# how many 90degrees are formed by the hour hand and minute hand in 12 hrs????

Nishant Vora IIT Patna
9 years ago
Hello Sudent

The continuous movement of the hour hand brings about something analogous to the "sidereal day" problem. Most of the time, there are 2 occurances per hour of 90 degrees between the hands. But after 2 PM, for example, there is an occurance before 2:30, and the next occurance is 3 PM. There is then one occurance after 3 PM, the next one being after 4 PM. The same thing happens again at 8 and 9. So it seems that a total of 44 is probably right. Another way to 'visualize' that 'twice per hour' can't always work is to see that occurances of 90 degrees must be farther apart than 30 minutes, because both hands are advancing, not just the minute hand. For some hours, the first occurance will be late enough in the hour that the next occurance is in the next hour.
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Let me show you a mathematical approach.
Common sense dictates that the minute hand moves at a faster rate of 5.5 degrees a minute (because the hour hand moves 0.5 degrees a min and the minute hand moves 6 degrees a minute).
We start at 12 midnight. The hands are together. For subsequent 90 degree angles to occur, the minute hand must "overtake" the hour hand by
90 degrees, then 270 degrees, then 360 + 90 degrees, then 360 + 270 degrees, then 360 + 360 +90 degrees.. and so on.
This can be re-expressed as: (1)90, 3(90), 5(90), 7(90), 9(90), 11(90)... n(90).
The number of minutes this takes to happen can be expressed as (1)90/5.5, 3(90)/5.5, 5(90)/5.5, 7(90)/5.5, 9(90)/5.5, 11(90)/5.5... n(90)/5.5.

In one day, there are 24 hr * 60 mins = 1440mins
To find the maximum value of n,
n(90)/5.5 = 1440
n = 88

but as seen from above, n must be an odd number (by pattern recognition and logic)
hence n must be the next smallest odd number (87)
counting 1,3,5,7,9,11......87, we see that the number of terms = (87-1)/2 +1 = 44.

In other words, the minute hand "overtakes" the hour hand on 44 occasions in 24 hours in order to give a 90 degree angle.