Circular Track Meeting Time Using LCM

 

❓ Question

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?

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🖼️ Solution Image

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✍️ Short Solution

This is an LCM-based application question.

They will meet again at the starting point after the Least Common Multiple (LCM) of their round times 💯

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🔹 Step 1 — Write the Given Times

Sonia’s time:

$$18 \text{ minutes}$$

Ravi’s time:

$$12 \text{ minutes}$$

We need:

$$\text{LCM}(12,18)$$

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🔹 Step 2 — Prime Factorisation

$$12 = 2^2 \times 3$$
$$18 = 2 \times 3^2$$

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🔹 Step 3 — Find LCM

Take highest powers of all prime factors:

$$\text{LCM} = 2^2 \times 3^2$$
$$= 4 \times 9$$
$$= 36$$

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🔹 Step 4 — Final Conclusion

After:

$$36 \text{ minutes}$$

both will again meet at the starting point.

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✅ Final Answer

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

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⭐ Key Insight

When two people repeat events at different intervals, common meeting time is found using:

$$\boxed{\text{LCM}}$$

🧠 Memory Line:

Repeating events together ⇒ use LCM