Question 7

Step 1 — Understand why we need to find the LCM

Sonia completes one round in 18 minutes. She will be back at the starting point at multiples of 18 minutes: $18, 36, 54, 72, \dots$

Ravi completes one round in 12 minutes. He will be back at the starting point at multiples of 12 minutes: $12, 24, 36, 48, 72, \dots$

To find the time when they both meet again at the starting point for the first time, we need to find the smallest common multiple of their individual times. Therefore, we need to calculate the Least Common Multiple (LCM) of 18 and 12.

Step 2 — Prime factors of Sonia's time

Sonia takes 18 minutes for one round. Let's find the prime factors of 18. $18 = 2 \times 9$ $= 2 \times 3 \times 3$

$\boxed{18 = 2^1 \times 3^2}$

Step 3 — Prime factors of Ravi's time

Ravi takes 12 minutes for one round. Let's find the prime factors of 12. $12 = 2 \times 6$ $= 2 \times 2 \times 3$

$\boxed{12 = 2^2 \times 3^1}$

Step 4 — Calculate LCM

To find the LCM, we take the product of the highest powers of all involved prime factors.

• The highest power of 2 is $2^2$.
• The highest power of 3 is $3^2$.

$LCM(12, 18) = 2^2 \times 3^2$ $= 4 \times 9$

$\boxed{36 \text{ minutes}}$

Answer

They will meet again at the starting point after 36 minutes.