Question

At what time between 7 and 8 on an analog clock do the hour and minute hand overlap?

Answer 1

At 7:00 the hour hand has gone through 210° , and angle between the two hands is 210°

the hour hand moves 30° per hour or 1/2 ° per minute
while the minute hand moves 6 degrees per minute
so the minute is gaining on the hour hand at the rate of 5 1/2 or 11/2 ° a minute
So to make up the 210° would take 210÷(11/2)
= 420/11 minutes or 38 2/11 minutes or 38 : 10.9 minutes

so the time is 7 : 38 : 10.9

The above becomes a general solution;
at any time between t and t+1 o'clock,
divide t by 11/2 to find the minutes past t o'clock when the hands overlap