How many times are the hands of a clock at $90$ degrees.

Question

How many times are the hands of a clock at right angle in a day?

Initially, I worked this out to be $2$ times every hour. The answer came to $48$.
However, in the cases of $3$ o'clock and $9$ o'clock, right angles happen only once.
So the answer came out to be $44$.
Is the approach correct?

Answer 1

Yes, but a more “mathematical” approach might be this: In a 12 hour period, the minute hand makes 12 revolutions while the hour hand makes one. If you switch to a rotating coordinate system in which the hour hand stands still, then the minute hand makes only 11 revolutions, and so it is at right angles with the hour hand 22 times. In a 24 hour day you get 2×22=44.

Answer 2

I think the total number of times, clock hands are perpendicular is 22. One of the solution converges to the other, 8:59 ~= 9:00 and 2:59 ~= 3:00, so there are 11 times per 12h.

Answer 3

Correct answer is 22*2. In every hour there are two instances of making 90 degree angle but in case of 3 and 9 there are exceptions. Between 2 and 4 there will be only three right angles because right angle at 2:59 and 3:00 is common and right angle at 8:59 and 9:00 is common. Total will be 22+22 = 44.