Question

At what time between 7 and 8 o'clock will both the hands of the clock coincide

Answer 1

7:40 is the time when they'll meet

Answer 2

Getting a relation between the hour hand & minute hand

Hour hand -> 60 min - 30 deg => 1 min - 30/60 deg => 1 min - 0.5 deg

Minute hand -> 60 min - 360 deg => 1min - 360/60 deg => 1 min - 6 deg

Hence, Minute hand movement = 12 * Hour hand movement.

Ideally, the hands of the clock would be together if both the hands are on 7 which is 7.35. But the hour clock would have made some ground beyond 7 which is 17.5 Deg. How 17.5 deg?From the hour hand relation 35 min - 35 * .5 deg = 17.5 deg

Now, if we consider the minute hand moves y deg past 7 and the hour hand moves x deg past 7, for them to coincide

1. y deg=17.5 deg+ x deg. This will be our Equation 1.

2. From the Hour & Minute hand relation(statements, in the beginning), the relation we can infer is y=12*x. This is our equation 2.

Now from the eq 1 & 2, 12x=17.5+x => 11x=17.5=>x = 17.5/11

3. This would lead to y Deg=12*17.5/11 Deg (from Eq 2)

4. From minute hand relation 1 min - 6 Deg => 1 Deg - 1/6 min => y Deg = y/6 min

From eq 3 & 4, 12*17.5/11*1/6= 35/11 = 3+2/11 min

Hence the minute hand would have moved 3+2/11 minutes pass 7. Therefore the time would be 7:35+3+2/11=7 hours and 38+2/11 min.