Question

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

Answer 1

Hint:
We can find the circumference of the track using the equation $C = 2\pi r$. Then we can find the time each cyclist takes to complete one round by dividing the circumference of the track with their respective velocity. After that we take the LCM of the time taken by both cyclists to complete one round to get the time at which they meet each other at the starting point.

Complete step by step solution:
We are given that the radius of the track is 63m.
$ \Rightarrow r = 63m$
Circumference of the track is given by the equation$C = 2\pi r$, on substituting the value of r we get,
$ \Rightarrow C = 2\pi \times 63$
On substituting the value of $\pi = \dfrac{{22}}{7}$, we get,
$ \Rightarrow C = 2 \times \dfrac{{22}}{7} \times 63$
On simplification we get,
$ \Rightarrow C = 44 \times 9 = 396m$
Now we can find the taken by X to complete one round and reach the starting point.
We know that time taken is the distance divided by the speed.
Speed of X is ${v_x} = 33{m {\left/
 \right.
} {\min }}$
Time taken by X to complete one round, ${t_x} = \dfrac{C}{{{v_x}}}$
Substituting the values, we get,
${t_x} = \dfrac{{396}}{{33}} = 12\min $
Similarly, speed of Y is \[{v_y} = 44{m {\left/
 \right.
} {\min }}\]
Time taken by Y to complete one round, ${t_y} = \dfrac{C}{{{v_y}}}$
Substituting the values, we get,
${t_y} = \dfrac{{396}}{{44}} = 9\min $
So, X reaches the starting point at intervals of 12 minutes and Y reaches the starting point at intervals of 9 minutes. We get the time at which they meet at the starting point by taking the LCM or least common multiple of 9 and 12.
LCM of 9 and 12 is given by long division method.

\[\text{LCM}(9,12) = 3 \times 3 \times 2 \times 2 = 36\]

Therefore, X and Y meet at the starting point after 36 minutes.

Note:
Alternate approach to find the LCM is to list the time interval at which each cyclist reaches the starting point. So, we can write,
The Times when X reaches the starting point are 9min, 18min, 27min, 36min, …
From the above values we can find that 36 minutes is common in both. So, X and Y meet at the starting point after 36 minutes.
LCM or lowest common multiple of 2 numbers is the lowest possible number which is divisible by both the numbers. We assume the speed is constant throughout this problem. We need to take the LCM of the time interval, not their speed.