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Worked Examples

Example-1

At what time, between seven o’clock and eight o’clock, will the hands of a clock be in the same straight line, but not together?

Solution
Figure (a) Figure (b)

 

Figure (a) shows the positions of hands of the clock at seven o’clock and Figure (b) shows the positions of hands of the clock when both the hands are in opposite direction in a straight line (i.e., the angle between the two hands is 180°).

 

The angle between the two hands at seven o’clock is 150°.

 

The two hands will be in a straight line, when the angle between them is 180°, i.e., when the angle increases by another 30°.

 

The angle changes at the rate of 5.5° per minute.

 

So, the time taken to change the angle by 30° is Description: 106505.png minutes.

 

Therefore, the hands are in the same straight line (but not together), at Description: 106512.png minutes past 7.

 

Answer: Description: 106516.png minutes past 7
 

 

Example-2

At what time, between four o’clock and five o’clock, the two hands of the clock overlap?

Solution

At four o’clock, the hour hand will be at 4 and the minute hand will be at 12. So, the angle between the two hands at four o’clock is 120°.

 

The hands overlap when the angle between them reduces to zero.

 

The angle changes by 5.5° in 1 minute.

 

So, time taken to change the angle by Description: 106526.png minutes.

 

Therefore, the hands overlap at Description: 106536.png minutes past 4.

Direct Formula:

 

Between t and (t + 1) o’clock, the two hands will overlap at Description: 106544.png minutes past t.

 

In this case, Description: 106554.png minutes past 4.

 
Note: Remember the direct formula and apply to get the answer much faster.

 

Example-3

At what time, between four o’clock and five o’clock, are the hands of the clock at right angle?

Solution

At four o’clock the angle between the minute and the hour hand is 120°. This angle reduces at the rate of 5.5° per minute till it reaches zero and thereafter the angle increases at the same rate. The two hands are at right angles when the angle between them changes by 30° and 210° (i.e., 120° - 90° and 120° + 90°).

 

The angle changes by 5.5° in 1 minute.

 

So, the angle changes by 30° in Description: 106588.png

 

The angle changes by 210° in Description: 106597.png

 

Thus, the hands are at right angle at Description: 106602.png minutes past 4 and again at Description: 106616.png minutes past 4.

 

[If we express the fraction of a minute in terms of seconds and round it off to the nearest integer, the two hands are at right angle once at 4 hours, 5 minutes, 27 seconds and again at 4 hours, 38 minutes, 2 seconds.]
 

Direct Formula:

 

Between t and (t + 1) o’clock, the two hands are at right angles at Description: 106633.png minutes past t.

 

Hence, in this case, they will be at right angles at Description: 106653.png and Description: 106668.png minutes past 4 or Description: 106684.png minutes and Description: 106693.png minutes past 4.
 

 

Example-4

A clock is set right at 5:00 a.m. The clock loses 16 minutes in 24 hours. What will be the right time when the clock indicates 10:00 p.m. on the fourth day?

Solution

Time from 5 a.m. of a particular day to 10 p.m. on the fourth day is 89 hours. (24 + 24 + 24 + 17.) The clock loses 16 minutes in 24 hours. In other words, 23 hours 44 minutes of this clock is equivalent to 24 hours of the correct clock or Description: 106704.png hours of this clock = 24 hours of the correct clock.

 

Thus, 89 hours of this clock Description: 106730.png = 90 hours of correct clock.

 

So, 89 hours of this clock = 90 hours of the correct clock.

 

Therefore, it is clear that in 89 hours this clock loses 1 hour and hence the correct time is 11:00 p.m., when this clock shows 10:00 p.m.

 

Answer: 11:00 p.m.
 

 

Example-5

At what time, between four o’clock and five o’clock, are the hands 2 minutes space apart?

Solution

The hands are two minutes space apart is the other way of telling that the angle between the hands is 12° (one minute space apart is equal to 6°).

 

At four o’clock, the angle between the hands is 120°. When this angle changes by (120° − 12°) = 108° and (120° + 12°) = 132°, the angle between the hands reduces to 12°.

The angle between the hands changes by 5.5° in 1 minute.

 

The angle changes by 108° in Description: 106747.png

 

The angle changes by 132° in Description: 106760.png

 

Thus, the hands are 2 minutes space apart, once at Description: 106767.png minutes past 4 and again at 24 minutes past 4 [i.e., approximately at 4 hours, 19 minutes, 38 seconds and at 4 hours, 24 minutes].
 

Direct Formula:

 

Between ‘t’ and ‘(t + 1)’ o’clock, the two hands will be ‘a’ minutes apart at Description: 106797.png or Description: 106815.png or 24 minutes.

 

Therefore, they will be 2 minutes spaces apart at Description: 106829.png minute past 4 and 24 minute past 4.
 

 

Example-6

At what time, between 5:30 and 6:00, will the hands of the clock be at right angles?

 

Figure (1) Figure (2) Figure (3)

 

Solution

At five o’clock, the angle between the minute and the hour hand is 150°.

 

In 1 minute, the angle changes by 5.5°.

 

In 30 minutes, the angle changes by 5.5 x 30 = 165°.

 

So, the angle between the minute and the hour hand at 5:30 is 165° – 150° = 15°.

 

The hands are at right angle when the angle between them further increases by another 75°.

Angle changes by 5.5° in 1 minute.

 

So, angle changes by 75° in Description: 106863.png minutes.

 

Therefore, the hands are at right angles at Description: 106873.png minutes past 5.

 

Answer: Description: 106876.png minutes past 5.
 

 

Example-7

A clock is set right at 4:00 a.m. on Sunday. The clock loses 20 minutes in 24 hours. What will be the correct time when this clock indicates 3:00 a.m. on Wednesday?

Solution

Time from 4:00 a.m. on Sunday to 3:00 a.m. on Wednesday = 24 × 3 – 1 = 72 – 1 = 71 hours.

 

Now, 23 hours 40 minutes of this clock = 24 hours of correct clock.

 

Description: 106892.png hours of this clock = 24 hours of correct clock.

 

So, 71 hours of this clock Description: 106901.png of correct clock.

 

Therefore, the correct time is 3:00 a.m. + (72 – 71) = 4:00 a.m.
 

 

Example-8

A minute hand of a clock overtakes the hour hand at intervals of 63 minutes of correct time. How much time does the clock lose or gain per day?

Solution

In any clock, the minute and hour hands overlap once in every Description: 106916.png minutes as per that clock.

 

In the given clock, the minute and hour hands are overlapping once in 63 minutes of correct time.

 

Thus, the clock is gaining Description: 106925.png minutes in 63 minutes.

 

So, time gained in 24 hours is Description: 106933.png minutes.
 

Direct Formula:

 

Gain or loss in 24 hours (1 day) = [Description: 106943.png - Given interval in minutes] × 60 × Description: 106954.png

(or) Description: 106962.png.

 

If the sign is positive (+), the clock is gaining time and if the sign is negative (-), the clock is losing time.

 

Here, Description: 106969.png

 

Since the sign is positive, there is a gain of Description: 106976.png minutes per day.
 

 

Example-9

What is the angle between the minute hand and the hour hand of a clock when the clock shows 3 hours 20 minutes?

Solution

The angle between the minute hand and the hour hand at three o’clock is 90°.

 

In 1 minute, the angle changes by 5.5°.

 

In 20 minutes, the angle changes by 5.5 x 20 = 110°.

 

Therefore, the angle between the two hands at 3:20 = 110° - 90° = 20°.
 

 

Example-10

At what time, between five o’clock and six o’clock, the minute and hour hands make an angle of 34° with each other?

Solution

The angle between the minute hand and the hour hand at five o’clock is 150°.

 

The angle between the hands becomes 34° when the angle changes by 116° and 184° i.e., (150° – 34°) and (150° +34°).

 

The angle changes by 5.5° in 1 minute.

 

The angle changes by 116° in Description: 106990.png minutes.

 

The angle changes by 184° in Description: 106998.png minutes.

 

Therefore, the angle between the two hands is 34° when the time is 5 hours Description: 107017.png minutes, and again at 5 hours Description: 107027.png minutes.
 

 

Example-11

At what time between five o’clock and six o’clock will the minute and the hour hand be perpendicular to each other?

Solution

When the angle between the two hands is 90°, the hands are perpendicular to each other.

 

The angle between the minute and the hour hand at five o’clock is 150°.

 

The angle becomes 90°, when it changes by 60° and 240°, i.e., (150° – 90°) and (150° + 90°).

 

The angle changes by 5.5° in 1 minute.

 

The angles changes by 60° in Description: 107045.png minutes.

 

The angle changes by 240° in Description: 107061.png minutes.

 

Therefore, the two hands are perpendicular at 5 hours Description: 107070.png minutes, and again at 5 hours Description: 107077.png minutes.
 

 

Example-12

At what angle are the hands of a clock inclined at 25 minutes past 5?

Solution

The angle between the minute and the hour hand at five o’clock is 150°.

 

In 1 minute the angle changes by 5.5°.

 

So, in 25 minutes the angle changes by 5.5 × 25 = 137.5°.

 

Thus, the angle between the two hands at 5:25 is 150° − 137.5° = 12.5°.
 





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