Question

A boat can travel 54km downstream and 72km upstream in 9 hours and 90km downstream and 84km upstream in 12 hours.Find the speed of the stream.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

The speed of boat be x km per hour

The speed of stream be y km per hour

So,

Speed of upstream = x - y km per hour

Speed of downstream = x + y km per hour

Case :- 1

A boat can travel 54 km downstream and 72 km upstream in 9 hours.

So, it means

$\rm :\longmapsto\:\dfrac{54}{x + y} + \dfrac{72}{x - y} = 9$

On divided both sides by 9, we get

$\rm :\longmapsto\:\boxed{\tt{ \dfrac{6}{x + y} + \dfrac{8}{x - y} = 1}} - - - - (1)$

Case :- 2

A boat can travel 90 km downstream and 84 km upstream in 12 hours.

So, it means

$\rm :\longmapsto\:\dfrac{90}{x + y} + \dfrac{84}{x - y} = 12$

So, on dividing both sides by 6, we get

$\rm :\longmapsto\:\boxed{\tt{ \dfrac{15}{x + y} + \dfrac{14}{x - y} = 2}} - - - - (2)$

Let further assume that

$\rm :\longmapsto\:\dfrac{1}{x + y} = u - - - (3)$

and

$\rm :\longmapsto\:\dfrac{1}{x - y} = v - - - (4)$

So, equation (1) and (2) can be rewritten as

$\rm :\longmapsto\:6u + 8v = 1 - - - (5)$

and

$\rm :\longmapsto\:15u + 14v = 2- - - (6)$

On multiply equation (5) by 5 and (6) by 2, we get

$\rm :\longmapsto\:30u + 28v = 4- - - (7)$

and

$\rm :\longmapsto\:30u + 40v = 5 - - - (8)$

On Subtracting equation (7) from (8), we get

$\rm :\longmapsto\:12v = 1$

$\bf\implies \:v = \dfrac{1}{12}$

$\bf\implies \:\dfrac{1}{x - y} = \dfrac{1}{12}$

$\bf\implies \:x - y = 12 - - - - (9)$

On substituting the value of v in equation (5), we get

$\rm :\longmapsto\:6u + 8 \times \dfrac{1}{12} = 1$

$\rm :\longmapsto\:6u +\dfrac{2}{3} = 1$

$\rm :\longmapsto\:6u = 1 - \dfrac{2}{3}$

$\rm :\longmapsto\:6u = \dfrac{3 - 2}{3}$

$\rm :\longmapsto\:6u = \dfrac{1}{3}$

$\bf\implies \:u = \dfrac{1}{18}$

$\bf\implies \:\dfrac{1}{x + y} = \dfrac{1}{18}$

$\bf\implies \:x + y = 18 - - - - (10)$

On Subtracting equation (9) from (10), we get

$\rm :\longmapsto\:2y = 6$

$\bf\implies \:y = 3$

Hence, Speed of stream is 3 km per hour

Answer 2

$\huge\purple{\mid{\fbox{\tt{Answer}}\mid}}$

Explanation:

Let speed of boat in still water = x

and speed of stream = y

54/(x + y) + 72/(x - y) = 9

$\frac{90}{x + y} + \frac{84}{x - y} = 12$

1st method : Solve the equations and get the values of x and y.

2nd method:

Take HCF of 54 & 90 = 18

& HCF of 72 and 84 = 12 it satisfies.

Therefore, x + y = 18 and x - y = 12

By solving these equations,

x = 15 km/hr and y = 3 km/h