Question

What is the angle between the minute hand and the hour hand of a clock when the clock shows $ 3 $ hours $ 20 $ minutes?

Answer 1

Hint: We know that the angle made by the clock to complete one rotation is $ 360{}^\circ $ . We know that the complete rotation of the clock takes $ 12 $ hours. Now the angle made by the clock to rotate one hour is given by $ \dfrac{360{}^\circ }{12}=30{}^\circ $ . From this value, we will calculate the angle made by the hour hand and minute hand when time shows $ 3 $ hours. After that, we will calculate the angle made by the hour hand and minute hand for one minute by using the ration $ 1 $ hour $ = $ $ 60 $ minutes. From this value, we will calculate the angle made by hour hand and minute hand when time is $ 20 $ minutes. By subtracting both the obtained values we will get the result.

Complete step by step answer:
We know the clock face is $ 360{}^\circ $ and the clock is divided into $ 12 $ hours. In the below figure, a clock face is showing $ 360{}^\circ $ and is also showing that the clock is divided into 12 hours. Also, the clock is showing a time of 3 hours 00 minutes.

In the above figure, the hour hand is at 3 and the minute hand is at 12. Now the hour hand moves an angle of $ \dfrac{360{}^\circ }{12}=30{}^\circ $ for one hour.
 $ \, therefore, $ The angle between hours hand and minutes hand when the time shows $ 3 $ hours is $ 3\times 30{}^\circ =90{}^\circ $.
We know that $ 1 $ hour $ = $ $ 60 $ minutes. For a minute the hour hand rotates by $ \dfrac{30{}^\circ }{60}=\dfrac{1}{2} $ degrees.
So, the angle changed in one minute is $ 5.5{}^\circ $.
In $ 20 $ minutes, the angle changes by $ 5.5{}^\circ \times 20=110{}^\circ $ .s
Now the angle between the hours hand and minute hand when the time shows $ 3:20 $ is the difference between the angle when the time shows is the difference between the angle when the time shows $ 3 $ hours and the angle when the time shows $ 20 $ minutes. Mathematically
 $ \alpha =110{}^\circ -90{}^\circ =20{}^\circ $
 $ \therefore $ The required angle is $ 20{}^\circ $ .

Note:
For this kind of problem we can use a simple formula i.e. $ \left| 30H-5.5M \right| $ where $ H $ is the hours and $ M $ is the minutes. Now the angle when the time is $ 3:20 $ from the above formula can be calculated as $ \left| 30\times 3-5.5\times 20 \right|=\left| 90-110 \right|=\left| -20 \right|=20{}^\circ $ .From both the methods we got the same result.