At what time between 10.00 am and 11.00 am, the minute hand and the hour hand of a watch would make an angle of 180°?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 10:22 am
The minute hand moves 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° every minute. At 10:00 the hour hand is 300° ahead of the minute hand (measured clockwise from 12); the hands are in a straight line (180° apart) when this lead has fallen to 180°.
Fall in the lead needed \(=300^{\circ}-180^{\circ}=120^{\circ}\)
Time taken \(\displaystyle =\frac{120}{5.5}=\frac{240}{11}=21\frac{9}{11}\) minutes
The hands are opposite at \(10{:}21\frac{9}{11}\), that is, at about 10:22 am
Tip: The angle between the hands at H hours M minutes is \(|30H-5.5M|\). Setting \(|300-5.5M|=180\) gives \(M=21\frac{9}{11}\) at once.