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CAT 2018 Slot 2 — QA Question 7

Relative SpeedEasy

Points A, P, Q and B lie on the same line such that P, Q and B are, respectively, 100 km, 200 km and 300 km away from A. Cars 1 and 2 leave A at the same time and move towards B. Simultaneously, car 3 leaves B and moves towards A. Car 3 meets car 1 at Q, and car 2 at P. If each car is moving in uniform speed then the ratio of the speed of car 2 to that of car 1 is

Answer & solution

  • A

    1 : 2

  • B

    2 : 7

  • 1 : 4

  • D

    2 : 9

Solution

Easy

All cars travel the same time until each meeting. At a meeting, the two cars' distances are in the same ratio as their speeds. Use car 3 vs car 1 (meet at Q) and car 3 vs car 2 (meet at P) to chain the three speeds, then read off car 2:car 1\text{car }2:\text{car }1.

A0 P100 Q200 B300 car1 & car2 → ← car3

Distances from A: P=100P=100, Q=200Q=200, B=300B=300 km. Cars 1, 2 start at A going east; car 3 starts at B going west. Car 3 meets car 1 at QQ, car 2 at PP.

1

Car 3 meets car 1 at Q. In the same time, car 1 covers A→Q=200A\to Q=200 km while car 3 covers B→Q=100B\to Q=100 km.

C3C1=100200=12(equal time)⇒ C3:C1=1:2\begin{aligned} &\frac{C_3}{C_1} = \frac{100}{200} = \frac{1}{2} \quad\text{(equal time)}\\ &\Rightarrow\ C_3:C_1 = 1:2 \end{aligned}
2

Car 3 meets car 2 at P. In the same time, car 2 covers A→P=100A\to P=100 km while car 3 covers B→P=200B\to P=200 km.

C2C3=100200=12(equal time)⇒ C3:C2=2:1\begin{aligned} &\frac{C_2}{C_3} = \frac{100}{200} = \frac{1}{2} \quad\text{(equal time)}\\ &\Rightarrow\ C_3:C_2 = 2:1 \end{aligned}
3

Chain the ratios. Scale to a common C3C_3. From step 1, C3:C1=2:4C_3:C_1=2:4; from step 2, C3:C2=2:1C_3:C_2=2:1.

C1:C2:C3=4:1:2⇒ C2C1=14\begin{aligned} &C_1:C_2:C_3 = 4:1:2\\ &\Rightarrow\ \frac{C_2}{C_1} = \frac{1}{4} \end{aligned}
C2:C1=1:4(option c)C_2:C_1 = 1:4\quad\text{(option c)}
CAT 2018 Slot 2 QA Q7: Points A, P, Q and B lie on the same line such that P, Q and B are, respectively, 100 km, 200 km and 300 km aw — Solution | TheCATExam