When the time shown by a clock is 4:20, what is the angle between the minute hand and the hour hand?

Introduction / Context:
This question again deals with finding the angle between the hour hand and the minute hand at a specific time, here 4:20. These problems test your ability to convert time into angular positions and then calculate their difference. By practising such questions at different times, you strengthen your understanding of how both hands move continuously.

Given Data / Assumptions:

• Time is 4:20.
• The minute hand moves 360 degrees in 60 minutes, that is, 6 degrees per minute.
• The hour hand moves 360 degrees in 12 hours, that is, 0.5 degrees per minute.
• Angles are measured from the 12 o clock position.

Concept / Approach:
We calculate the position of the minute hand by multiplying the number of minutes by 6 degrees. To find the position of the hour hand, we start with the base angle for the hour and add the extra motion caused by the minutes past the hour. Then we take the absolute difference between the two angles to obtain the angle between the hands. If needed, we can use the general formula |30H - 5.5M| for quick computation.

Step-by-Step Solution:
Step 1: Find the angle of the minute hand at 4:20. It is at 20 minutes, so its angle from 12 o clock is 20 * 6 = 120 degrees. Step 2: Find the angle of the hour hand. At 4:00, the hour hand is at 4 * 30 = 120 degrees. Step 3: From 4:00 to 4:20, the hour hand moves further at 0.5 degrees per minute. In 20 minutes it moves 20 * 0.5 = 10 degrees. Step 4: Thus, at 4:20 the hour hand angle is 120 + 10 = 130 degrees. Step 5: The angle between the hands is the absolute difference: difference = |130 - 120| = 10 degrees. Step 6: Since 10 degrees is less than 180 degrees, this is the required angle.

Verification / Alternative check:
Use the general formula angle = |30H - 5.5M| with H = 4 and M = 20. Compute: |30 * 4 - 5.5 * 20| = |120 - 110| = 10 degrees. This matches the detailed angle calculations, confirming the result.

Why Other Options Are Wrong:
0 degrees: This would correspond to the hands coinciding, which does not happen at 4:20.

30 degrees: This is a common angle at some times but not at 4:20 based on the actual positions.

60 degrees: Much larger than the actual separation; might come from a rough guess rather than calculation.

20 degrees: Close but still double the correct value, often arising from mixing up angles or misusing the formula.

Common Pitfalls:
One common error is to forget the additional movement of the hour hand during the 20 minutes and assume it is still at 120 degrees. Another pitfall is incorrectly substituting into the formula, for example using minutes only and ignoring the hour. Always compute each hand position carefully and then take the absolute difference.

Final Answer:
At 4:20, the angle between the hands is 10 degrees.