At what angle the hands of a clock are inclined at 15 minutes past 5?

Angle traced by hour hand in	13	hrs =	❨	360	x	13	❩	°	= 130°.
	3			12		3

Angle traced by min. hand in 20 min. =	❨	360	x 20	❩	°	= 120°.
		60

∴ Required angle = (130 - 120)° = 10°.

Mother's age when Ayesha's brother was born = 36 years.

Father's age when Ayesha's brother was born = (38 + 4) years = 42 years.

∴ Required difference = (42 - 36) years = 6 years.

Let Ronit's present age be x years. Then, father's present age =(x + 3x) years = 4x years.

∴	(4x + 8) =	5	(x + 8)
		2

⟹ 8x + 16 = 5x + 40

⟹ 3x = 24

⟹ x = 8.

Hence, required ratio =	(4x + 16)	=	48	= 2.
	(x + 16)		24

Let the present ages of son and father be x and (60 -x) years respectively.

Then, (60 - x) - 6 = 5(x - 6)

⟹ 54 - x = 5x - 30

⟹ 6x = 84

⟹ x = 14.

∴ Son's age after 6 years = (x+ 6) = 20 years..

Let C's age be x years. Then, B's age = 2x years. A's age = (2x + 2) years.

∴ (2x + 2) + 2x + x = 27

⟹ 5x = 25

⟹ x = 5.

Hence, B's age = 2x = 10 years.

Time from 7 a.m. to 4.15 p.m. = 9 hrs 15 min. =	37	hrs.
	4

3 min. 5 sec. of this clock = 3 min. of the correct clock.

⟹	37	hrs of this clock =	1	hrs of the correct clock.
	720		20

⟹	37	hrs of this clock =	❨	1	x	720	x	37	❩	hrs of the correct clock.
	4			20		37		4

      = 9 hrs of the correct clock.

∴ The correct time is 9 hrs after 7 a.m. i.e., 4 p.m.

In 12 hours, the hands coincide or are in opposite direction 22 times.

∴ In 24 hours, the hands coincide or are in opposite direction 44 times a day.

At 4 o'clock, the hands of the watch are 20 min. spaces apart.

To be in opposite directions, they must be 30 min. spaces apart.

∴ Minute hand will have to gain 50 min. spaces.

55 min. spaces are gained in 60 min.

50 min. spaces are gained in	❨	60	x 50	❩min. or 54	6	min.
		55			11

∴ Required time = 54	6	min. past 4.
	11

When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.

At 7 o'clock, they are 25 min. spaces apart.

∴ Minute hand will have to gain only 5 min. spaces.

55 min. spaces are gained in 60 min.

5 min. spaces are gained in	❨	60	x 5	❩min	= 5	5	min.
		55				11

∴ Required time = 5	5	min. past 7.
	11

In 12 hours, they are at right angles 22 times.

∴ In 24 hours, they are at right angles 44 times.