How many times do the hands of a clock point in exactly opposite directions between 3 p.m. and 10 p.m.?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
The hands are opposite once in every hour except between 5 and 6, because the opposite position for that hour comes exactly at 6:00. Measured from 12, the minute hand gains 5.5° a minute on the hour hand.
At 3:00 the minute hand is 90° behind the hour hand. To be opposite it must gain \(90^\circ+180^\circ=270^\circ\), which takes \(\frac{270}{5.5}=49\frac{1}{11}\) minutes: 3:49\(\frac{1}{11}\)
Between 4 and 5: at 4:54\(\frac{6}{11}\)
Between 5 and 6 none; the next one is at 6:00 (hour hand on 6, minute hand on 12)
Then once in each of the hours 7–8, 8–9 and 9–10 (at about 7:05, 8:11 and 9:16)
At 10 p.m. the hands are 60° apart, and the next opposite position comes at \(21\frac{9}{11}\) minutes past 10, after 10 p.m.
Number of times \(=2+1+3=6\)