APMOPS SMOPS Solution

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?

Full Solution:

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:

2x=(x+110)/6

12x=x+110

11x=110

x=10

Since x deg — 2x min, hence the answer is

2x=20 min after 12 o’clock

Ans: 12.20