CAT 2023 Slot 1 QA Question 14

Type-in-the-answer (no negative marking) · Arithmetic · Time, Speed & Distance · Try it, then check the answer and solution below.

CAT 2023 Slot 1QAArithmetic • Time, Speed & DistanceModerate

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

TITA Answer:

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⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: 972

Answer and solution

Answer: 972

📌 Core Concept

When two objects move towards each other, their relative speed is the sum of their individual speeds. After meeting, the time taken to reach their destinations can be used to find the distance.

🔢 Step-by-Step Solution

Define Variables:

Let

D

be the distance between towns

A

and

B

Let

S

be Surbhi's speed in km/h.

Let

T

be the time taken for Arvind and Surbhi to meet.

Distance Covered Before Meeting:

Arvind's distance:

54T

Surbhi's distance:

ST

Total distance:

54T + ST = D

Equation:

T(54 + S) = D

...(1)

Remaining Distance After Meeting:

Arvind's remaining distance:

ST = 54 \times 6 = 324

Equation:

S = \frac{324}{T}

...(2)

Surbhi's Remaining Distance:

Surbhi's remaining distance:

54T = S \times 24

Equation:

S = \frac{54T}{24} = \frac{9T}{4}

...(3)

Equating Speeds:

From (2) and (3):

\frac{324}{T} = \frac{9T}{4}

Solving:

324 \times 4 = 9T^2

1296 = 9T^2

T^2 = 144

T = 12

hours

Calculate Surbhi's Speed:

S = \frac{324}{12} = 27

km/h

Calculate Total Distance:

Using equation (1):

12(54 + 27) = D

12 \times 81 = D

D = 972

⚡ 30-Second Shortcut

After meeting, the time taken by Arvind and Surbhi to reach their destinations is in the ratio of 6:24, which simplifies to 1:4. This ratio is the square root of the ratio of their speeds. Thus, the speeds are in the ratio 1:4, leading to the calculation of the total distance.

🎯 Final Answer

The distance between town

A

and town

B

is \boxed{972} km.

CAT 2023 Slot 1 QA Question 14

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