Question

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

• A. 4
• B. 3
• C. 2
• D. 5

Answer 1

The Correct Option is B

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

• Ram's speed = \( \frac{15 \times 1000}{60 \times 60} = \frac{15 \times 5}{18} = \frac{75}{18} \) m/s
• Rahim's speed = \( \frac{5 \times 1000}{60 \times 60} = \frac{5 \times 5}{18} = \frac{25}{18} \) m/s

Step 2: Find circumference of each circle

• Ram's path: \( C_R = 2\pi \times 100 = 200\pi \) meters
• Rahim's path: \( C_H = 2\pi \times 20 = 40\pi \) meters

Step 3: Time to complete one round

• Ram's time per round = \( \frac{200\pi}{\frac{75}{18}} = \frac{200\pi \times 18}{75} = 48\pi \) seconds
• Rahim's time per round = \( \frac{40\pi}{\frac{25}{18}} = \frac{40\pi \times 18}{25} = 28.8\pi \) seconds

Step 4: LCM of the two times

The Least Common Multiple (LCM) of \( 48\pi \) and \( 28.8\pi \) is: \[ \text{LCM}(48\pi, 28.8\pi) = 144\pi \text{ seconds} \]

Step 5: Number of rounds Ram completes in that time

\[ \text{Rounds} = \frac{144\pi}{48\pi} = \boxed{3} \]

Answer:

Ram will meet Rahim after completing 3 full rounds.

Answer 2

Step 1: Convert Speeds to m/s

\[ \text{Ram's speed} = \frac{15 \times 1000}{3600} = \frac{75}{18} \text{ m/s} \] \[ \text{Rahim's speed} = \frac{5 \times 1000}{3600} = \frac{25}{18} \text{ m/s} \]

Step 2: Calculate Circumference of Each Track

\[ \text{Ram's circumference} = 2\pi \times 100 = 200\pi \text{ m} \] \[ \text{Rahim's circumference} = 2\pi \times 20 = 40\pi \text{ m} \]

Step 3: Time to Complete One Round

\[ \text{Ram's time per round} = \frac{200\pi}{75/18} = \frac{200\pi \times 18}{75} = 48\pi \text{ seconds} \] \[ \text{Rahim's time per round} = \frac{40\pi}{25/18} = \frac{40\pi \times 18}{25} = 28.8\pi \text{ seconds} \]

Step 4: Find LCM of Times

To meet at the starting point again, the time must be LCM of \( 48\pi \) and \( 28.8\pi \): \[ \text{LCM}(48\pi, 28.8\pi) = 144\pi \text{ seconds} \]

Step 5: Number of Rounds by Ram

\[ \text{Rounds Ram completes} = \frac{144\pi}{48\pi} = \boxed{3 \text{ rounds}} \]

Answer:

Ram completes 3 full rounds before meeting Rahim again at the starting point.