How many times in a continuous period of 24 hours, are the 2 hands of an ordinary clock at a right angle?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 44 times
The minute hand gains one full lap (360°) on the hour hand about every \(1\frac{1}{11}\) hours, so in 12 hours it laps the hour hand 11 times, not 12. In every lap the angle between the hands grows from 0° to 360° and passes through 90° twice (once at 90° and once at 270°).
In 12 hours: 11 laps \(\times\) 2 right angles \(=22\) times
In 24 hours: \(2\times22=44\) times
Tip: Most hours have two right angles, but 3 o’clock and 9 o’clock are each shared by two hours: between 2 and 4 the hands are at right angles only three times (about 2:27, 3:00 and 3:33), and the same happens between 8 and 10. That is why 12 hours give 22, not 24, and why 48 in Option (4) is too many.