Question

At 6:00 pm, the hour hand and the minute hand of an analog clock are at 180 degrees with each other. After approximately how much time will they be at 180 degrees with each other again?

• A. 48 minutes, 40 seconds
• B. 54 minutes, 33 seconds
• C. 60 minutes
• D. 65 minutes, 27 seconds

Answer 1

The Correct Option is D

To determine when the hour and minute hands of a clock will next be 180 degrees apart after 6:00 pm, we can use the following explanation and calculations:

The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute.

The hour hand moves 30 degrees in 60 minutes, so it moves 0.5 degrees per minute.

At 6:00 pm, the two hands are 180 degrees apart. We need to find the time \(t\) minutes after 6:00 pm when this condition will be true again.

The angle moved by the minute hand in \(t\) minutes is:

\(6t\) degrees

The angle moved by the hour hand in \(t\) minutes is:

\(0.5t\) degrees

Initially, the minute hand is at 180 degrees relative to the hour hand. For them to be 180 degrees apart again, the difference in angle between them must reach 360 degrees (a full circle plus another half circle or a zero-degree overlap plus half circle):

\(|6t - 0.5t| = 360\)

This simplifies to:

\(5.5t = 360\)

Solve for \(t\):

\(t = \frac{360}{5.5} = \frac{720}{11}\) or approximately 65.4545 minutes

Convert the decimal time into minutes and seconds:

\(0.4545 \times 60 \approx 27\) seconds

Thus, the hands will again be at 180 degrees after approximately:

65 minutes and 27 seconds

Option	Time
1	48 minutes, 40 seconds
2	54 minutes, 33 seconds
3	60 minutes
4	65 minutes, 27 seconds

The correct answer is: 65 minutes, 27 seconds