincide 3 times.

**The answer is a) 3 times.**