Question

Q7. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.​

Answer 1

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

Let assume that

Speed of boat in still water be 'x' km per hour

and

Speed of stream be 'y' km per hour.

So,

Speed of boat in upstream = x - y km per hour

and

Speed of boat in downstream = x + y km per hour.

According to first condition

A boat covers 32 km upstream and 36 km downstream in 7 hours.

Time taken to cover 32 km in upstream with the speed of x - y km per hour is

$\rm \: = \:\dfrac{32}{x - y}$

and

Time taken to cover 36 km in downstream with the speed of x + y km per hour is

$\rm \: = \:\dfrac{36}{x + y}$

Since, total time taken is 7 hours.

Thus,

$\rm :\longmapsto\:\dfrac{32}{x - y} + \dfrac{36}{x + y} = 7 - - - (1)$

According to second condition

It covers 40 km upstream and 48 km downstream in 9 hours.

Time taken to cover 40 km in upstream with the speed of x - y km per hour is

$\rm \: = \:\dfrac{40}{x - y}$

and

Time taken to cover 48 km in downstream with the speed of x + y km per hour is

$\rm \: = \:\dfrac{48}{x + y}$

Since, total time taken is 9 hours.

Thus,

$\rm :\longmapsto\:\dfrac{40}{x - y} + \dfrac{48}{x + y} = 9 - - - (2)$

So, Now we have two equations

$\rm :\longmapsto\:\dfrac{32}{x - y} + \dfrac{36}{x + y} = 7- - - (1)$

and

$\rm :\longmapsto\:\dfrac{40}{x - y} + \dfrac{48}{x + y} = 9 - - - (2)$

Let assume that,

$\boxed{ \bf{ \: \frac{1}{x - y} = u}} \: \: \: \: and \: \: \: \: \boxed{ \bf{ \: \frac{1}{x + y} = v}}$

So, above equations can be rewritten as

$\rm :\longmapsto\:32u + 36v = 7 - - - (3)$

and

$\rm :\longmapsto\:40u + 48v = 9 - - - (4)$

On multiply equation (3) by 5 and equation (4) by 4, we get

$\rm :\longmapsto\:160u + 180v = 35 - - - (5)$

and

$\rm :\longmapsto\:160u + 192v = 36 - - - (6)$

On Subtracting equation (5) from equation (6), we get

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

$\bf\implies \:v = \dfrac{1}{12} - - - (7)$

On substituting the value of v, in equation (3), we get

$\rm :\longmapsto\:32u + 36 \times \dfrac{1}{12} = 7$

$\rm :\longmapsto\:32u + 3 = 7$

$\rm :\longmapsto\:32u = 7 - 3$

$\rm :\longmapsto\:32u = 4$

$\bf\implies \:u = \dfrac{1}{8} - - - (8)$

Now,

$\boxed{ \bf{ \: \frac{1}{x - y} = u}} \: \: \: \: and \: \: \: \: \boxed{ \bf{ \: \frac{1}{x + y} = v}}$

$\rm :\longmapsto\:{ \bf{ \: \dfrac{1}{x - y} = u}} \: \: \: \: and \: \: \: \: { \bf{ \: \dfrac{1}{x + y} = v}}$

$\rm :\longmapsto\:{ \bf{ \: \dfrac{1}{x - y} = \dfrac{1}{8} }} \: \: \: \: and \: \: \: \: { \bf{ \: \dfrac{1}{x + y} = \dfrac{1}{12} }}$

$\rm :\longmapsto\:x - y = 8 - - - - (9)$

and

$\rm :\longmapsto\:x + y = 12 - - - - (10)$

On adding equation (9) and (10), we get

$\rm :\longmapsto\:2x = 20$

$\bf :\longmapsto\:x = 10$

On substituting value of x in equation (10), we get

$\rm :\longmapsto\:10 + y = 12$

$\rm :\longmapsto\:y = 12 - 10$

$\bf :\longmapsto\:y = 2$

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{Speed \: of \: boat \: in \: still \: water\: = \: 10 \: km \: per \: hr} \\ &\sf{Speed \: of \: stream \: = \: 2 \: km \: per \: hr} \end{cases}\end{gathered}\end{gathered}$