At what time between 8 and 9 o’clock will the hands of a clock be in the same straight line but not together?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8:10\(\frac{10}{11}\)
In a straight line but not together means the hands point opposite ways, 180° apart. Measured from 12, the hour hand is at \(240^\circ+0.5^\circ\) per minute after 8 and the minute hand at \(6^\circ\) per minute.
At 8:00 the hour hand is at 240° and the minute hand at 0°: the minute hand is 240° behind
Each minute the minute hand gains \(6^\circ-0.5^\circ=5.5^\circ\) on the hour hand
The gap must come down from 240° to 180°, a gain of \(240^\circ-180^\circ=60^\circ\)
Minutes needed \(\displaystyle =\frac{60}{5.5}=\frac{120}{11}=10\frac{10}{11}\)
So the hands are opposite at \(10\frac{10}{11}\) minutes past 8
