Question

6. Explain why?
7
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round
of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the
same point and at the same time, and go in the same direction. After how many minutes
will they meet again at the starting point?​

Answer 1

Working out:

The above question says that, two children Sonia and Ravi takes 18 minutes and 12 minutes respectively to cover one round of a circular path.

We need to find the time after which they will meet again at the starting point if they started together.

For this, we need to find the common multiple for the time taken by the children. Infact, the lowest common multiple will be the least time after which they will meet. So, let's express 18 and 12 into their prime factors:

• 18 = 2 × 3²
• 12 = 2² × 3

We know,

• HCF = product of the least powers of common factors in both the numbers.
• LCM = product of the highest powers of all the factors involved in both the numbers$.$

Highest powers

• 2 ➝ Highest power (2)
• 3 ➝ Highest power (3)

⇛ Lowest common multiple = 2² × 3² = 36

So, Sonia and Ravi will meet after 36 minutes at the starting point after completing their revolution around the circular path.

And we are done !!

Answer 2

Answer:

$\begin{tabular}{r|l}2&18 \\\cline{1-2}3&9\\\cline{1-2}3&3\\\cline{1-2}&1\\\end{tabular} \qquad \begin{tabular}{r|l}2&12 \\\cline{1-2}2&6\\\cline{1-2}3&3\\\cline{1-2}&1\\\end{tabular}$

18 = 2 × 3 × 3 × 1

⠀ = 2 × 3² × 1

⠀

12 = 2 × 2 × 3 × 1

⠀ = 2² × 3 × 1

⠀

• According to the Question :

They'll again meet after the Least Common Multiple of the time they both take to complete one round.

⠀⠀⠀⠀Understand it like One had already travelled 12 minutes and another is 18 minutes ahead. Now Both will meet when that Least Common Multiple of Both Time Happens.

• We always take Highest Power of any of Multiple while we doing LCM.

⠀

$:\implies\sf LCM(18,12) = 2^2\times 3^2\times1\\\\\\:\implies\sf LCM(18,12) = 4\times9\times1\\\\\\:\implies\underline{\boxed{\sf LCM(18,12) = 36\:minutes}}$

⠀

∴ Both will meet after 36 minutes.