**at noon the minute and the hour hands overlap. when will it happen next?

1 hour and 60/11 minutes.

Explanation:

How many degrees the Minute Hand Travels in 1 minute: 6º

How many degrees the Hour Hand Travels in 1 minute: 0.5º

If x is the number of minutes the clock traveled after 1 o'clock to reach the next meeting point,

The degrees that the minute hand traveled is 6 * x = 6xº

The degrees that the hour hand traveled is 0.5 * x = 0.5xº

In 12 hours the hour hand travels 360º,

So, in 1 hour, it travels 30º

Total number of degrees the clock traveled after 1 o'clock to reach the next meeting point,

6x = 30 + 0.5x

6x because the minute hand starts at the beginning.

30 because that counts for the number of degrees covered until 1 o'clock.

0.5x because that counts for the number of degrees the hour hand travels after 1 o'clock.

6x = 30 + 0.5x

5.5x = 30

55x = 300

x = 300/55 = 60/11

If you add back that one hour, then it becomes 1 hour and 60/11 minutes.