Two circular tracks T1 and T2 of radii 100 m and 20 m, respectively touch at a point A. Starting from A at the same time, Ram and Rahim are walking on track T1 and track T2 at speeds 15 km/hr and 5 km/hr respectively. The number of full rounds that Ram will make before he meets Rahim again for the first time is

The correct answer is 3.

Since the two circular tracks touch at exactly one point — point A — Ram and Rahim can only meet again at point A. We need to find the earliest time when both are simultaneously back at A, and then count how many full rounds Ram has completed by that time.

Step 1: Find the circumference of each track.

Track T₁ has radius 100 m, so its circumference is:

$C_1=2\pi \times 100=200\pi \text{m}$

Track T₂ has radius 20 m, so its circumference is:

$C_2=2\pi \times 20=40\pi \text{m}$

Step 2: Convert speeds to metres per hour.

Ram's speed: 15 km/hr = 15,000 m/hr
Rahim's speed: 5 km/hr = 5,000 m/hr

Step 3: Find the time each person takes to complete one full round.

Time for Ram to complete one full round of T₁:

$t_{\text{Ram}}=\frac{200\pi }{15000}=\frac{\pi }{75}\text{hr}$

Time for Rahim to complete one full round of T₂:

$t_{\text{Rahim}}=\frac{40\pi }{5000}=\frac{\pi }{125}\text{hr}$

Step 4: Find the first time both are back at A simultaneously — this is the LCM of their lap times.

We need:

$T_{\text{meet}}=\text{LCM}(\frac{\pi }{75},\frac{\pi }{125})$

Using the LCM rule for fractions:

$\text{LCM}(\frac{a}{b},\frac{c}{d})=\frac{\text{LCM}(a,c)}{\text{GCD}(b,d)}$

Here, a = c = π, b = 75, d = 125.
LCM(��, π) = π
GCD(75, 125) = 25   (since 75 = 3 × 25 and 125 = 5 × 25)

Therefore:

$T_{\text{meet}}=\frac{\pi }{25}\text{hr}$

Step 5: Count the number of full rounds Ram completes in this time.

$\text{Rounds by Ram}=\frac{T_{\text{meet}}}{t_{\text{Ram}}}=\frac{\pi /25}{\pi /75}=\frac{75}{25}=3$

(We can verify: Rounds by Rahim = (π/25) ÷ (π/125) = 125/25 = 5, which is also a whole number — confirming they both return to A at exactly this time.)

Conclusion: Ram completes exactly 3 full rounds on track T₁ before he meets Rahim again for the first time at point A.