To answer this question, let's analyze the movement of the hour and minute hands on a clock.

The minute hand completes a full revolution every 60 minutes, while the hour hand completes a full revolution every 12 hours. Therefore, for every hour, the minute hand moves 12 times faster than the hour hand.

Let's consider a scenario where the minute hand is ahead of the hour hand, and they are about to cross each other. As time passes, the minute hand continues to move ahead, while the hour hand slowly catches up. Eventually, they will cross each other, and the minute hand will be behind the hour hand.

The only times when the minute and hour hands cross each other between noon and midnight are when the minute hand is ahead of the hour hand. From noon to 1 o'clock, the minute hand crosses the hour hand once. From 1 o'clock to 2 o'clock, the minute hand crosses the hour hand twice. This pattern continues until 11 o'clock, where the minute hand crosses the hour hand 11 times.

Therefore, the correct answer is C) 10. The hour and minute hands cross each other 10 times between noon and midnight.

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To find the number of degrees between the clock hands at 3:15 PM, we need to determine the positions of the hour and minute hands.

At 3:00 PM, the hour hand is exactly on the 3, and the minute hand is pointing at the 12.

Between 3:00 PM and 3:15 PM, the hour hand moves a fraction of the way between the 3 and the 4. Since there are 12 hour marks on the clock, the hour hand moves 1/12th of the way between the 3 and the 4.

Between 3:00 PM and 3:15 PM, the minute hand moves to the 3 (15 minutes past the hour).

Therefore, the angle between the minute hand and the hour hand at 3:15 PM is the difference between the positions of the hour and minute hands.

To calculate the angle, we can use the formula:

Angle = |(30 * H) - (11/2 * M)|

Where: H = Hour hand position M = Minute hand position

Substituting the values: H = 3 + 1/12 = 3.0833 (approx.) M = 15

Angle = |(30 * 3.0833) - (11/2 * 15)|

Angle = |92.4985 - 82.5|

Angle = |9.9985|

Since we need the absolute value of the angle, the answer is approximately 10 degrees.

Therefore, the correct answer is Option E) 7.5.

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