Clock Minute Hand Position Calculation

This solution determines the exact position of the minute hand when both the hour and minute hands of a clock overlap between the 5 and 6 hour markings.

Calculating Hands Coincidence Time

Let $M$ be the minutes past 5:00 when the hands coincide. The angle of the minute hand from the 12 is $ \theta_M = 6M $ degrees. The angle of the hour hand from the 12 is $ \theta_H = (5 \times 30) + (0.5M) $ degrees.

At coincidence, $ \theta_M = \theta_H $:

$ 6M = 150 + 0.5M $

Solving for $M$:

$ 5.5M = 150 $

$ M = \frac{150}{5.5} = \frac{300}{11} $

Converting to minutes and seconds:

$ M = 27 \frac{3}{11} \text{ minutes} \approx 27.27 \text{ minutes} $

Minute Hand Position Relative to Markings

The minute hand is located at the $\frac{300}{11}$ minute mark on the clock dial.

The clock markings relevant to this position are:

• The marking for 5, which represents 25 minutes past the hour.
• The marking for 6, which represents 30 minutes past the hour.

Determining Closest Clock Marking

We compare the minute hand's position ($ M = \frac{300}{11} $) to the positions of the markings (25 and 30 minutes).

Distance from the marking for 5:

$ \left| \frac{300}{11} - 25 \right| = \left| \frac{300 - 275}{11} \right| = \frac{25}{11} \text{ minutes} \approx 2.27 \text{ minutes} $

Distance from the marking for 6:

$ \left| \frac{300}{11} - 30 \right| = \left| \frac{300 - 330}{11} \right| = \frac{30}{11} \text{ minutes} \approx 2.73 \text{ minutes} $

Since $ \frac{25}{11} < \frac{30}{11} $, the minute hand's tip is closer to the marking for 5.