At which one of the following times, do the hour hand and the minute hand of the clock make an angle of 180° with each other?
Between 7:05 hours and 7:10 hours
The hour hand and the minute hand of the clock make an angle of 180° with each other 11 times during 12 hours.
Options clearly mentioned we need to find the time where hour hand and minute hand make an angle of 180° after 7 o'clock.
Angle covered by minute hand in 1 minute\(= ({360 \over 60})^\circ = 6^\circ\)
Angle covered by hour hand in 1 hour \(= ({360 \over 12})^\circ = 30^\circ\)
Angle covered by hour hand in 1-minute \(= ({30 \over 60})^\circ = ({1 \over 2})^\circ\)
From the above equation, we can say that the minute hand goes ahead by \(5 {1\over 2}^\circ\) in comparison to the hour hand in 1 minute.
At, 7 o'clock angle made between the hour hand and minute hand = 7 × 30° = 210°. (or it is 360° - 210° = 150°)
For making an angle of 180° between the hour hand and minute hand, the Minute hand needs to cover 30° with respect to the hour hand.
So, the time required to travel 30° by minute hand with respect to hour hand past 7 o'clock \(= {30^\circ \over ({11\over 2})^\circ} minutes = {60\over 11}minutes = 5 {5 \over 11} minutes\)
Therefore, the hour hand and the minute hand of the clock make an angle of 180° with each other Between 7:05 hours and 7:10 hours.
Shortcut Trick
- (5n - 30) ×\(12\over11\) minutes past n, when n > 6
- (5n + 30) × \(12\over11\) minutes past n, when n < 6
we simply put n = 7, we get
((5 × 7) - 30) × \(12\over11\) minutes past 7 o'clock
= (35 - 30) × \(12\over11\) minutes past 7 o'clock
= \(5 ×12\over11\) minutes past 7 o'clock
= \(60 \over 11\) minutes past 7 o'clock
= \(5{5 \over 11} minutes\) past 7 o'clock .
Therefore, the hour hand and the minute hand of the clock make an angle of 180° with each other Between 7:05 hours and 7:10 hours.
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1. Between 3:16 p.m. and 3:17 p.m., both hour hand and minute hand coincide.
2. Between 4:58 p.m. and 4:59 p.m., both minute hand and second hand coincide.
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