Clocks

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

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  𝟅 = 30(H) -   \({11 \over {2}}\) (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.?

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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?

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  𝟅 =   \({11 \over {2}}\) (M) -  30(H)
  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?

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In clock, a ll angles except 0 ˚ and 180 ˚ are formed 22 times in 12 hours.   Hence (c).

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

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    𝟅 = 30(H) -   \({11 \over {2}}\) (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?

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Speed of the minute hand = 6 ˚ per minute

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 1/11th of the time already completed in that day. What is the exact time?

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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?

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    At 6 o’ clock it rings 6 times.
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?

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The minute hand takes one hour to cover 360 ˚ .

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?

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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?

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  The clock was 8 minutes late at 2pm on Thursday  and 8 minutes ahead at 4 pm on Satur day
  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?

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At exactly 3 o’ clock the hour hand is 15 minutes spaces ahead of the minute hand. 

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 5th day, then what time will the clock show?

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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.

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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?

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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).