What is the angle made by the hour hand and the minute hand, if the clock shows 9:15 pm ?

Sure, here's how you can calculate the angle between the hour and minute hand at 9:15 PM:

1. First, we need to understand that in a clock, each hour makes up 30 degrees (360 degrees divided by 12 hours) and each minute makes up 6 degrees (360 degrees divided by 60 minutes).

2. At 9:15, the minute hand is at 3 (each 'hour' position on the clock represents 5 minutes, so 15 minutes is 3 'hours' on the clock face).

3. The hour hand at 9:15 isn't on the 9 exactly, it's a quarter of the way between the 9 and the 10 because it's a quarter past 9. So, we calculate the position of the hour hand as follows: 9 hours * 30 degrees/hour = 270 degrees. But since it's a quarter past 9, we add another quarter of the 30 degrees, which is 7.5 degrees. So, the total position of the hour hand is 270 degrees + 7.5 degrees = 277.5 degrees.

4. The minute hand at 15 minutes is at 3, so its position is: 3 * 30 degrees = 90 degrees.

5. Now, subtract the smaller angle from the larger angle to find the angle between the two hands: 277.5 degrees - 90 degrees = 187.5 degrees.

6. However, the angle between clock hands is typically taken as the smaller angle at the intersection, so we subtract this from 360 degrees: 360 degrees - 187.5 degrees = 172.5 degrees.

So, the angle between the hour hand and the minute hand at 9:15 PM is 172.5 degrees.