Question

The radius of a circular track is $63{\text{ m}}$. Two cyclists $X$ and $Y$ start together from the same position at the same time and in the same direction with speeds $33{\text{ m/min}}$ and ${\text{44 m/min}}$ respectively. After how many minutes will they meet again at the starting point?

Answer 1

Hint: Compare the ratio of the speeds of two cyclists. Since they are moving in the same direction, they will meet again once the faster one will cover exactly one extra revolution than the slower one. And if this meeting point is the starting point then this time will be our answer.

Complete step-by-step answer:
According to the question, the radius of the circular track is $63{\text{ m}}$. So, the length of the circular track is the circumference of the circle. Let it be denoted as $L$.
Thus, applying the formula of circumference of circle i.e. $2\pi r$ and taking $\pi = \dfrac{{22}}{7}$, we’ll get
$
   \Rightarrow L = 2 \times \dfrac{{22}}{7} \times 63 \\
   \Rightarrow L = 396{\text{ m}} \\
$
Thus the length of the circular track is $396{\text{ m}}$.
The speed of the cyclists is given as $33{\text{ m/min}}$ and ${\text{44 m/min}}$. Thus the ratio of their speeds is:
$ \Rightarrow $Ratio of Speed $ = \dfrac{{33}}{{44}} = \dfrac{3}{4} = 3:4$
So, this ratio suggests to us that the time in which the cyclist $X$ will complete 3 revolutions, $Y$ will complete 4 revolutions. Thus they will meet at this instant on the starting point and since $Y$ has completed exactly one extra revolution than $X$, this is also their first meeting.
Therefore the time of occurrence of this instant will be our answer.
Time taken by $X$ to complete one revolution is given by the formula:
$ \Rightarrow {\text{Time = }}\dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$, where distance is the length of track i.e. $L = 396{\text{ m}}$ and speed of $X$ is $33{\text{ m/min}}$.
Putting these values, we’ll get:
$ \Rightarrow $Time taken by $X$ for one revolution $ = \dfrac{{396}}{{33}} = 12{\text{ m}}$
So the time taken by $X$ to complete 3 revolutions $ = 12 \times 3 = 36{\text{ m}}$.

Thus, the cyclists will meet again at the starting point after 36 minutes.

Note: As we have calculated the time taken by $X$ to complete 3 revolutions for our answer, we can also calculate the time taken by $Y$ to complete 4 revolutions to obtain the same result because they are meeting at this point.
$ \Rightarrow $Time taken by $Y$ for one revolution $ = \dfrac{{396}}{{44}} = {\text{ 9 m}}$.
So the time taken by $Y$ to complete 4 revolutions $ = 9 \times 4 = 36{\text{ m}}$.