Two runners, Alice and Bob, are racing on two different circular tracks, T1 and T2, with radii 60 \ m and 40 \ m respectively. They both start from point P at the same moment. Alice runs on track T1 at a constant speed of 12 \ km / hr and Bob runs on track T2 at a constant speed of 8 \ km / hr. After how many full rounds of Alice does she meet Bob again for the first time since the race started? Option 1: 1 Option 2: 2 Option 3: 3 Option 4: 4

Answer: 1. Solution: Alice and Bob start at the same point P but run on different circular tracks. We need to find after how many complete rounds Alice makes before they meet again at point P. First, convert speeds to consistent units since track radii are given in meters. Alice's speed: 12 km/hr = 12 x 1000/3600 = 10/3 m/s Bob's speed: 8 km/hr = 8 x 1000/3600 = 20/9 m/s Calculate the circumferences of both tracks. Track T1 (Alice): C_1 = 2 r_1 = 2 x 60 = 120 meters Track T2 (Bob): C_2 = 2 r_2 = 2 x 40 = 80 meters Find the time each runner takes to complete one full round. Alice's time per round: t_1 = (120)/(10/3) = 120 x 3/10 = 36 seconds Bob's time per round: t_2 = (80)/(20/9) = 80 x 9/20 = 36 seconds Since both Alice and Bob take exactly 36 seconds to complete one round on their respective tracks, they will return to point P at exactly the same time after each round. Therefore, Alice meets Bob again for the first time after 1 full round.

Arithmetic

Time, Speed & Distance

Hard

Two runners, Alice and Bob, are racing on two different circular tracks, $T1$ and $T2$ , with radii $60 \mathrm{\ m}$ and $40 \mathrm{\ m}$ respectively. They both start from point $P$ at the same moment. Alice runs on track $T1$ at a constant speed of $12 \mathrm{\ km} / \mathrm{hr}$ and Bob runs on track $T2$ at a constant speed of $8 \mathrm{\ km} / \mathrm{hr}$ . After how many full rounds of Alice does she meet Bob again for the first time since the race started?

✅ Correct Option: 1

Alice and Bob start at the same point $P$ but run on different circular tracks. We need to find after how many complete rounds Alice makes before they meet again at point $P$ .

First, convert speeds to consistent units since track radii are given in meters.

Alice's speed: $12 \text{ km/hr} = 12 \times \frac{1000}{3600} = \frac{10}{3} \text{ m/s}$

Bob's speed: $8 \text{ km/hr} = 8 \times \frac{1000}{3600} = \frac{20}{9} \text{ m/s}$

Calculate the circumferences of both tracks.

Track $T1$ (Alice): $C_1 = 2\pi r_1 = 2\pi \times 60 = 120\pi \text{ meters}$

Track $T2$ (Bob): $C_2 = 2\pi r_2 = 2\pi \times 40 = 80\pi \text{ meters}$

Find the time each runner takes to complete one full round.

Alice's time per round:

$t_1 = \frac{120\pi}{\frac{10}{3}} = 120\pi \times \frac{3}{10} = 36\pi \text{ seconds}$

Bob's time per round:

$t_2 = \frac{80\pi}{\frac{20}{9}} = 80\pi \times \frac{9}{20} = 36\pi \text{ seconds}$

Since both Alice and Bob take exactly $36\pi$ seconds to complete one round on their respective tracks, they will return to point $P$ at exactly the same time after each round.

Therefore, Alice meets Bob again for the first time after $1$ full round.

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