How many times are the hands of a clock at right angles between 1 p.m. and 6 p.m. on the same day?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
The hands are at right angles twice in most hours: once with the minute hand 90° behind the hour hand and once with it 90° ahead. But the right angle at 3 o’clock is the second one of the hour 2 to 3 and the first one of the hour 3 to 4 at the same moment, so it is counted only once. The minute hand gains 5.5° a minute.
1 to 2: 1:21\(\frac{9}{11}\) and 1:54\(\frac{6}{11}\) (2 times)
2 to 3: 2:27\(\frac{3}{11}\) and 3:00 (2 times)
3 to 4: 3:00 is already counted; 3:32\(\frac{8}{11}\) (1 new time)
4 to 5: 4:05\(\frac{5}{11}\) and 4:38\(\frac{2}{11}\) (2 times)
5 to 6: 5:10\(\frac{10}{11}\) and 5:43\(\frac{7}{11}\) (2 times)
Total \(=2+2+1+2+2=9\)
At 1 p.m. (30°) and 6 p.m. (180°) the hands are not at right angles, so the end points add nothing
Why the other options are incorrect
Option (1) 8 — Incorrect. This leaves out the right angle at 3:00 altogether.
Option (3) 10 — Incorrect. This takes 2 for each of the 5 hours and counts 3:00 twice.
Option (4) 12 — Incorrect. This counts 2 for each of 6 hours; from 1 p.m. to 6 p.m. is only 5 hours.