`{:("Column A"," ","Column B"),("The clockwise angle made by the hour hand and the minute hant at 12:15 pm",,"The anticlockwise angle made by the hour hand and the minute hand at 12:45 pm"):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to calculate the angles made by the hour and minute hands of a clock at two different times: 12:15 PM and 12:45 PM.

### Step-by-Step Solution:

**Step 1: Calculate the angle for Column A (12:15 PM)**

1. **Position of the Hour Hand at 12:15 PM:**
- The hour hand moves 30 degrees for each hour (360 degrees / 12 hours).
- At 12:15 PM, the hour hand is a quarter of the way between 12 and 1.
- Therefore, the hour hand's position is:
\[ \text{Angle of hour hand} = 0 + \left(\frac{15}{60} \times 30\right) = 0 + 7.5 = 7.5 \text{ degrees} \]

2. **Position of the Minute Hand at 12:15 PM:**
- The minute hand moves 6 degrees for each minute (360 degrees / 60 minutes).
- At 15 minutes, the minute hand's position is:
\[ \text{Angle of minute hand} = 15 \times 6 = 90 \text{ degrees} \]

3. **Calculate the Clockwise Angle:**
- The clockwise angle between the hour hand and minute hand is:
\[ \text{Clockwise angle} = \text{Angle of minute hand} - \text{Angle of hour hand} = 90 - 7.5 = 82.5 \text{ degrees} \]

**Step 2: Calculate the angle for Column B (12:45 PM)**

1. **Position of the Hour Hand at 12:45 PM:**
- At 12:45 PM, the hour hand is three-quarters of the way between 12 and 1.
- Therefore, the hour hand's position is:
\[ \text{Angle of hour hand} = 0 + \left(\frac{45}{60} \times 30\right) = 0 + 22.5 = 22.5 \text{ degrees} \]

2. **Position of the Minute Hand at 12:45 PM:**
- At 45 minutes, the minute hand's position is:
\[ \text{Angle of minute hand} = 45 \times 6 = 270 \text{ degrees} \]

3. **Calculate the Anticlockwise Angle:**
- The anticlockwise angle between the hour hand and minute hand is:
\[ \text{Anticlockwise angle} = 360 - (\text{Angle of minute hand} - \text{Angle of hour hand}) = 360 - (270 - 22.5) = 360 - 247.5 = 112.5 \text{ degrees} \]

### Conclusion:
- Column A (12:15 PM) = 82.5 degrees
- Column B (12:45 PM) = 112.5 degrees

Thus, the angles are not equal.