Two test pilots, $A$ and $B$, run on a circular test track of length $1{,}200\text{ m}$ starting from the same point at the same time in the same direction. $A$ runs at a constant speed of $6\text{ m/s}$ and $B$ runs at a constant speed of $10\text{ m/s}$. After how many seconds from the start will they meet for the first time at the original starting point?

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### 1. Strategy & Theoretical Principles When two runners start simultaneously from the same point on a circular track of circumference $L$: - Time taken by $A$ to complete 1 full lap: $T_A = \frac{L}{V_A}$ - Time taken by $B$ to complete 1 full lap: $T_B = \frac{L}{V_B}$ To meet specifically at the **starting point**, the elapsed time must be an integer multiple of both lap times. Therefore, the time of their first meeting at the starting point is: $$T_{\text{start}} = \text{LCM}(T_A, T_B)$$ (Note: Meeting anywhere on the track occurs at $T = \frac{L}{V_B - V_A}$, but meeting at the *starting point* requires the LCM of lap times). ### 2. Analytical Formulation & Calculation Given: - Track length $L = 1{,}200\text{ m}$ - Speed of $A$: $V_A = 6\text{ m/s} \implies T_A = \frac{1200}{6} = 200\text{ seconds}$ - Speed of $B$: $V_B = 10\text{ m/s} \implies T_B = \frac{1200}{10} = 120\text{ seconds}$ ### 3. Step-by-Step Calculation Find the least common multiple of 200 and 120: - $200 = 2^3 \times 5^2$ - $120 = 2^3 \times 3 \times 5$ $$\text{LCM}(200, 120) = 2^3 \times 3 \times 5^2 = 8 \times 3 \times 25 = 600\text{ seconds}$$ At $t = 600\text{ s}$, $A$ has completed exactly $\frac{600}{200} = 3$ laps and $B$ has completed $\frac{600}{120} = 5$ laps, so both are simultaneously at the starting point. ### 4. Distractor Forensics - Calculating meeting anywhere on the track gives $T = \frac{1200}{10 - 6} = 300\text{ seconds}$, which is the time of their first meeting on the track, NOT at the starting point. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Distinguish "first meeting anywhere on the track" ($L/\Delta V = 300\text{ s}$) from "first meeting at the starting point" ($\text{LCM}(T_A, T_B) = 600\text{ s}$). - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*, S. Chand.

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