PRMO 2019 question 7

7. On a clock, there are two instants between 12 noon and 1 PM, when the hour hand and the minute hand are at right angles. The difference in minutes between these two instants is written as a + b/c , where a, b, c are positive integers, with b < c and b/c in the reduced form. What is the value of a + b + c ?

Solution:
Assume that clock is made up of 360 degrees.
Hour hand moves 360/12 = 30 degrees per hour => 30/60 degrees per minute => 0.5 degrees per minute.
Minute hand moves 360 degrees per hour => 360/60 degrees per minute => 6 degrees per minute.
Speed of minute hand relative to hour hand  = 6 - 0.5 = 5.5 degrees per minute.
The angle between them would become 90 after 90/5.5 minutes.
And it would become 270 after 270/5.5 minutes.
Difference between those 2 instants  = (270-90)/5.5 = 180/5.5 = 32 + 8/11
a = 32, b = 8, c = 11
a + b + c = 51

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