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CAT 2019 Slot 2 — QA Question 21

Circular RaceEasy

Two ants A and B start from a point P on a circle at the same time, with A moving clock-wise and B moving anti-clockwise. They meet for the first time at 10:00 am when A has covered 60% of the track. If A returns to P at 10:12 am, then B returns to P at

Answer & solution

Correct answer: 10 : 27 am

  • 10 : 27 am

  • B

    10 : 25 am

  • C

    10 : 45 am

  • D

    10 : 18 am

Solution

Easy

At the first meeting the two have together covered one full lap, so the arcs they cover are in the ratio of their speeds. From A's 60%60\% at meeting and A's full-lap time we get A's lap time; the speed ratio then gives B's lap time, from which B's remaining 60%60\% pins down the return time.

1

Speed ratio from the meeting. By 10:00 A covers 60%60\% and B covers 40%40\% of the track in the same time, so speeds are in ratio 60:40=3:260:40=3:2, and lap times are in inverse ratio 2:32:3.

vAvB=6040=32⇒ TATB=23(time∝1/speed)\begin{aligned} &\frac{v_A}{v_B}=\frac{60}{40}=\frac{3}{2}\\ &\Rightarrow\ \frac{T_A}{T_B}=\frac{2}{3} \quad\text{(time}\propto 1/\text{speed)} \end{aligned}
2

Find A's full lap time. A is at 60%60\% at 10:00 and back at P (full lap) at 10:12, so A covers the last 40%40\% in 1212 min. At a constant pace, 40%→1240\%\to12 min means the full 100%100\% takes:

TA=10040×12=30 min\begin{aligned} &T_A=\frac{100}{40}\times 12=30\ \text{min} \end{aligned}
3

B's lap time and return. Using TA:TB=2:3T_A:T_B=2:3, then B still needs the remaining 60%60\% after 10:00.

TB=32×TA=32×30=45 min⇒ B’s remaining 60%=0.6×45=27 min⇒ return=10 ⁣: ⁣00+27 min=10 ⁣: ⁣27 am\begin{aligned} &T_B=\frac{3}{2}\times T_A=\frac{3}{2}\times 30=45\ \text{min}\\ &\Rightarrow\ \text{B's remaining }60\%=0.6\times 45=27\ \text{min}\\ &\Rightarrow\ \text{return}=10\!:\!00+27\ \text{min}=10\!:\!27\ \text{am} \end{aligned}
B returns at 10 ⁣: ⁣27 am\text{B returns at }10\!:\!27\ \text{am}

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CAT 2019 Slot 2 QA Q21: Two ants A and B start from a point P on a circle at the same time, with A moving clock-wise and B moving anti — Solution | TheCATExam