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
- 4
- 3
- 2
- 5
The Correct Option is B
Approach Solution - 1
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.
Approach Solution -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.
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