Arithmetic

Time, Speed & Distance

Medium

Points $A, P, Q$ and $B$ lie on the same line such that $P, Q$ and $B$ are, respectively, $100 \mathrm{~km}, 200 \mathrm{~km}$ and $300 \mathrm{~km}$ 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

$1:2$

$2:9$

$1:4$

$2:7$

✅ Correct Option: 3

We have four points on a straight line:

A (starting point)

P (100 km from A)

Q (200 km from A)

B (300 km from A)

Cars 1 and 2 start from A and move toward B. Car 3 starts from B and moves toward A. All three cars start simultaneously.

Car 3 meets Car 1 at point Q (200 km from A) and Car 3 meets Car 2 at point P (100 km from A).

Since Car 3 meets Car 2 at P, which is closer to A than Q, Car 2 is slower than Car 1.

When Car 3 meets Car 1 at Q:

Car 1 traveled: 200 km (from A to Q)

Car 3 traveled: 300 - 200 = 100 km (from B to Q)

Since both cars started simultaneously and met at Q, they took the same time to reach Q.

When time is constant, speed is directly proportional to distance traveled.

Speed of Car 1 : Speed of Car 3 = 200 : 100 = 2 : 1

When Car 3 meets Car 2 at P:

Car 2 traveled: 100 km (from A to P)

Car 3 traveled: 300 - 100 = 200 km (from B to P)

Again, both took the same time to reach P.

Speed of Car 2 : Speed of Car 3 = 100 : 200 = 1 : 2

We have two relationships:

Car 1 : Car 3 = 2 : 1

Car 2 : Car 3 = 1 : 2

From the first ratio: Car 3 = $\dfrac{\text{Car 1}}{2}$

From the second ratio: Car 3 = $2 \times \text{Car 2}$

Setting them equal:

$\dfrac{\text{Car 1}}{2} = 2 \times \text{Car 2}$

$\text{Car 1} = 4 \times \text{Car 2}$

Therefore: Car 2 : Car 1 = 1 : 4

The ratio of the speed of Car 2 to Car 1 is 1:4.

This makes intuitive sense since Car 2 is much slower than Car 1, which is why Car 3 (coming from the opposite direction) meets Car 2 first at point P, then continues and meets the faster Car 1 later at point Q.

CAT 2017 Slot 2

In a $10 \mathrm{~km}$ race. A, B, and C , each running at uniform speed, get the gold, silver, and bronze medals, respectively. If A beats B by $1 \mathrm{~km}$ and B beats C by $1 \mathrm{~km}$ , then by how many metres does A beat C?

CAT 2019 Slot 2

A cyclist leaves $A$ at $10$ am and reaches $B$ at $11$ am. Starting from $10:01$ am, every minute a motor cycle leaves $A$ and moves towards $B$ . Forty-five such motor cycles reach $B$ by $11$ am. All motor cycles have the same speed. If the cyclist had doubled his speed, how many motor cycles would have reached $B$ by the time the cyclist reached $B$ ?

CAT 2020 Slot 3

Anil, Sunil, and Ravi run along a circular path of length $3 \mathrm{~km}$ , starting from the same point at the same time, and going in the clockwise direction. If they run at speeds of $15 \mathrm{~km} / \mathrm{hr}, 10 \mathrm{~km} / \mathrm{hr}$ , and $8 \mathrm{~km} / \mathrm{hr}$ , respectively, how much distance in km will Ravi have run when Anil and Sunil meet again for the first time at the starting point?

CAT 2020 Slot 3

$A$ and $B$ are two railway stations $90 \mathrm{~km}$ apart. A train leaves $A$ at $9:00$ am, heading towards $B$ at a speed of $40 \mathrm{~km} / \mathrm{hr}$ . Another train leaves $B$ at $10:30$ am, heading towards $A$ at a speed of $20 \mathrm{~km} / \mathrm{hr}$ . The trains meet each other at

CAT 2022 Slot 1

Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in $10$ minutes while train B takes $9$ minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train $B$ to travel from station $Y$ to station $X$ is

CAT 2023 Slot 1

Arvind travels from town $A$ to town $B$ , and Surbhi from town B to town $A$ , both starting at the same time along the same route. After meeting each other, Arvind takes $6$ hours to reach town $B$ while Surbhi takes $24$ hours to reach town $A$ . If Arvind travelled at a speed of $54 \mathrm{~km} / \mathrm{h}$ , then the distance, in km, between town $A$ and town $B$ is

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