Special Angles (0 ° and 180 ° and 90 ° )
FOOD FOR THOUGHT : 0 ° and 180 ° are the two angles which don't form at all between the duration of 1 Hour.
For 0 ° : (i.e. between 11 ‘O Clock to 12 ‘O Clock and Between 12 ‘O Clock to 1 ‘O Clock ). During this duration, only one time both the Hands coincides sharp at 12 ‘O Clock.
For 180 ° : ((i.e. between 5 ‘O Clock to 16 ‘O Clock and Between 6 ‘O Clock to 7’ O Clock ). During this duration, only one time both the Hands are opposite to each other sharp at 6 ‘O Clock.
Q 1. How many times are the hands together in a day ?
Sol : In Hour = 1 time.
In 12 hours = 12 - 1 = 11 times
Between 11 'O'Clock to 1 ‘ O'clock, the time duration is 2 Hours. But the hands coincide at
one time only i.e. right at 12 ‘ O Clock. Therefore,In 12 hours Duration Hands coincide 1 time
less in the complete duration of 12 hours.
Q2. How many times are the hands opposite each other in a day ?
Sol : In 1 hour = 1 time
In 12 hours = 12 - 1 = 11 time
Between 5 'O'Clock to 7 ‘ O'clock, the time duration is 2 Hours. But the hands are opposite to
each other at one time only i.e. right at 6‘ O Clock. Therefore,In 12 hours Duration Hands are
1 time less at 180 ° in the complete duration of 12 hours.
Similarly, In 24 hours = 22 times Hands coincides. (Forms 180 ° angle )
Q3. How many times are the hands be right angled in a day ?
Sol : In 1 hour = 2 times.
In 12 hours = 24 - 2 times = 22 times (At 3 ‘O clock and 9 ‘ O clock , only 1 time right angle forms. At sharp 3 ‘ O clock and 9’ O Clock only.)
In 24 hours = 48 - 4 times = 44 times
Q 4. How many times are the hands together from 10 am to 3 pm ?
Sol : No. of Hours from 10 am to 3 pm. = 5 hours
No. of times coincide = 4 times (12 O’Clock is included)
Q 5. How many times are the hands together from 5 : 30 pm to 9 : 30 pm ?
Sol : No. of Hours from 5 : 30 pm to 9 : 30 pm. = 4 hours
Between 5 to 6 pm. = the hands coincide before 5 : 30
Between 9 to 10 pm. = the hands coincide after 9 : 30
Hence we need to check the duration 6 pm to 9 pm = 3 hours
No. of times coincide = 3 times (12 O’Clock is not included)
Q 6. How many times are the hands opposite to each other from 10 am to 4 pm ?
Sol : No. of Hours from 10 : 00 am to 4 : 00 pm. = 6 hours
No. of times opposite to each other = 6 times. (Because 6 O’Clock is not included.)
Q 7. How many times are the hands opposite to each other from 4 pm to 10 pm ?
Sol : No. of Hours from 4 : 00 pm to 10 : 00 pm. = 6 hours
No. of opposite to each other = 6 - 1 = 5 times. (Because 6 O’Clock is included.)
Q 8. How many times do the hands coincide and opposite to each other from 11 am to 5 pm ?
Sol : Coincide : No. of Hours from 11 : 00 am to 5 : 00 pm. = 6 hours
No. of times coincide = 6 - 1 times (12 O’Clock is included)
Opposite to each other : No. of Hours from 11 : 00 am to 5 : 00 pm. = 6 hours
No. of opposite to each other = 6 times. (Because 6 O’Clock is not included.)
Q 9. How many times are the hands be in a straight line for Sunday 5 PM to Monday 7 PM ?
Sol : FOOD FOR THOUGHT : Straight line = 0 ° (coincide) and 180 ° (opposite to each other) (Mistake committed by students in the exam frequently.)
No. of Hours from Sunday 5 PM to Monday 7 PM. = 24 hours + 2 hours
HANDS COINCIDE
No. of Hours from Sunday 5 PM to Monday 5 PM = 24 hours = 22 times
Between 5 PM to 7 PM = 2 hours = 2 times (Because 12 O’Clock is not included.)
HANDS OPPOSITE TO EACH OTHER
No. of Hours from Sunday 5 PM to Monday 5 PM = 24 hours = 22 times
Between 5 PM to 7 PM = 2 hours = 1 time (Because 6 O’Clock is included.)
No of times = (22 + 1 ) +( 22 + 2 ) = 24 + 23 = 47 times
Hence the answer is 47 times hands be in a straight line.
Q 10. How many times are the hands at right angles from 1 pm to 5 pm ?
Sol : No. of Hours from 1 : 00 pm to 5 : 00 pm. = 4 hours
No of times Right angle forms from 1 : 00 to 5 : 00 = 2 * 4 - 1 = 8 - 1 = 7 times.
(Because 3 PM is included in the duration)
Q 11. How many times are the hands coincide, opposite to each other and are at right angles from 5 pm to 11 pm ?
Sol : No. of Hours from 5 : 00 pm to 11 : 00 pm. = 6 hours
No. of times Hands coincides = 6 times (Because 12 O’Clock is not included.)
No.of times Hands Opposite to each other = 6 - 1 = 5 times (Because 6 O’Clock is included.)
No of times Hands are at right angle = 2*6 - 1 = 12 - 1 = 11 times
(Because 9 PM is included in the duration)
Q 12. How many times are the hands coincide, opposite to each other and are at right angles from 2 pm to 10 pm ?
Sol : No. of Hours from 2 : 00 pm to 10 : 00 pm. = 8 hours
No. of times Hands coincides = 8 times (Because 12 O’Clock is not included.)
No.of times Hands Opposite to each other = 8 - 1 = 7 times (Because 6 O’Clock is included.)
No of times Hands are at right angle = 2*8 - 2 = 16 - 2 = 14 times
(Because 3 PM and 9 PM is included in the duration)
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