# A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.

Given:

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours.

To do:

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

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 $30\ km$ upstream and $28\ km$ downstream in $7\ hours$.

Time taken $=\frac{30}{y−x} +\frac{28}{y+x}$

​$\Rightarrow 7= \frac{30}{y−x} +\frac{28}{y+x}$......(i)

The boat goes $21\ km$ upstream and return in 5 hours.

Time taken $=\frac{21}{y-x}+\frac{21}{y+x}$

​$\Rightarrow 5 =\frac{21}{y-x}+\frac{21}{y+x}$........(ii)

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

From (i) and (ii),

$30u+28v=7$......(iii)

$21u+21v=5$.......(iv)

Multiply equation (iii)$by$7$and equation (iv) by$10$, we get,$7(30u+28v)=7(7)210u+196v=49$......(v)$10(21u+21v)=10(5)210u+210v=50$......(vi) Subtracting equation (v) from equation (vi), we get,$210v−196v=50−4914v=1v=\frac{1}{14}\Rightarrow \frac{1}{y+x}=\frac{1}{14}y+x=14$.......(vii) From equation (iii),$30u=7−28v30u=7−28\times \frac{1}{14}30u=7−2=5\Rightarrow u=\frac{5}{30}\Rightarrow u=\frac{1}{6}\Rightarrow y−x=6$.....(viii) Adding equations (vii) and (viii), we get,$2y=20y=10$From equation (vii),$x=14−yx=14−10=4$Hence, the speed of the stream is$4\ km/hr$and the speed of the boat in still water is$10\ km/hr\$ .

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Updated on: 10-Oct-2022

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