When an analog clock shows 4:20 p.m., what is the angle between the hour hand and the minute hand?

Introduction / Context:
Clock angle problems require you to convert a given time into positions of the hour and minute hands measured in degrees. The problem here asks for the angle between the two hands at 4:20 p.m., which is a time where the hour hand is between the 4 and 5 marks on the dial. Understanding the rates at which each hand moves allows us to compute the exact angle and distinguish between closely spaced options like 5, 10, 15, and 25 degrees.

Given Data / Assumptions:

- The time on the clock is 4:20 p.m. - The minute hand completes a full circle of 360 degrees in 60 minutes. - The hour hand completes a full circle of 360 degrees in 12 hours. - We want the smaller angle between the hour and minute hands. - The clock is assumed to be accurate and circular with 12 equal hour divisions.

Concept / Approach:
The minute hand moves 6 degrees per minute because 360 divided by 60 equals 6. The hour hand moves 0.5 degrees per minute because it moves 360 degrees in 12 hours, which is 720 minutes, giving 360 divided by 720 equals 0.5 degrees per minute. To find the angles of both hands from the 12 o clock position, we use these rates and then subtract the smaller from the larger to find the angle between them. If that angle exceeds 180 degrees, we subtract it from 360 degrees to get the smaller interior angle, but in this case that will not be necessary.

Step-by-Step Solution:
Step 1: At 4:20, the minute hand is at 20 minutes. Step 2: Angle of the minute hand is 20 * 6 = 120 degrees from 12. Step 3: At 4:00, the hour hand is at 4 * 30 = 120 degrees. Step 4: In the 20 minutes after 4:00, the hour hand moves extra degrees equal to 20 * 0.5 = 10 degrees. Step 5: Therefore, the hour hand angle from 12 at 4:20 is 120 + 10 = 130 degrees. Step 6: The difference between the positions of the minute and hour hands is |130 - 120| = 10 degrees. Step 7: Since 10 degrees is less than 180 degrees, it is already the smaller angle between the hands.

Verification / Alternative check:
We can visualize the clock: the minute hand points at the 4 on the dial (representing 20 minutes) which is at 120 degrees, while in twenty minutes the hour hand has moved one third of the way between the 4 and 5 marks. Each hour division is 30 degrees, so one third of that is 10 degrees beyond the 4. The hour hand is thus slightly ahead of the minute hand. The visual picture agrees with our calculation that the difference in their positions is 10 degrees.

Why Other Options Are Wrong:
An angle of 5 degrees would require the hour hand to be much closer to the minute hand, which does not match the one third of an hour movement. Fifteen degrees or twenty five degrees would overstate the movement of the hour hand away from the minute hand. The calculation based on exact rates yields 10 degrees, so the other numerical values do not satisfy the actual geometry of the clock at 4:20.

Common Pitfalls:
A common mistake is to forget the continuous movement of the hour hand and simply keep it fixed at the 4 mark, which would give 120 degrees and lead people to incorrectly think that the hands coincide. Another pitfall is miscalculating the hour hand movement as 20 degrees instead of 10 degrees by confusing the rates. Always remember that the hour hand moves only 0.5 degrees per minute, which is much slower than the minute hand, and use the correct rates in your calculations.

Final Answer:
At 4:20 p.m., the angle between the hour and minute hands of the clock is 10 degrees.