At what time between 4 and 5 o'clock, will the hands of a watch point in opposite directions?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(54 \frac{6}{11}\) min past 4
The minute hand moves 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° on the hour hand every minute. The hands point in opposite directions when they are 180° apart.
At 4:00 the hour hand is at \(4\times30^\circ=120^\circ\) and the minute hand at 0°, so the minute hand is 120° behind
To be 180° ahead, it must gain \(120^\circ+180^\circ=300^\circ\)
Time taken \(\displaystyle =\frac{300}{5.5}=\frac{600}{11}=54\frac{6}{11}\) min
Tip: The angle between the hands at H hours M minutes is \(|30H-5.5M|\). Setting \(|120-5.5M|=180\) gives \(M=54\frac{6}{11}\) at once; the other root is negative, so there is only one such time between 4 and 5.