When the time shown by a clock is 4:15, what is the angle between the minute hand and the hour hand?

Introduction / Context:
Angle at a given time questions require converting time into the angular positions of each hand on the clock. Here, the clock shows 4:15, and you need to find the angle between the minute hand and the hour hand. This involves calculating both positions from the 12 o clock mark and then finding the difference between them. It is a standard and important concept in aptitude questions on clocks.

Given Data / Assumptions:

• Time is 4:15.
• The minute hand moves 360 degrees in 60 minutes, so its speed is 6 degrees per minute.
• The hour hand moves 360 degrees in 12 hours, that is, in 720 minutes, so its speed is 0.5 degrees per minute.
• We measure angles from the 12 o clock position in a clockwise direction.

Concept / Approach:
The minute hand angle is straightforward: multiply the number of minutes by 6 degrees. The hour hand angle depends on both the hour and the minutes past the hour. Once the separate angles are known, take the absolute difference between them to find the angle between the hands. If needed, we may also choose the smaller of the two possible angles, but at 4:15 the direct difference is already less than 180 degrees.

Step-by-Step Solution:
Step 1: Compute the minute hand angle. At 4:15, the minute hand points at 15 minutes, so its angle is 15 * 6 = 90 degrees from 12 o clock. Step 2: Compute the hour hand angle. At 4:00, the hour hand is at 4 * 30 = 120 degrees. Step 3: From 4:00 to 4:15, the hour hand moves further because it moves continuously at 0.5 degrees per minute. In 15 minutes, it moves 15 * 0.5 = 7.5 degrees. Step 4: Therefore, at 4:15, the hour hand angle from 12 is 120 + 7.5 = 127.5 degrees. Step 5: Find the angle between the hands by taking the absolute difference: difference = |127.5 - 90| = 37.5 degrees. Step 6: Since 37.5 degrees is less than 180 degrees, it is already the smaller angle between the hands, and no further adjustment is needed.

Verification / Alternative check:
We can use the general formula for the angle between hands: angle = |30H - 5.5M|, where H is the hour and M is minutes. Here H = 4 and M = 15. Compute: |30 * 4 - 5.5 * 15| = |120 - 82.5| = 37.5 degrees. This matches the value obtained from direct calculations, verifying that the answer is correct.

Why Other Options Are Wrong:
0 degrees: This would mean the hands coincide, which clearly does not happen at 4:15.

27 degrees: Not obtained from any correct use of the formula or from the actual angles 90 and 127.5 degrees.

15 degrees: This is too small and might result from incorrectly ignoring the continuous motion of the hour hand.

52.5 degrees: Slightly larger than the correct value and likely arises from miscalculating either the hour hand or minute hand angle.

Common Pitfalls:
One frequent mistake is to assume that at 4:15 the hour hand is still exactly at 120 degrees, which ignores its movement during the 15 minutes. Another pitfall is misusing the formula and substituting the wrong values for H and M. Careful conversion of time into angles and simple absolute differences are enough to avoid these errors.

Final Answer:
The angle between the hands at 4:15 is 37.5 degrees.