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