We show how to solve “Clock” olympiad questions, which appear often in APMOPS/ SMOPS Olympiad questions.
Question: (SMOPS 2001, Q9)
Between 12 o’clock and 1 o’clock, at what time will the hour hand and minute hand make an angle of 110 degrees?
We first analyse the hour hand:
It takes 60 min for the hour hand to move 360/12=30 degrees.
30 deg — 60 min
1 deg — 60/30=2 min
x deg — 2x min
(We measure the degree from the 12 o’clock vertical position.)
Next we analyse the minute hand:
It takes 5 min for the minute hand to move 30 degrees.
30 deg — 5 min
1 deg — 5/30=1/6 min
(x+110) deg — (x+110)/6 min
Now, we want the hour hand to be at x deg, and the minute hand to be at (x+110) deg simultaneously:
Since x deg — 2x min, hence the answer is
2x=20 min after 12 o’clock