Find the angle between the minute hand and hour hand of a clock, when the time is 7:30 pm.

Answer

To find the angle between the minute hand and hour hand of a clock when the time is 7:30 PM, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the positions of the hour and minute hands:**
- At 7:30, the minute hand points at the 6 on the clock.
- The hour hand is positioned between 7 and 8.

2. **Calculate the angle for each hour:**
- Each hour on the clock represents an angle of \(30\) degrees (since \(360\) degrees divided by \(12\) hours equals \(30\) degrees per hour).

3. **Determine the angle of the hour hand:**
- The hour hand at 7:00 is at \(7 \times 30 = 210\) degrees.
- Since the hour hand moves as time passes, we need to find its position at 7:30.
- The hour hand moves \(30\) degrees in one hour, so in half an hour (from 7:00 to 7:30), it moves \(15\) degrees (which is \(30/2\)).
- Therefore, the angle of the hour hand at 7:30 is \(210 + 15 = 225\) degrees.

4. **Determine the angle of the minute hand:**
- The minute hand at 30 minutes (or 6 on the clock) is at \(6 \times 30 = 180\) degrees.

5. **Calculate the angle between the hour and minute hands:**
- The angle between the hour hand and the minute hand is the absolute difference between their angles:
\[ \text{Angle} = |225 - 180| = 45 \text{ degrees} \]

### Final Answer:
The angle between the minute hand and hour hand at 7:30 PM is \(45\) degrees.

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