# Worked Examples

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

Figure (a) | Figure (b) |

Figure (a) shows the positions of hands of the clock at seven oâ€™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 180Â°).

The angle between the two hands at seven oâ€™clock is 150Â°.

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

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

So, the time taken to change the angle by 30Â° is minutes.

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

Answer: minutes past 7

At what time, between four oâ€™clock and five oâ€™clock, the two hands of the clock overlap?

At four oâ€™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 oâ€™clock is 120Â°.

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

The angle changes by 5.5Â° in 1 minute.

So, time taken to change the angle by minutes.

Therefore, the hands overlap at minutes past 4.

Direct Formula:

Between ** t** and (

**+ 1) oâ€™clock, the two hands will overlap at minutes past**

*t***.**

*t*In this case, minutes past 4.

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

At what time, between four oâ€™clock and five oâ€™clock, are the hands of the clock at right angle?

At four oâ€™clock the angle between the minute and the hour hand is 120Â°. This angle reduces at the rate of 5.5Â° 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 30Â° and 210Â° (i.e., 120Â° - 90Â° and 120Â° + 90Â°).

The angle changes by 5.5Â° in 1 minute.

So, the angle changes by 30Â° in

The angle changes by 210Â° in

Thus, the hands are at right angle at minutes past 4 and again at 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 (

**+ 1) oâ€™clock, the two hands are at right angles at minutes past**

*t***.**

*t*Hence, in this case, they will be at right angles at and minutes past 4 or minutes and minutes past 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?

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 hours of this clock = 24 hours of the correct clock.

Thus, 89 hours of this clock = 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.

At what time, between four oâ€™clock and five oâ€™clock, are the hands 2 minutes space apart?

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

At four oâ€™clock, the angle between the hands is 120Â°. When this angle changes by (120Â° âˆ’ 12Â°) = 108Â° and (120Â° + 12Â°) = 132Â°, the angle between the hands reduces to 12Â°.

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

The angle changes by 108Â° in

The angle changes by 132Â° in

Thus, the hands are 2 minutes space apart, once at 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 â€˜** t**â€™ and â€˜(

**+ 1)â€™ oâ€™clock, the two hands will be â€˜**

*t***â€™ minutes apart at or or 24 minutes.**

*a*Therefore, they will be 2 minutes spaces apart at minute past 4 and 24 minute past 4.

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) |

At five oâ€™clock, the angle between the minute and the hour hand is 150Â°.

In 1 minute, the angle changes by 5.5Â°.

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

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

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

Angle changes by 5.5Â° in 1 minute.

So, angle changes by 75Â° in minutes.

Therefore, the hands are at right angles at minutes past 5.

Answer: minutes past 5.

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?

Time from 4:00 a.m. on Sunday to 3:00 a.m. on Wednesday = 24 Ã— 3 â€“ 1 = 72 â€“ 1 = 71 hours.

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

hours of this clock = 24 hours of correct clock.

So, 71 hours of this clock of correct clock.

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

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?

In any clock, the minute and hour hands overlap once in every 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 minutes in 63 minutes.

So, time gained in 24 hours is minutes.

Direct Formula:

Gain or loss in 24 hours (1 day) = [ - Given interval in minutes] Ã— 60 Ã—

(or) .

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

Here,

Since the sign is positive, there is a gain of minutes per day.

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

The angle between the minute hand and the hour hand at three oâ€™clock is 90Â°.

In 1 minute, the angle changes by 5.5Â°.

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

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

At what time, between five oâ€™clock and six oâ€™clock, the minute and hour hands make an angle of 34Â° with each other?

The angle between the minute hand and the hour hand at five oâ€™clock is 150Â°.

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

The angle changes by 5.5Â° in 1 minute.

The angle changes by 116Â° in minutes.

The angle changes by 184Â° in minutes.

Therefore, the angle between the two hands is 34Â° when the time is 5 hours minutes, and again at 5 hours minutes.

At what time between five oâ€™clock and six oâ€™clock will the minute and the hour hand be perpendicular to each other?

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

The angle between the minute and the hour hand at five oâ€™clock is 150Â°.

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

The angle changes by 5.5Â° in 1 minute.

The angles changes by 60Â° in minutes.

The angle changes by 240Â° in minutes.

Therefore, the two hands are perpendicular at 5 hours minutes, and again at 5 hours minutes.

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

The angle between the minute and the hour hand at five oâ€™clock is 150Â°.

In 1 minute the angle changes by 5.5Â°.

So, in 25 minutes the angle changes by 5.5 Ã— 25 = 137.5Â°.

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