What is the reflex angle between the hands of a clock when the time shown is 10:25?

Introduction / Context:
Here we are asked specifically for the reflex angle between the clock hands at 10:25. A reflex angle is the larger angle formed between the two hands, that is an angle greater than 180 degrees but less than 360 degrees. We first compute the smaller angle between the hands, then subtract that from 360 degrees to get the reflex angle.

Given Data / Assumptions:

• The time is 10:25.
• Minute hand speed: 6 degrees per minute.
• Hour hand speed: 30 degrees per hour plus 0.5 degree per minute.
• Reflex angle means we finally take 360 minus the acute or smaller angle.

Concept / Approach:
Use: minute hand angle = 6 * minutes, hour hand angle = 30 * hours + 0.5 * minutes. The smaller angle between them is the absolute difference of these two angles. Then the reflex angle is: reflex angle = 360 - smaller angle.

Step-by-Step Solution:
Step 1: Compute the minute hand angle. Minutes = 25. Minute hand angle = 6 * 25 = 150 degrees. Step 2: Compute the hour hand angle. Hours = 10, minutes = 25. Hour hand angle = 30 * 10 + 0.5 * 25. Hour hand angle = 300 + 12.5 = 312.5 degrees. Step 3: Find the smaller angle between the hands. Difference = |312.5 - 150| = 162.5 degrees. Step 4: Find the reflex angle. Reflex angle = 360 - 162.5 = 197.5 degrees. In mixed form, this is 197 1/2 degrees. So the reflex angle between the hands is 197 1/2 degrees.

Verification / Alternative check:
We can verify quickly: if the smaller angle is less than 180 degrees and is 162.5 degrees, then the larger angle must be more than 180 degrees and less than 360 degrees, which is exactly 360 - 162.5 = 197.5 degrees. This matches the calculations and the given answer option.

Why Other Options Are Wrong:
Options b (167 1/2), c (157 1/2), and d (187 1/2) differ from 197 1/2 by 10 or 20 degrees and are not equal to 360 minus the correct smaller angle of 162.5 degrees. They would come from incorrect arithmetic or incorrect initial angle calculations.

Common Pitfalls:
Sometimes students confuse the smaller and larger angles and stop at 162.5 degrees, forgetting that the question asks for the reflex angle. Another pitfall is to assume the hour hand is fixed at 300 degrees for 10 oclock and ignore its progress during the 25 minutes, which leads to a wrong difference.

Final Answer:
The reflex angle between the hands of the clock at 10:25 is 197 1/2 degrees.