How many times do the hour hand and minute hand of a clock coincide in a day?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 22
The hands coincide each time the minute hand gains one full round (360°) on the hour hand. This takes a little more than an hour, so the hands meet fewer times than the number of hours.
The minute hand turns 6° a minute and the hour hand 0.5° a minute, so the minute hand gains 5.5° a minute
Time to gain 360°: \(\displaystyle\frac{360}{5.5}=65\frac{5}{11}\) minutes between two meetings
Meetings in 12 hours (720 minutes): \(\displaystyle 720\div65\frac{5}{11}=11\)
Meetings in a day (24 hours): \(2\times11=\) 22
Between 11 and 1 o’clock the hands meet only once, at 12, which is why there are 11 meetings in 12 hours and not 12.
Tip: In a day the hands coincide 22 times, point in opposite directions 22 times and are at right angles 44 times.