Time, Speed and Distance Question 6 - CAT 2020
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:
Step-by-Step Explanation
Calculate track circumferences:
Circumference of T₁ = 2πR₁ = 2π(100) = 200π meters
Circumference of T₂ = 2πR₂ = 2π(20) = 40π meters
Determine time taken for a single round:
Convert speeds to meters per hour, or leave them as ratios since the time units cancel out:
Time for 1 round of Ram = 200π/15
Time for 1 round of Rahim = 40π/5 = 8π
Find the first meeting time at point A:
Because the tracks only touch at point A, they can only meet at point A. The meeting time must be the LCM of their individual round completion times:
LCM(200π/15, 8π/1) = LCM(40π/3, 8π/1) = LCM(40π, 8π)/HCF(3, 1) = 40π/1 = 40π
Calculate rounds completed by Ram:
Rounds completed = Total Meeting Time / Time for 1 round
= 40π/(40π/3) = 3 rounds
(Note: Because the first meeting occurs exactly at the completion of the 3rd round, the number of full rounds completed before that moment can be interpreted as 2 or 3 depending on the boundary condition; CAT accepted both 2 and 3 as correct answers).
Correct Option: B
💡 FundaMakers Tip:
Remember the fraction LCM rule:
LCM(a/b, c/d) = LCM(a, c)/HCF(b, d)
When managing complex track interactions that link together at a single node, finding the LCM of the independent loop periodicities tells you exactly when they sync up.