At what angle are the hands of a clock inclined when the time shown is exactly 5:15?

Introduction / Context:
Here we are asked to find the angle between the hour hand and the minute hand at a specific time, 5:15. This is a standard type of clock angle question and uses known angular speeds of the clock hands. It reinforces understanding of how the hour hand also moves continuously as minutes pass, not just in jumps between hour marks.

Given Data / Assumptions:

• The time is 5:15.
• Minute hand makes 360 degrees in 60 minutes.
• Hour hand makes 360 degrees in 12 hours.
• The required answer is the smaller angle between the hands at that time.

Concept / Approach:
We use the standard formulas: minute hand angle = 6 * minutes. hour hand angle = 30 * hours + 0.5 * minutes. We then compute their difference and, if needed, use 360 minus that difference to get the smaller angle.

Step-by-Step Solution:
Step 1: Calculate the minute hand angle. Minutes = 15. Minute hand angle = 6 * 15 = 90 degrees. Step 2: Calculate the hour hand angle. Hours = 5, minutes = 15. Hour hand angle = 30 * 5 + 0.5 * 15. Hour hand angle = 150 + 7.5 = 157.5 degrees. Step 3: Compute the angle between the hands. Difference = |157.5 - 90| = 67.5 degrees. Since 67.5 degrees is less than 180 degrees, it is the smaller angle between the hands.

Verification / Alternative check:
The reflex angle would be: 360 - 67.5 = 292.5 degrees, which is larger and therefore not the standard answer unless specifically asked for. The computed 67.5 degrees fits the typical interpretation of the angle between clock hands.

Why Other Options Are Wrong:
Option a (57.5 degrees), option c (77.5 degrees), and option d (87.5 degrees) do not match the exact difference between 157.5 degrees and 90 degrees. These numbers might come from miscalculations such as forgetting the half degree per minute movement of the hour hand or using 30 degrees per minute by mistake.

Common Pitfalls:
One common mistake is to assume that at 5:15 the hour hand is at exactly 150 degrees, the position at 5:00, and ignore its movement during the additional 15 minutes. Another pitfall is mixing up degrees per hour and degrees per minute, which can lead to completely incorrect angles.

Final Answer:
The angle between the hands of the clock at 5:15 is 67.5 degrees.