Question

Find the angle between the minute hand and the hour hand of a clock when the time is $7:20$A.M.

Answer 1

Hint: At first find the angle at least should be there by using fact that each hour contains ${{30}^{\circ }}$ so $\left( 7-4 \right)=3$ contains $3\times 3{{0}^{\circ }}$ or ${{90}^{\circ }}$. Then apply the fact that hour hand advances one full number for each complete rotation of the minute hand through ${{360}^{\circ }}$ which means minute hand advances 20 minutes or $\dfrac{1}{3}$ of an hour.

Complete step-by-step answer:
In the question we have to find the angle between the minute hand and the hour hand of the clock if the clock’s hour $7:20$A.M.
We know that angle in a clock is ${{360}^{\circ }}$. In a clock number are labelled from 1-12 in the perimeter of circle, thus dividing the clock into 12 sections so we can say that as the sections are equally divided then angle subtended by each section is $\dfrac{{{360}^{\circ }}}{12}$ which is ${{30}^{\circ }}$.
As we know the number represented on the clock, in every successive section the difference has 5 minutes in between. So, we can say for 5 minutes that it contains ${{30}^{\circ }}$. So, in one minute there contains $\left( \dfrac{{{30}^{\circ }}}{5} \right)$ or ${{6}^{\circ }}$.
In the question we are told that the clock is reading $7:20$ which means that minute hand is at ‘4’ and hour hand is a slight part the ‘7’ assuming the hands of the clock rotate relatively smoothly. So, if $\left( 7-4 \right)$ is 3 then $3\times 3{{0}^{\circ }}$ (${{30}^{\circ }}$ for each hour interval) or ${{90}^{\circ }}$ will be at least angle between hands but the answer is incomplete.
Now, we know that the hour hand advances one full number for each hour. So, each hour is represented by a complete rotation of the minute hand through ${{360}^{\circ }}$. This means that when the minute hand advances 20 minutes or $\dfrac{1}{3}$ of an hour, the hour hand also advances by one-third of those ${{30}^{\circ }}$ or $\dfrac{1}{3}\times 3{{0}^{\circ }}$ which is ${{10}^{\circ }}$.
So, the angle between ${{90}^{\circ }}+{{10}^{\circ }}\ =\ {{100}^{\circ }}$.
Hence the angle is ${{100}^{\circ }}$.

Note: There is a formula to do these kinds of problems in shortcut method which is $\left| \dfrac{11\text{M}}{2}-30\text{H} \right|$ here M means reading of minute hand and H means reading of hour hand.