Between 6 PM and 7 PM the minute hand of a clock will be ahead of the hour hand by 3 minutes at

The minute hand moves 6 degrees per minute, and the hour hand moves 0.5 degrees per minute. The relative speed of the minute hand with respect to the hour hand is 5.5 degrees per minute.

At 6:00 PM, the hour hand is at 6 (180 degrees from 12), and the minute hand is at 12 (0 degrees). So, the minute hand is 180 degrees behind the hour hand.

We want the minute hand to be 3 minutes ahead of the hour hand. Since 1 minute on the clock face corresponds to 6 degrees, 3 minutes corresponds to 3 * 6 = 18 degrees.

For the minute hand to be 18 degrees ahead, it must first cover the initial 180-degree gap and then move an additional 18 degrees ahead. Total relative distance to be covered = 180 degrees (to catch up) + 18 degrees (to be 3 minutes ahead) = 198 degrees.

Time taken = Total relative distance / Relative speed Time taken = 198 degrees / 5.5 degrees/minute = 1980 / 55 minutes = 36 minutes.

Therefore, the minute hand will be 3 minutes ahead of the hour hand at 36 minutes past 6 PM, which is 6:36 PM.

Analyzing the options: A) 6:15 PM: At this time, the minute hand is still significantly behind the hour hand. B) 6:18 PM: The minute hand is still behind the hour hand. C) 6:36 PM: This matches our calculation. At 6:36 PM, the minute hand is at 36 * 6 = 216 degrees. The hour hand is at 180 + (36 * 0.5) = 180 + 18 = 198 degrees. The minute hand is 216 - 198 = 18 degrees ahead, which is 3 minutes. D) 6:48 PM: At this time, the minute hand would be much more than 3 minutes ahead of the hour hand.

The final answer is C.