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A person can row 8 km upstream and 24 km downstream in 4 hours. He can row 12 km downstream and 12 km upstream in 4 hours. Find the speed of the boat in current and in still water respectively.

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3 km/h and 8 km/h

4 km/h and 6 km/h

4 km/h and 8 km/h

4 km/h and 5 km/h

answer is A.

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Detailed Solution

It is given that a boat goes 8 km upstream and 24 km downstream in 4 hours.
In 4 hours, it can go 12 km upstream and 12 km downstream.
We know that the

s p e e d = d i s t a n c e t i m e

.
Hence,

d i s t a n c e = s p e e d × t i m e

.
Let the speed of the boat in still water be x km/h and speed of the boat in current be y km/h.
Time taken to cover 8 km upstream is

8 x − y

.
Time taken to cover 24 km downstream is

24 x + y

.
Similarly, time taken to cover 12 km upstream is

\frac{12}{x - y}

.
And, time taken to cover 12 km downstream is

\frac{12}{x + y}

.
Total time taken in both the cases is 4 hours.
Therefore,

24 x + y + 8 x − y = 4 12 x + y + 12 x − y = 4

Simplifying the above equations,

6 x + y + 2 x − y = 1 3 x + y + 3 x − y = 1

Let

p = 1 x + y

and

q = 1 x − y

.
Then

6 p + 2 q − 1 = 0

and

3 p + 3 q − 1 = 0

will be the required equations.
The solution of the equations

a_{1} p + b_{1} q + c_{1} = 0 and a_{2} p + b_{2} q + c_{1} = 0

by cross multiplication method is given by the formula,

p b 1 c 2 − b 2 c 1 = q a 2 c 1 − a 1 c 2 = 1 a 1 b 2 − a 2 b 1

Here,

a 1 = 6, b 1 = 2, c 1 = − 1, a 2 = 3, b 2 = 3, c 2 = − 1

.
Using the cross-multiplication method to find the value of p and q,

⇒ p 2 (− 1) − 3 (− 1) = q (− 1) 3 − (− 1) 6 = 1 6 (3) − 3 (2) ⇒ p − 2 + 3 = q − 3 + 6 = 1 18 − 6 ⇒ p 1 = q 3 = 112

Solving for p and q,

\Rightarrow p = \frac{1}{12}

and

q = \frac{1}{4}

Putting values of p and q,

And,

Solving the pair of equations by elimination method.

x + y = 12 … … (1)

x − y = 4 … … (2)

Adding equations 1 and 2,

⇒ 2 x = 16 ⇒ x = 8

Putting value of x in equation 1 and solving for y,

⇒ 8 + y = 12 ⇒ y = 4

Hence, the speed of the boat in current is 4km/h and in still water is 8 km/h respectively.
Hence, the correct option is 1.

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