Hint

In 1 hour, the minute hand gains 330° over the hour hand. i.e. in 60 minute, the minute hand gains 330° over the hour hand.
∴ In 16 minutes, the minute hand gains over the hour hand by

Hint

In a clock, each minute makes 6°
∴ 3 minutes will make 6 × 3 = 18°

Hint

At 2’O Clock, Minute Hand will be 10 × 6 = 60° behind the Hour Hand.
In 30 minutes, Minute Hand will gain 5 ½° × 30 = 150 + 15 = 165°
∴Angle between Hour Hand and Minute Hand = 165 – 60 = 105°

Hint

Required angle = 240 – 24 × (11/2) = 240 –132 = 108°.

Hint

At 15 minutes past 5, the minute hand is at 3 and hour hand slightly advanced from 5. Angle between their 3rd and 5th position.
Angle through which hour hand shifts in 15 minutes is
∴Required angle =

Hint

Angle traced by the hour hand in 12 hours = 360°
Angle traced by it in 3 hrs 25 min. i.e.
Angle traced by it in 25 min.
Required angle

Hint

When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.
At 7 o’clock, they are 25 min. spaces apart.
∴ Minute hand will have to gain only 5 in. spaces.
55 min. spaces are gained in 60 min.
5 min. spaecs are gained in

Hint

Hint

At 5 o’clock, the hands are 25 min. spaces apart.
To be at right angles and that too between 5.30 and 6, the minute hand has to gain (25 + 15) = 40 min. spaces
55 min. spaces are gained in 60 min.
40 min. spaces are gained in

Hint

At 4 o’clock, the hands of the watch are 20 min. spaces apart.
To be in opposite directions, they must be 30 min. spaces apart.
∴ Minute hand will have to gain 50 min. spaces
55 min. spaces are gained in 60 min.
50 min. spaces are gained in

Hint

At 9’o clock, the Minute Hand is ahead of Hour Hand by 15 minutes. The hands will be opposite to each other when there is a space of 30 minutes between them.
This will happen when the Minute Hand gains 15 minutes’ space over Hour Hand.
Time taken by Minutes Hand to gain 15 minutes

Hence the Hands are opposite to each other at minutes past 9.

Hint

Since, in one hour, two hands of a clock coincide only once, so, there will be value.
Required time T = minutes past H.
Here H = initial position of hour hand = 3(Since 3 o’clock)
A° = required angle = 0°(Since it coincides)
T = minutes past 3 = minutes past 3.

Hint

55 min spaces are gained in 60 min
⇒35 min spaces will be gained in 38.18 min.
⇒Answer = 7 hrs + 38.18 min.

Hint

55 min. spaces are covered in 60 min.
60 min. spaces are covered in
Loss in 64 min.
Loss in 24 hrs.

Hint

Angle traced by the hour hand in 6 hours

Hint

Time from 5 a.m. on a day to 10 p.m. on 4th day is 89 hours.
Now, 23 hrs. 44 min. of this clock are the same as 24 hours of the correct clock.
i.e., ³⁵⁶⁄₁₅ hrs. of this clock = 24 hrs. of correct clock.
∴89 hrs. of this clock =hrs. of correct clock
= 90 hrs of correct clock.
So, the correct time is 11 p.m.

Hint

Time from 7 a.m. to quarter pas 4
= 9 hours 15 min.=555 min.
Now, 37⁄12 min. of this watch = 3 min. of the correct watch.
555 min. of this watch = min.
= of the correct watch.
Correct time is 9 hours after 7 a.m. i.e., 4 p.m.

Hint

The clock gains 15 min in 24 hours.
Therefore, in 16 hours, it will gain 10 minutes.
Hence, the time shown by the clock will be 4.10 am.

Hint

Time interval indicated by incorrect clock = 6 p.m – 1 p.m = 5 hrs.
Time gained by incorrect clock in one hour = + 1 min.= + 1/60 hr.
Using the formula,


⇒ True time interval
∴ True time = 1 p.m. + hrs. = 5 p.m. + hrs.
= 5 p.m. + × 60 min = minutes past 5.

Hint

Time from noon on Sunday to 3 pm on Wednesday = 75 hours.
24 hours 2 minutes of the first clock= 24 hours of the correct one.
⇒ 1 hour of the first clock= 24 × (30/721) hours of correct one.
⇒ 75 hours of the first clock = 24 × 30 × (75/721) hours of correct one
= 54000/721 hours = 74 hours 53.7 min.
Hence the answer is 2:54 pm.

Hint

In a watch than is running correct the minute hand should cross the hour hand once in every min. So they should ideally cross 3 times once in min = 196.36 minutes.
But in the watch under consideration, they meet after every 3hr, 18 min and 15 seconds,
i.e. min.
Thus, our watch is actually losing time (as it is slower than the normal watch). Hence when our watch elapsed = 1426.27.
Hence the amount of time lost by our watch in one day i.e., 13 min and 50s (approx).

Hint

Time between 10 a.m. on Mon to 10:30 a.m. on Sun = 144 ½ hrs.
24 ½ hours of incorrect clock = 24 hours of correct time.
∴144 ½ hours of incorrect clock = x hours of correct time.
∴x = = 144 hours i.e., the true time is 10 a.m. on Sunday.

Hint

Time from 12 p.m. on Monday to 2 p.m. on the following Monday = 7 days 2 hours = 170 hours.
∴ The watch gains
Now, min. are gained in 170 hrs.
∴ 2 min. are gained in
∴ Watch is correct 2 days 2 hrs. after 12 p.m. on Monday i.e. it will be correct at 2 p.m. on Wednesday.

Hint

15th August, 1947 = (1946 years + Period from 1st Jan., 1947 to 15th Aug., 1947)
Counting of odd days :
1600 years have 0 odd day. 300 years have 1 odd day.
47 years = (11 leap years + 36 ordinary years)
= [(11 × 2) + (36 × 1)] odd days = 58 odd days
⇒ 2 odd days.

= 227 days = (32 weeks + 3 days) = 3 odd days.
Total number of odd days = (0 + 1 + 2 + 3) odd days = 6 odd days.
Hence, the required day was ‘Friday’.

Hint

26th Jan., 1950 = (1949 years + Period from 1st Jan., 1950 to 26th Jan., 1950)
1600 years have 0 odd day. 300 years have 1 odd day.
49 years = (12 leap years + 37 ordinary years)
= [(12 × 2) + (37 × 1)] odd days = 61 odd days = 5 odd days.
Number of days from 1st Jan. to 26th Jan. = 26 = 5 odd days
Total number of odd days = (0 + 1 + 5 + 5) = 11 = 4 odd days
∴ The required days was ‘Thursday’

Hint

2000 years have 2 odd days.

= 11 odd days = 4 odd days.
1st January, 2010 has 1 odd day. Total number of odd days
= (2 + 4 + 1) = 7 = 0.
∴ 1st January, 2010 will be Sunday.

Hint

16th July, 1776 mean (1775 years + 6 months + 16 days)
Now, 1600 years have 0 odd days.
100 years have 5 odd days
75 years contain 18 leap years and 57 ordinary years and therefore (36 + 57) or 93 or 2 odd days.
∴1775 years given 0 + 5 + 2 = 7 and so 0 odd days.
Also number of days from 1st Jan. 1776 to 16th July, 1776
Jan. Feb. March April May June July
31 + 29 + 31 + 30 + 31 + 30 + 16
= 198 days = 28 weeks + 2 days = 2 odd days
∴Total number of odd days = 0 + 2 = 2.
Hence the day on 16th July, 1776 was ‘Tuesday’.

Hint

Count the number of days from 2005 onwards to get 0 odd day.

= 7 or 0 odd day.
∴ Calender for the year 2005 is the same as that for the year 2011.

Hint

We go on counting the odd days from 1991 onwards till the sum is divisible by 7. The number of such days are 14 upto the year 2001. So, the calender for 1991 will be repeated in the year 2002.