How many times are the hands of a clock positioned such that there is an angle of $33°$ between them in a day ?
My trying:
The minute hand moves 360 degrees in 60 minutes. This means that the angle of the minute hand is given by 6t, where t is number of minutes past midnight.
The hour hand moves 30 degrees in 60 minutes. This means that the angle of the hours hand is given by 0.5t.
The hands start together at midnight. The first time they make a 33 degree angle is when the minute hand has moved 33 degrees further than the hour hand, so this is given by the equation:
$ 6t = 0.5t + 33 \Rightarrow 5.5t = 33 \Rightarrow t = 6 minutes$ . In other words about 6 minutes past midnight.
I have found that 22 times the hour and minutes hand are in 33 degree angle . How can I find this number ? How can I find the 22 times ?
Best Answer
This problem does not require any complex math to solve: during one day, the hour hand goes around twice, and the minute hand 24 times, in the same direction. As such, the two hands will be in exactly the same position 22 times. That is, there will be a difference of $0$ degrees between them $22$ times. But, any other difference of degree, say $x$ degrees, will happen exactly that many times as well, because that difference will happen exactly once between two successive times when the difference is $0$. So, no matter what difference of degrees we are talking about, it will happen $22$ times a day.