It is 7.30 now. How much time will it pass until the clock hands form the same angle again?
Student tip
Suppose that minute and hour hands move at a constant velocity each.Solution
At 7.30, the minute hand and the hour hand form an angle of 45o, and they are 7,5 minutes apart. If the hour hand does not move, then after 15 minutes the angle between the minute hand and the hour hand will be 45o again. But the hour hand also moves with a velocity equal to 5 minutes per hour (60 minutes), so the angle between the minute hand and the hour hand will be the same again, after 15+t minutes, for some t>0.
We are going to determine the value of t from the fact that for 15+t minutes, the hour hand should move t minutes. We have the following proportion
5:60=t:(15+t),
which can be solved in the following way
5(15+t)=60t
75+5t=60t
75=55t,
from where
t=7555=1511minutes.
So, the answer is that the clock hands will form the same angle after
15+t=15+1511=16411minutes≈16min22sec.
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