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 the 6, and the minute hand is at the 12.

The minute hand moves at 6° per minute (360° in 60 minutes).

The hour hand moves at 0.5° per minute (360° in 12 hours = 30° per hour).

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Let $t$ be the number of minutes after 6:00 pm.

Position of minute hand from 12 o'clock: $6t$ degrees

Position of hour hand from 12 o'clock: $180 + 0.5t$ degrees (starting at 6, which is 180°)

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The minute hand is ahead by 3 minutes means it is ahead by 3 minute divisions on the clock face.

Each minute division represents 6°, so 3 minutes = $3 \times 6° = 18°$

The equation for the minute hand being ahead of the hour hand by 18°:

$6t - (180 + 0.5t) = 18$

$6t - 180 - 0.5t = 18$

$5.5t = 198$

$t = \dfrac{198}{5.5}$

$t = 36$

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The time is 6:36 pm.

At 6:36 pm:

• Minute hand position: $36 \times 6° = 216°$
• Hour hand position: $180° + 36 \times 0.5° = 198°$
• Difference: $216° - 198° = 18°$ (which equals 3 minute divisions)

Therefore, the answer is Option 3: 6:36 pm.