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

• At 6:00 PM, the hour hand is at 30 degrees (6 * 30) and the minute hand is at 0 degrees, so the hour hand is 30 degrees ahead.

• The minute hand moves at 6 degrees per minute, and the hour hand moves at 0.5 degrees per minute, giving a relative speed of 5.5 degrees per minute.

• For the minute hand to be ahead by 3 minutes (18 degrees), it must first cover the initial 30-degree gap and then gain an additional 18 degrees, totaling 48 degrees of relative movement.

• The time taken to cover 48 degrees at 5.5 degrees per minute is 48 / 5.5 = 96/11 minutes, which is approximately 8.72 minutes.

• Adding this to 6:00 PM gives 6:08:43 PM, but this is when the minute hand is *behind* or just catching up? No, let's re-evaluate the 'ahead by 3 minutes' condition.

• Actually, 'ahead by 3 minutes' means the minute hand is 18 degrees *in front* of the hour hand. The total angle to cover is 360 - 30 + 18 = 348 degrees? No, simpler: The minute hand gains 5.5 deg/min. At 6:00, it is 30 deg behind. To be 18 deg ahead, it must gain 30 + 18 = 48 degrees. Time = 48/5.5 = 8.72 min. This is 6:08:43. This is not an option. Let's check the options directly.

• Let's check 6:36 PM. Minute hand is at 36 min = 216 degrees. Hour hand is at 6.6 hours = 6.6 * 30 = 198 degrees. Difference = 216 - 198 = 18 degrees. 18 degrees corresponds to 3 minutes on the clock face (18/6 = 3). Thus, at 6:36 PM, the minute hand is exactly 3 minutes ahead of the hour hand.