# A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the steam.

Given:

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time.

To do:

We have to find the speed of the stream and the speed of the boat in still water.

Solution:

Let the speed of the stream be $x\ km/hr$

Let the speed of the boat in still water be $y\ km/hr$

Upstream speed $=y−x\ km/hr$

Downstream speed $=y+x\ km/hr$

We know that,

$Time=\frac{Speed}{Distance}$

The boat goes $12\ km$ upstream and $40\ km$ downstream in $8\ hours$.

Time taken $=\frac{12}{y−x} +\frac{40}{y+x}$

​$\Rightarrow 8= \frac{12}{y−x} +\frac{40}{y+x}$......(i)

The boat goes $16\ km$ upstream and $32\ km$ downstream in $8\ hours$.

Time taken $=\frac{16}{y-x}+\frac{32}{y+x}$

​$\Rightarrow 8 =\frac{16}{y-x}+\frac{32}{y+x}$........(ii)

Let $\frac{1}{y-x}=u$ and $\frac{1}{y+x}=v$

From (i) and (ii),

$12u+40v=8$......(iii)

$16u+32v=8$.......(iv)

Multiply equation (iii) by $4$ and equation (iv) by $3$, we get,

$48u+160v=32$......(v)

$48u+96v=24$......(vi)

Subtracting  equation (vi) from equation (v), we get,

$160v−96v=32−24$

$64v=8$

$v=\frac{1}{8}$

$\Rightarrow \frac{1}{y+x}=\frac{1}{8}$

$y+x=8$.......(vii)

From equation (iii),

$12u=8−40v$

$12u=8−40\times \frac{1}{8}$

$12u=8−5=3$

$\Rightarrow u=\frac{3}{12}$

$\Rightarrow u=\frac{1}{4}$

$\Rightarrow y−x=4$.....(viii)

Adding equations (vii) and (viii), we get,

$2y=12$

$y=6$

From equation (vii),

$x=8−y$

$x=8−6=2$

Hence, the speed of the stream is $2\ km/hr$ and the speed of the boat in still water is $6\ km/hr$ .

Updated on: 10-Oct-2022

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