How many times are the hands of a clock at right angle in a day?

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

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

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.

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.

Angle traced by hour hand in	21	hrs =	❨	360	x	21	❩	°	=	157	1	°
	4			12		4					2

Angle traced by min. hand in 15 min. =	❨	360	x 15	❩	°	= 90°.
		60

∴ Required angle =	❨	157	1	❩	°	- 90° = 67	1	°
			2				2

At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.

To be coincident, it must gain 15 min. spaces.

55 min. are gained in 60 min.

15 min. are gained in	❨	60	x 15	❩min	=	16	4	min.
		55					11

∴ The hands are coincident at 16	4	min. past 3.
	11

Angle traced by hour hand in	17	hrs =	❨	360	x	17	❩	°	= 255°.
	2			12		2

Angle traced by min. hand in 30 min. =	❨	360	x 30	❩	°	= 180°.
		60

∴ Required angle = (255 - 180)° = 75°.

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock).

AM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

PM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

The hands overlap about every 65 minutes, not every 60 minutes.

∴ The hands coincide 22 times in a day.

Time from 12 p.m. on Monday to 2 p.m. on the following Monday = 7 days 2 hours = 170 hours.

∴ The watch gains	❨	2 + 4	4	❩min.	or	34	min. in 170 hrs.
			5			5

Now,	34	min. are gained in 170 hrs.
	5

∴ 2 min. are gained in	❨	170 x	5	x 2	❩hrs	= 50 hrs.
			34

∴ Watch is correct 2 days 2 hrs. after 12 p.m. on Monday i.e., it will be correct at 2 p.m. on Wednesday.

Angle traced by the hour hand in 6 hours =	❨	360	x 6	❩	°	= 180°.
		12