Question

A, B and C starts cycling around a circular path in the same direction at same time. Circumference of the path is

1980 m

. If the speed of A is

330 m / \min

, speed of B is

198 m / \min

and C is

220 m / \min

and they start from the same point, then after what time interval they will be together at the starting point?

Accepted Answer

Let's assume that A, B and C meet after time $t$ and they would have respectively completed, $n_{A}$ , $n_{B}$ and $n_{C}$ rounds and they would be $S$ distance away from the start point.

From the given condition of the question, it can be written that

$330 t = S + 1980 n_{A} . . . (1) 198 t = S + 1980 n_{B} . . . (2) 220 t = S + 1980 n_{C} . . . (3)$

From equation (1) and (2), it can be written that

$\begin{array}{rcl} 198 t & = & 330 t + 1980 (n_{B} - n_{A}) \\ \Rightarrow & 132 t = 1980 (n_{A} - n_{B}) \\ \Rightarrow & t = 15 (n_{A} - n_{B}) . . . (4) \end{array}$

Similarly, from equation (1) and (3), it can be written that

$\begin{array}{rcl} 220 t & = & 330 t + 1980 (n_{C} - n_{A}) \\ \Rightarrow & 110 t = 1980 (n_{A} - n_{C}) \\ \Rightarrow & t = 18 (n_{A} - n_{C}) . . . (5) \end{array}$

As the time at which the objects meet is the same, from equation (4) and 95), it can be concluded that both $(n_{A} - n_{B})$ and $(n_{A} - n_{C})$ must be whole numbers.

The values of the differences can be found out by taking the LCM of $15$ and $18$ , which is $90$ .

Hence, all teh objects meet after $90 \min$ .

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