How many times are the hour hand and the minute hand of a clock at right angles during their motion from 1.00 p.m. to 10.00 p.m.?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 16
The minute hand gains 330° on the hour hand every hour, so the hands are usually at right angles twice in each hour. The exceptions are the right angles at 3 o’clock and 9 o’clock: each falls on the hour itself and is shared by two neighbouring hours, so those two hours give only three right angles between them instead of four.
1 p.m. to 10 p.m. is 9 hours, which would give \(9\times2=18\) right angles
2 p.m. to 4 p.m. gives only three, at about 2:27, 3:00 and 3:33, so one less
8 p.m. to 10 p.m. gives only three, at about 8:27, 9:00 and 9:33, so one less
Right angles from 1 p.m. to 10 p.m.: \(18-2=16\)
At 1:00 p.m. and at 10:00 p.m. the hands are 30° and 60° apart, so neither end of the period adds a right angle.
Tip: In 12 hours the hands are at right angles 22 times (not 24), because the right angles at 3 and 9 o’clock are each shared by two hours; in a day, 44 times.