At what time between 7:30 and 8:00 are the hands of a clock at right angles?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 7:54\(\frac{6}{11}\)
The minute hand gains 5.5° a minute on the hour hand. At 7:00 it is 210° behind the hour hand. It is 90° behind once (early in the hour); the hands are at right angles again when it has passed the hour hand and gone 90° ahead of it.
At 7:00 the hour hand is at \(7\times30^\circ=210^\circ\) and the minute hand at 0°
90° behind: a gain of \(210^\circ-90^\circ=120^\circ\), which takes \(\displaystyle\frac{120}{5.5}=21\frac{9}{11}\) minutes; 7:21\(\frac{9}{11}\) is before 7:30
90° ahead: a gain of \(210^\circ+90^\circ=300^\circ\)
Minutes needed \(\displaystyle =\frac{300}{5.5}=\frac{600}{11}=54\frac{6}{11}\)
The time is \(54\frac{6}{11}\) minutes past 7, which lies between 7:30 and 8:00
