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

The correct option is 6 : 36 PM.

To understand why this is correct, we can analyze the movement of the hands of a clock using relative speed.

Let's define the movement of the clock hands in terms of "minute spaces" on the clock dial (which has 60 minute spaces in total):
In 60 minutes, the minute hand traverses 60 minute spaces, while the hour hand traverses 5 minute spaces (for example, from the 12 o'clock mark to the 1 o'clock mark).
Therefore, in 60 minutes, the minute hand gains $60-5=55$ minute spaces over the hour hand.

This means the relative speed of the minute hand with respect to the hour hand is:
$\frac{55}{60}=\frac{11}{12}$ minute spaces gained per minute.

Alternatively, the minute hand gains 1 minute space in:
$\frac{60}{55}=\frac{12}{11}$ minutes.

Now, let's look at the initial position at 6 PM:
At exactly 6 PM, the hour hand is at the 6 o'clock mark (30 minute spaces ahead of the 12 o'clock mark), and the minute hand is at the 12 o'clock mark (0 minute spaces).
So, at 6 PM, the hour hand is 30 minute spaces ahead of the minute hand.

We want to find the time when the minute hand is ahead of the hour hand by 3 minutes (which corresponds to 3 minute spaces on the dial).
For this to happen, the minute hand must first catch up the 30 minute space gap to the hour hand, and then move an additional 3 minute spaces ahead of it.
Thus, the total spaces that the minute hand needs to gain over the hour hand is:
$30+3=33$ minute spaces.

Since the minute hand gains 1 minute space in $\frac{12}{11}$ minutes, the time required to gain 33 minute spaces is:
$33\times \frac{12}{11}$ minutes.

Calculating this value gives:
$3\times 12=36$ minutes.

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