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CAT 2020 Slot 2 — QA Question 13

Circular RaceEasy

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

Answer & solution

  • 3

  • B

    5

  • C

    4

  • D

    2

Solution

Easy

The two circles touch only at AA, so the walkers can only meet at AA. Find how long each takes to return to AA (one lap), then the LCM of those times gives the next common visit to AA. The number of laps Ram has done by then is the answer.

A T1 (r=100) T2 (r=20)
1

Time for one lap each. Use radii in km (100 m=0.1 km100\,\text{m}=0.1\,\text{km}, 20 m=0.02 km20\,\text{m}=0.02\,\text{km}). Lap time = circumference / speed.

tRam=2π(0.1)15=π75 hrtRahim=2π(0.02)5=π125 hr\begin{aligned} &t_{\text{Ram}} = \frac{2\pi(0.1)}{15} = \frac{\pi}{75}\ \text{hr}\\ &t_{\text{Rahim}} = \frac{2\pi(0.02)}{5} = \frac{\pi}{125}\ \text{hr} \end{aligned}
2

They meet only at AA. Distinct circles meeting at a single point means AA is the only shared spot. They are together again when both are back at AA simultaneously, i.e. at LCM⁡\operatorname{LCM} of the lap times.

LCM⁡ ⁣(π75,π125)=πgcd⁡(75,125)=π25 hr(LCM⁡ of fractions=LCM(num)⁡gcd⁡(den))\begin{aligned} &\operatorname{LCM}\!\left(\tfrac{\pi}{75},\tfrac{\pi}{125}\right) = \frac{\pi}{\gcd(75,125)} = \frac{\pi}{25}\ \text{hr}\\ &\quad\text{(}\operatorname{LCM}\text{ of fractions} = \tfrac{\operatorname{LCM(num)}}{\gcd(\text{den})}\text{)} \end{aligned}
3

Count Ram's laps. Divide the meeting time by Ram's lap time (step 1).

Rounds=π/25π/75=7525=3\begin{aligned} &\text{Rounds} = \frac{\pi/25}{\pi/75} = \frac{75}{25} = 3 \end{aligned}
Ram completes 3 full rounds\text{Ram completes } 3 \text{ full rounds}
CAT 2020 Slot 2 QA Q13: Two circular tracks T 1 and T 2 of radii 100 m and 20 m, respectively touch at a point A. Starting from A at t — Solution | TheCATExam