Clocks

Answer: 75°

Angle made by the minute hand with vertical at 3 o'clock is 30 x 3 = 90°
In 30 minutes angle traversed by hour hand = 30 x .5 = 15°
Hence, at 3:30 angle made by the hour hand with the vertical 90 + 15 = 105°

In 30 minutes angle made by the minute hand with the vertical = 30 x 6 = 180°

Now, the angle between hour hand and minute hand is 180 - 105 = 75°

Answer: 22.5°

Angle made by the minute hand with vertical at 9 o'clock is 30 x 9 = 270°
In 45 minutes angle traversed by hour hand = 45 x .5 = 22.5°
Hence, at 9:45 angle made by the hour hand with the vertical 270 + 22.5 = 292.5°

In 45 minutes angle made by the minute hand with the vertical = 45 x 6 = 270°

Now, the angle between hour hand and minute hand is 292.5 - 270 = 22.5°

Answer: 190°

The number of minutes traversed by the hour hand in 6 hours 20 minutes is 60 x 6 + 20 = 380
Since, the hour hand traverses .5° every minute,
the angle traversed by the hour hand in 380 minutes = 280 x 0.5 = 190°

Answer: 112.5°

When the hands of a clock interchange their position, the time would be 12:45 pm. The number of minutes from 9 am to 12 45 pm is 225. Angle traversed by the hour hand during this period is .5 x 225 = 112.5°.

Answer: 44

In 12 hours, the hands of a clock coincide or are opposite in direction 22 times.
Hence, in 24 hours the hands of a clock coincide or are opposite in direction 44 times.

Answer: 5:27

Let the hands of a clock meet each other 't' minutes after 5 o'clock. Then,

Angle made by hour hand with the vertical = 30x5 + 0.5 x t = 150 + .5t
Angle made by minute hand with the vertical = 6t

When the two hands meet each other, the angle between them will be 0.

Hence,

150 + .5t - 6t = 0
t = $\frac{150}{5.5}$ = 27$\frac{3}{11}$
The two hands of a clock meet at 5 : 27 : 3/11

Answer: 1:38

Let the hands of a clock meet each other 't' minutes after 1 o'clock. Then,

Angle made by hour hand with the vertical = 30x1 + 0.5 x t = 30 + .5t
Angle made by minute hand with the vertical = 6t

When the two hands are opposite to each other, the angle between them will be 180.

Hence,

6t - (30 + .5t) = 180
5.5t = 210
t = $\frac{210}{5.5}$ = 38$\frac{2}{11}$
The two hands of a clock are oppposite between 1 and 2 o' clock at 1 : 38 : 2/11

Answer: 3 : 32

Let the hands of a clock meet perpendicular to each other 't' minutes after 3. Then,

Angle made by hour hand with the vertical = 30x3 + 0.5 x t = 90 + .5t
Angle made by minute hand with the vertical = 6t

Between 3 and 4, the two hands can be perpendicular before or after they meet. Hence,

Case 1 : Before the hands meet
90 + .5t - 6t = 90
t = 0 minutes.
Hence, the hands are perpendicular at 3 o'clock

Case 1 : After the hands meet
90 + .5t - 6t = 90
t = 0 minutes.
Hence, the hands are perpendicular at 3 o'clock

6t - (90 + .5t) = 90
5.5t = 180
t = $\frac{180}{5.5}$ = 32.72
The two hands of a clock are perpendicular 32.72 minutes after 3 o'clock

Answer: 65$\frac{5}{11}$ minutes

In one minute hand covers 6°, while hour hand covers .5°. Hence the minute hand gains 5.5° on hour hand ever minute.
The minute hand will meet hour hand again when it gains 360° over the minute hand. The number of minutes taken to gain 360° = $\frac{360}{5.5}$ = 65$\frac{5}{11}$ minutes

Answer: 32 minutes after 6 o' clock

Let the minute hand cross hour hand 't' minutes after 6 o'clock. Then,

Angle made by hour hand with the vertical = 30x6 + 0.5 x t = 180 + .5t
Angle made by minute hand with the vertical = 6t

When the minute hands crosses the hour hand, the angle made by both the hands with the vertical will be same.

Hence,

180 + .5t = 6t
t = $\frac{180}{5.5}$ = 32.72
The minute hands crosses the hour hand 32.72 minutes after 6 o' clock

Answer: 12 : 18 PM

Since the watch gains 3 seconds in 2 minutes, in 1 hour it will gain 3 x $\frac{60}{2}$ = 90 seconds

From 12:00 AM to 12:00 PM, i.e in 12 hours, the watch gains

90 x $12$ = 1080 seconds or 18 minutes,

Hence at 12 noon, the time on the watch is 12:18 pm

Answer: 6 : 00 PM

Since the watch loses 10 seconds in 3 minutes, it will lose 1 minute in 3 x $\frac{60}{10}$ = 18 mins.

i.e when the incorrect watch completes 17 minutes, the correct watch completes 18 minutes.
Hence, if the incorrect watch takes 1 minute, the correct watch takes $\frac{18}{17}$ minutes

When the incorrect watch moves from 9:00 AM to 5:30 PM, i.e 8 and half hours or 510 minutes, the correct watch moves 510 x $\frac{18}{17}$ = 540 minutes or 9 hours.

Hence when the incorrect watch shows 5 : 30 pm, the correct time is 6 pm

Answer: 12 minutes

In a correct watch, the hands coincide every 65$\frac{5}{11}$ minutes.

If the hands coincide every 66 minutes, then
the gain in 66 minutes is 66 - 65$\frac{5}{11}$ = $\frac{6}{11}$

Gain in 24 hours = $\frac{6}{11}$ x $\frac{1}{66}$ x 24 x 60 = 11.9 minutes ~ 12 minutes

Answer: 6:00 am on Wednesday

For the watch to show correct time again, it has to lose 12 hours.

Since the watch loses 10 minutes in 1 hour, it will lose 12 hours or 720 minutes in $\frac{720}{10}$ i.e 72 hours or 3 days.

Hence the watch will show correct time at 6 am on wednesday.

Answer: 9 AM Thurday

Since the first clock loses 5 minutes every hour and second clock gains 5 minute every hour, the first clock is behind the second clock by 10 minutes at the end of an hour.