A clock is started at noon. By 20 minutes past 5 PM on the same day, the angle through which the hour hand has turned is

Options

A. `150^@`
B. `160^@`
C. `145^@`
D. `155^@`

Explanation

To solve the problem of finding the angle through which the hour hand of a clock has turned from noon to 20 minutes past 5 PM, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Time Interval**:
- The clock starts at noon (12:00 PM).
- We need to find the angle at 5:20 PM.
- The total time from noon to 5:20 PM is 5 hours and 20 minutes.

2. **Convert Time to Hours**:
- Convert 20 minutes into hours:
\[ 20 \text{ minutes} = \frac{20}{60} \text{ hours} = \frac{1}{3} \text{ hours} \]
- Therefore, the total time in hours is:
\[ 5 \text{ hours} + \frac{1}{3} \text{ hours} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \text{ hours} \]

3. **Calculate the Degrees Covered by the Hour Hand**:
- The hour hand of a clock moves 360 degrees in 12 hours.
- Therefore, in 1 hour, it moves:
\[ \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees/hour} \]
- For \(\frac{16}{3}\) hours, the angle turned by the hour hand is:
\[ \text{Angle} = \left(\frac{16}{3} \text{ hours}\right) \times (30 \text{ degrees/hour}) = \frac{16 \times 30}{3} = \frac{480}{3} = 160 \text{ degrees} \]

4. **Final Answer**:
- The angle through which the hour hand has turned by 20 minutes past 5 PM is **160 degrees**.