#### Clocks

**1)** Find the angle between the hands of a clock when the time is 7: 40 p.m.

= 30(7) - \({11 \over {2}}\) (40)

= 210 - 220 = |- 10˚ |= 10˚ Hence (a).

**2)**

By what angle will the hour hand rotate between 8: 30 a.m. and 9: 55 a.m.?

The speed of the hour hand = \({1 \over {2}}˚\) per minute

Total time = 85 minutes

So, the hour hand will rotate through = \({85 \over {2}}\) = 42.5 ˚ . Hence (b).

**3)** At what time will the hands of the clock form 110˚ between 2 and 3 o’ clock?

110 ˚ = \({11 \over {2}}\) (M ) - 30(2)

110 + 60 = \({11 \over {2}}\) (M)

M = 30\({10 \over {11}}\)

Required time = 2: 30\({10 \over {11}}\) . Hence (d).

**4)** How many times do the hands of a clock make 110˚ in 12 hours?

**5)** What is the reflex angle between the two hands of a clock when the time is 8:16 a.m.?

= 30(8 ) - \({11 \over {2}}\) (16 )

= 240 - 88 = 152 ˚

Reflex angle = 360 – 152 = 208˚ . Hence (b).

**6)** If the minute hand of a clock moves 234˚, then how many degrees will the hour hand move?

To travel 234 ˚ it will take 39 minutes

Speed of the hour hand = \({1 \over {2}}˚\) per minute

For 39 minutes it will move 19.5 ˚ . Hence (c).

**7)**

A asked a girl, “What is the time now?”. The girl replied that the time left is 1/11^{th} of the time already completed in that day. What is the exact time?

Let the time completed be x hours. Time left = 24 – x = \({x \over {11}}\) à x = 22 hours The exact time => 10 p.m. Hence (b).

**8)**At 6 o’ clock, the clock rings 6 times. The time taken for the six rings is 55 seconds. How long will the clock take to ring at 10 o’clock?

Time lapse between the two rings = \({55 \over {5}}\) = 11

At 10 o’ clock, it rings 10 times. After first ring, the clock will ring 9 times.

Required time = 9 *11 = 99 seconds. Hence (c).

**9)** A clock shows 11 a.m. When the minute hand covers 6480˚, what is the angle between the hour and the minute hands?

For 6480 ˚ it travels 18 hours and time will be 5 p.m.

𝟅 = 30(H) - \({11 \over {2}}\) (M)

𝟅 = 30(5) - \({11 \over {2}}\) (0) = 150˚ . Hence (d).

**10)**

A clock was set right at 11 a.m. on a Saturday. It loses 3% time during the first week and gains 6% in the next week. After 14 days, what will be the time that the clock will show from the time was set right?

There are 7 * 24 = 168 hours in a week. If the clock loses 3% time during the first week, then it will show 3% of 168 hours less than 11 a.m. at the end of the first week à 5.04 hours less After that, the clock gains 6% during the next week. The second week has 6% of 168 hours à more than the actual time. à 10.08 hours more It lost 5.04 hours during the first week and then gained 10.08 hours during the next week, net gain = - 5.04 + 10.08 = 5.04 hours gain in time. Clock will show a time which is 5.04 hours more than 11 a.m. two weeks from the time it was set right. 5.04 hours = 5 hours 2 minutes 24 seconds. Hence (a).

**11)** A clock was 8 minutes behind the actual time on 2 p.m. on Thursday and 8 minutes ahead of actual time on 4 p.m. Saturday. When will it show the correct time?

Thus, the clock has gained 16 minutes in 10(from Thursday after 2 pm) + 24(complete Fr iday) + 16 ( 12 p m to 4 pm on Satur day) = 50 hours

S o 8 minutes will be gained in (8/16)*50 = 25 hours

Thus the clock will show the correct time at near about 3 : 00 pm on Fri day. He nce (d).

**12)** The hour hand and the minute hand of a clock are 18 minutes apart. What will be the exact time, if the hour hand is between 3 and 4 o’ clock?

To be 1 8 minute spaces apart, the minute hand has to gain (15 + 18) = 3 3 minute spaces over the hour hand.

In 1 hour 55 minute spaces are gained.

For gaining 3 3 minute spaces = \({60 \over {55}}\) * 3 3 = 36 minutes.

Required time = 3: 36 . Hence (c).

**13)**

An inaccurate clock shows 6 p.m. and the clock gains 8 minutes for every 12 hours. If the actual time shows 7 p.m. on the 5^{th} day, then what time will the clock show?

Time from 6 p.m. on the first day to 7 p.m. on the fifth day = 97 hours

For every 12 hours clock gains 8 minutes

For every 1 hour 8/12 = 2/3 minutes gain

Total gain from 1 ^{st} day to at 6 p.m. to 5 ^{th} day 7 p.m. => 97 * \({2 \over {3}}\) = 64.6 minutes gain

So, when the actual time is 7 p.m. the clock shows 8: 05 p.m. Hence (a).

**14)**

A clock is set right at 7 a.m. in the morning. It gains 4 seconds every 15 minutes. If the actual time is 4 p.m. on the same day, find the angle between the hour hand and the minute hand the clock shows.

Clock gains 4 seconds in every 15 minutes. So in one hour, it gains 16 seconds. From 7 a.m. to 4 p.m. à 9 hours The clock gains 144 seconds, i.e. \({12 \over {5}}\) minutes 𝟅 = 30(H) ~ \({11 \over {2}}\)(M) = 30(4) ~ \({11 \over {2}}\)(\({12 \over {5}}\)) = 120 – 13.2 = 106.8˚ Hence (b).

**15)**

Meenu’s office time is 9: 30 a.m. to 6: 30 p.m. One day, Meenu works overtime. When she reaches home, her watch shows a reflex angle of 200˚. During the extended hours of work, the minute hand rotates three full cycles. At what time does Meenu reach home?

Reflex angle = 200˚, Normal angle = 360˚ - 200˚ = 160˚ Office ends at 6: 30 p.m. Extended time is 3 hours (since the minute hand rotates three full cycles) and she left office at 9: 30 p.m. 160 = 30(9) ~ \({11 \over {2}}\)(M) \({11 \over {2}}\)(M) = 110 à M = 20 minutes Meenu reaches home at 9: 50 p.m. Hence (d).