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CAT 2021 Slot 3 — QA Question 10

Circular RaceEasy

Mira and Amal walk along a circular track, starting from the same point at the same time. If they walk in the same direction, then in 45 minutes, Amal completes exactly 3 more rounds than Mira. If they walk in opposite directions, then they meet for the first time exactly after 3 minutes. The number of rounds Mira walks in one hour is

Answer & solution

Answer: 8

Solution

Easy

Translate the two meeting conditions into equations relating the track length LL to the two speeds. Opposite directions: combined distance =L=L in 33 min. Same direction: Amal gains 33 full laps (3L3L) over Mira in 4545 min. Eliminate LL to get Mira's lap time, then convert to laps per hour.

1

Opposite directions. Let Mira's and Amal's speeds be mm and aa (m/min). They first meet when their combined distance equals one lap.

3(m+a)=L(meet in 3 min)⇒ 3m+3a=L...(1)\begin{aligned} &3(m+a)=L \quad\text{(meet in 3 min)}\\ &\Rightarrow\ 3m+3a=L \quad\text{...(1)} \end{aligned}
2

Same direction. In 4545 min Amal completes 33 more laps than Mira, i.e. covers 3L3L extra distance.

45a−45m=3L...(2)\begin{aligned} &45a-45m=3L \quad\text{...(2)} \end{aligned}
3

Eliminate LL. Compute 15×15\times(1) −- (2) to cancel the aa-terms and isolate mm.

15(3m+3a)−(45a−45m)=15L−3L[15⋅(1)−(2)]⇒ 45m+45a−45a+45m=12L⇒ 90m=12L ⇒ Lm=7.5 min\begin{aligned} &15(3m+3a)-(45a-45m)=15L-3L \quad\text{[15·(1)−(2)]}\\ &\Rightarrow\ 45m+45a-45a+45m=12L\\ &\Rightarrow\ 90m=12L\ \Rightarrow\ \frac{L}{m}=7.5\text{ min} \end{aligned}
4

Laps per hour. Mira takes Lm=7.5\tfrac{L}{m}=7.5 min per lap, so in 6060 min she completes 607.5\tfrac{60}{7.5} laps.

Rounds=607.5=8\begin{aligned} &\text{Rounds}=\frac{60}{7.5}=8 \end{aligned}
Mira completes 8 rounds per hour\text{Mira completes }8\text{ rounds per hour}
CAT 2021 Slot 3 QA Q10: Mira and Amal walk along a circular track, starting from the same point at the same time. If they walk in the — Solution | TheCATExam