Solving the Clock Problem

To find the exact time when the hour and minute hands of a clock coincide between 3 and 4 o'clock, we can use the concept of relative speed.

1. Understanding the Speeds of the Hands

• The minute hand moves 360 degrees in 60 minutes, so its speed is 6 degrees per minute.
• The hour hand moves 360 degrees in 12 hours (720 minutes), so its speed is 0.5 degrees per minute.
• The relative speed at which the minute hand gains on the hour hand is 6 - 0.5 = 5.5 degrees per minute.

2. Initial Position at 3 o'clock

At exactly 3:00, the minute hand is at the 12 (0°) and the hour hand is at the 3 (90°). Therefore, the minute hand is 90 degrees behind the hour hand.

3. Calculating the Time to Coincide

For the hands to coincide, the minute hand must gain 90 degrees on the hour hand. We can calculate the time this takes using the relative speed:

Time = Total Angle to Gain / Relative Speed

Time = 90° / 5.5° per minute

Time = 90 / (11/2) minutes

Time = 180 / 11 minutes

4. Converting to a Mixed Fraction

To convert 180/11 into a more readable format, we divide 180 by 11:

180 ÷ 11 = 16 with a remainder of 4.

This gives us the mixed fraction: $16\frac{4}{11}$ minutes.

Conclusion

The needles will coincide at $16\frac{4}{11}$ minutes past 3 o'clock.