Find the angle between the hour hand and the minute hand of a clock when the time shown is exactly 3:25.

Introduction / Context:
This question asks for the angle between the hour and minute hands at a given time. It is a basic clock angle problem that uses standard formulas for the angular positions of the two hands. These questions are common in aptitude tests as they test accuracy in applying simple arithmetic and geometric reasoning.

Given Data / Assumptions:

• The time is 3:25.
• The minute hand completes 360 degrees in 60 minutes.
• The hour hand completes 360 degrees in 12 hours.
• We need the smaller angle between the two hands.

Concept / Approach:
The angle of the minute hand from 12 oclock is: minute hand angle = 6 * minutes. The angle of the hour hand from 12 oclock is: hour hand angle = 30 * hours + 0.5 * minutes. We compute both angles and then take the absolute difference. If necessary, we choose the smaller of the two possible angles (theta and 360 minus theta).

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 = 3, minutes = 25. Hour hand angle = 30 * 3 + 0.5 * 25. Hour hand angle = 90 + 12.5 = 102.5 degrees. Step 3: Find the absolute difference. Difference = |150 - 102.5| = 47.5 degrees. This is less than 180 degrees, so it is already the smaller angle between the hands.

Verification / Alternative check:
If we take the reflex angle, it would be: 360 - 47.5 = 312.5 degrees. Since questions about angle between clock hands usually refer to the acute or smaller angle unless the reflex angle is specifically mentioned, 47.5 degrees is the correct interpretation here.

Why Other Options Are Wrong:
Options b (57.5 degrees), c (45.5 degrees), and d (55.5 degrees) do not match the calculated difference between 102.5 degrees and 150 degrees. They may arise from approximate or incorrect calculations such as ignoring the movement of the hour hand or adding or subtracting an incorrect value.

Common Pitfalls:
A common error is to assume the hour hand is exactly at 90 degrees at 3:25, which ignores its extra movement during the 25 minutes. Another mistake is to compute the angle as 30 * 3 - 6 * 25, which uses the wrong formula. Being careful with decimal values like 12.5 is also important to avoid rounding errors.

Final Answer:
The angle between the hour and minute hands at 3:25 is 47.5 degrees.