A boat goes 24 km upstream and 28 km downstream in 6 hrs. It goes 30 km upstream and 21 km downstream in $6\frac{1}{2}$ hrs. Find the speed of the boat in still water and also speed of the stream.

Given:

A boat goes 24 km upstream and 28 km downstream in 6 hrs. It goes 30 km upstream and 21 km downstream in $6\frac{1}{2}$ hrs.

To do:

We have to find the speed of the boat in still water and also 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 $24\ km$ upstream and $28\ km$ downstream in $6\ hours$.

Time taken $=\frac{24}{y−x} +\frac{28}{y+x}$

​$\Rightarrow 6= \frac{24}{y−x} +\frac{28}{y+x}$......(i)

The boat goes $30\ km$ upstream and $21\ km$ downstream in $6\frac{1}{2}=\frac{6\times2+1}{2}=\frac{13}{2}\ hours$.

Time taken $=\frac{30}{y-x}+\frac{21}{y+x}$

​$\Rightarrow \frac{13}{2} =\frac{30}{y-x}+\frac{21}{y+x}$........(ii)

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

From (i) and (ii),

$24u+28v=6$......(iii)

$30u+21v=\frac{13}{2}$.......(iv)

Multiply equation (iii)$ by $5$ and equation (iv) by $4$, we get,

$120u+140v=30$......(v)

$120u+84v=26$......(vi)

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

$140v−84v=30−26$

$56v=4$

$v=\frac{4}{56}=\frac{1}{14}$

$\Rightarrow \frac{1}{y+x}=\frac{1}{14}$

$y+x=14$.......(vii)

From equation (iii),

$24u=6−28v$

$24u=6−28\times \frac{1}{14}$

$24u=6−2=4$

$\Rightarrow u=\frac{4}{24}$

$\Rightarrow u=\frac{1}{6}$

$\Rightarrow y−x=6$.....(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 $10\ km/hr$ . 

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Updated on: 10-Oct-2022

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