Question

A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. After what time they will meet again at the starting point.
(a) 26 minutes and 18 seconds
(b) 42 minutes and 36 seconds
(c) 45 minutes
(d) 46 minutes and 12 seconds
(e) None of these

Answer 1

Hint: First of all, visualize the question and understand it. Now, take the lowest common multiple that is LCM of 252, 308 and 198 seconds to find the time when they will meet for the first time at the starting point.

Complete step by step answer:
In this question, we are given that A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. We have to find the time when they will meet again at the starting point.
Here, we are given that A completes a round in 252 seconds that means A passes from the starting point every 252 seconds. Similarly, B passes from the starting point every 308 seconds and C passes from the starting point every 198 seconds.
Since we need to calculate the time when they would meet at the starting point again, we actually will calculate the time when they would pass from the starting point at the same time. So, we have to calculate the common multiple of 252, 308 and 198 because that would be the time when all three of them would be passing from the starting point.
So, for this, let us calculate the LCM or the lowest common multiple of 252, 308 and 198 which is as follows
\[\begin{align}
  & 2\left| \!{\underline {\,
  252,308,198 \,}} \right. \\
 & 2\left| \!{\underline {\,
  126,154,99 \,}} \right. \\
 & 3\left| \!{\underline {\,
  63,77,99 \,}} \right. \\
 & 3\left| \!{\underline {\,
  21,77,33 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7,77,11 \,}} \right. \\
 & 11\left| \!{\underline {\,
  1,11,11 \,}} \right. \\
 & \text{ 1,1,1} \\
\end{align}\]
So, we get the LCM of 252, 308 and 198 as
\[2\times 2\times 3\times 3\times 7\times 11=2772\text{ seconds}\]
This means that A, B and C will meet at the starting point every 2772 seconds and will meet again after 2772 seconds from the start for the first time.
We can also write 2772 seconds as
\[2772\text{ seconds =}\dfrac{2772}{60}\text{ minutes}\]
\[\Rightarrow 2772\text{ seconds = }46.2\text{ minutes}\]
\[\Rightarrow 2772\text{ seconds = }46\text{ minutes}+0.2\times 60\text{ seconds}\]
\[\Rightarrow 2772\text{ seconds = }46\text{ minutes}+12\text{ seconds}\]
So, after 46 minutes and 12 seconds, they will meet at the starting point.

So, the correct answer is “Option d”.

Note: In this question, many students find the time when they would meet for the first time but they must keep in mind that they have to find the time when they would meet for the time at the starting point and not at some other point on the circular track. Also, students must remember that if they are not given the time directly, then they can find the time taken to complete one round by using \[\text{Time}=\dfrac{\text{Length of circular track}\left( L \right)}{\text{Speed of the person}}.\]