When Is The Right Time?
How many times in a day do the hour and minute hands of a clock form a right angle?
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2 solutions
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I really like this solution! Another way of phrasing this solution is as follows: Imagine being on the hour hand and think about what the minute hand does relative to you (so this is the "let the hour hand stand still" part) Since the minute hand makes 12 revolutions and the hour hand makes 1 revolution, the minute hand makes (12 - 1) = 11 revolutions relative to the hour hand. During each such revolution, 2 right angles are made, and this happens twice in a 24 hour period as mentioned above. Hopefully this helps if someone is confused! @Kenneth Choo
Nice way of the rephrasing the solution! (+1)
Great interpretation!
I understand that the answer is 44 according to the other solutions or you can count it manually, but I don't quite understand a part in your explanation.
"If you let the hour hand stand still, you can see that the minute hand will only make 11 revolutions,"
Relevant wiki: Circles - Problem Solving - Medium
The minute hand moves 360 degrees in 60 minutes. This means that the angle of the minute hand is given by 6 t , where t is number of minutes past midnight.
The hour hand moves 30 degrees in 60 minutes. This means that the angle of the minutes hand is given by 0 . 5 t .
The hands start together at midnight. The first time they make a 90 degree angle is when the minute hand has moved 90 degrees further than the hour hand, so this is given by the equation:
6
t
=
0
.
5
t
+
9
0
5
.
5
t
=
9
0
t
=
1
6
1
1
4
minutes
In other words about 16 minutes past midnight.
The next time is when the minutes hand has gained another 180 degrees on the hour hand, and is 90 degrees behind it:
6
t
=
0
.
5
t
+
2
7
0
5
.
5
t
=
2
7
0
t
=
4
9
1
1
1
minutes
.
At about 11 minutes to 1 o'clock.
For every 180 degrees that the minute hand gains on the hour hand there will be one 90 degree angle, so every 4 9 1 1 1 − 1 6 1 1 4 = 3 2 1 1 8 minutes
24 hours is 1440 minutes.
3 2 1 1 8 1 4 4 0 = 4 4
So every 24 hours there are 44 right angles between the minute hand and second hand.
Good way of looking at the problem.
Another approach is to note that between n and n+1 o'clock the hands for a right angle once coming nearer and once parting. So there will be 24 * 2=48 right angles. But between 3 and 4, and 9 and 10, coming near is not possible. So every 12 hours there are two misses, 4 misses in 24 hour. So total is 48-4=44.
The hours that begin with a 90-degree angle create the two misses; therefore, between 2 & 3 and 8 & 9 there is only one 90-degree angle.
http://www.visnos.com/demos/clock
Did the same!
wonderful approach , like it !
@Vedant Kushe this is my approach . As we rotate the hour hand to the 4 and minute to the 1 .So we also encouter a right angle .In the same passion if we do it consecutively till it comes to the 11 hours and minute hand on 2 we complete the 1 hrs . In a day we got 24 hrs so the number of time right angle is formed =24*2=44.
Relevant wiki: Circles - Problem Solving - Medium
The minute hand makes 12 revolutions every 12 hours while the hour hand makes only one revolution. If you let the hour hand stand still, you can see that the minute hand will only make 11 revolutions, so it creates a right angle with the hour hand 22 times and in a 24 hour period the answer would be 2 × 2 2 = 4 4 .