In 16 minutes, by how many degrees does the minute hand of a clock gain over the hour hand?

Introduction / Context:
This question concerns the relative speed of the minute hand and the hour hand on a clock. Instead of asking for a specific time of coincidence, it asks directly by how many degrees the minute hand advances ahead of the hour hand during a period of 16 minutes. Such questions emphasise the concept of relative angular speed, which is fundamental to many clock problems.

Given Data / Assumptions:

• Time interval considered: 16 minutes.
• The minute hand moves 360 degrees in 60 minutes.
• The hour hand moves 360 degrees in 12 hours, that is, 720 minutes.
• We must find the angle through which the minute hand gains over the hour hand in this period.

Concept / Approach:
First, we determine the angular speed of each hand. The minute hand has a higher angular speed than the hour hand. The difference between these speeds is the relative angular speed, that is, the rate at which the minute hand gains on the hour hand. Once we know this relative speed, we multiply it by the time interval to find the net angular gain of the minute hand over the hour hand in that period.

Step-by-Step Solution:
Step 1: Compute the speed of the minute hand. It completes 360 degrees in 60 minutes, so its speed is 360 / 60 = 6 degrees per minute. Step 2: Compute the speed of the hour hand. It completes 360 degrees in 12 hours, or 12 * 60 = 720 minutes, so its speed is 360 / 720 = 0.5 degrees per minute. Step 3: Determine the relative speed of the minute hand compared with the hour hand: relative speed = 6 - 0.5 = 5.5 degrees per minute. Step 4: In 16 minutes, the gain in angle is: gain = 5.5 * 16 degrees. Step 5: Multiply 5.5 by 16: 5.5 * 16 = (5 * 16) + (0.5 * 16) = 80 + 8 = 88 degrees. Step 6: Therefore, in 16 minutes the minute hand gains 88 degrees over the hour hand.

Verification / Alternative check:
To verify, separately compute the actual rotation of each hand in 16 minutes. The minute hand rotates 6 * 16 = 96 degrees. The hour hand rotates 0.5 * 16 = 8 degrees. The gain is 96 - 8 = 88 degrees, which matches the result from the relative speed method. This cross check confirms that the calculation is correct.

Why Other Options Are Wrong:
16 degrees: This would be the rotation of a hand moving at 1 degree per minute over 16 minutes, which is not the relative speed here.

80 degrees: This is the approximate rotation of the minute hand minus almost no movement of the hour hand, but the hour hand does move and the precise difference is 88 degrees.

94 degrees: Larger than the correct gain and not supported by the exact speeds of the hands.

72 degrees: This might come from incorrectly using 4.5 degrees per minute as the relative speed, which is not correct for clock hands.

Common Pitfalls:
Students sometimes forget that the hour hand moves during the interval and incorrectly take the gain as equal to the full movement of the minute hand. Others miscalculate the speed of the hour hand, using 30 degrees per hour but forgetting to convert this correctly into degrees per minute. Always calculate both individual speeds and then subtract to get the relative speed for these types of questions.

Final Answer:
In 16 minutes, the minute hand gains 88 degrees over the hour hand.