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 ${\displaystyle \frac{360{}^{\circ }}{12}}=30{}^{\circ }$ . 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 ${\displaystyle \frac{360{}^{\circ }}{12}}=30{}^{\circ }$ 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 ${\displaystyle \frac{30{}^{\circ }}{60}}={\displaystyle \frac{1}{2}}$ 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. $|30H-5.5M|$ 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 $|30\times 3-5.5\times 20|=|90-110|=|-20|=20{}^{\circ }$ .From both the methods we got the same result.