Two runners, Sony and Mony, start running on a circular track of length 200 m at speeds of 18 and 24 km/h, respectively, in the same direction. After how much time from the start will they meet again at the starting point?

The correct answer is 120 s.

To find when both runners meet again at the starting point, we need to find when both Sony and Mony complete a whole number of laps at the same time — that is, the Least Common Multiple (LCM) of their individual lap times.

Step 1: Convert speeds from km/h to m/s

Sony's speed:
18\text{km/h}=18\times \frac{1000}{3600}=5\text{m/s}

Mony's speed:
24\text{km/h}=24\times \frac{1000}{3600}=\frac{20}{3}\text{m/s}

Step 2: Find the time for each runner to complete one full lap

The track length is 200 m. Time = Distance ÷ Speed.

Time for Sony to complete one lap:
T_{\text{Sony}}=\frac{200}{5}=40\text{s}

Time for Mony to complete one lap:
T_{\text{Mony}}=\frac{200}{\frac{20}{3}}=200\times \frac{3}{20}=30\text{s}

Step 3: Find the LCM of 40 and 30

Both runners will be at the starting point simultaneously only when both have completed an integer number of laps. This happens at the LCM of their lap times.

Prime factorizations:
40=2^3\times 5
30=2\times 3\times 5

LCM takes the highest power of each prime factor:
\text{LCM}(40,30)=2^3\times 3\times 5=8\times 3\times 5=120\text{s}

Step 4: Verify

At t = 120 s:
Sony completes: \frac{120}{40}=3 laps ✓
Mony completes: \frac{120}{30}=4 laps ✓

Both return to the starting point exactly — Sony after 3 laps and Mony after 4 laps — so they meet at the starting point after 120 seconds.