# A motorboat whose speed in still water is $18km/hr$, takes $1$ an hour more to go $24km$upstream than to return downstream to the spot. Find the speed of the stream.

Last updated date: 22nd Mar 2023

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Answer

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Hint: Speed of downstream is always greater than the speed of the upstream.

Given that,

Distance covered in upstream and in downstream is $ = 24km$

Speed of the boat in still water $ = 18km/hr$

Let the speed of the stream $ = xkm/hr$

Now,

Speed of the boat in upstream $ = $ speed of the boa in still water $ - $ speed of the stream

$

= 18km/hr - x km/hr \\

= (18 - x)km/hr \\

$

Speed of the boat in downstream $ = $ speed of the boat in still water $ + $ speed of the stream

$

= 18km/hr + x km/hr \\

= \left( {18 + x} \right)km/hr \\

$

We know that,

Time taken for the upstream $ = $ Time taken to cover downstream \[ + 1\]$

\dfrac{{{\text{Distance of upstream}}}}{{{\text{Speed of upstream}}}} = \dfrac{{{\text{Distance of downstream}}}}{{{\text{Speed of downstream}}}} + 1 \\

\\

\dfrac{{24}}{{18 - x}} = \dfrac{{24}}{{18 + x}} + 1 \\

\\

24\left( {18 + x} \right) = 24\left( {18 - x} \right) + \left( {18 - x} \right)\left( {18 + x} \right) \\

\\

432 + 24x = 432 - 24x + 324 - {x^2} \\

\\

24x + 24x = 324 - {x^2} \\

\\

{x^2} + 48x - 324 = 0 \\

$

By solving the quadratic equation, we get

$

{x^2} + 48x - 324 = 0 \\

{x^2} + 54x - 6x - 324 = 0 \\

\left( {x + 54} \right)\left( {x - 6} \right) = 0 \\

$

Thus, we have two values of $x$ i.e. $x = 6, - 54$

Therefore, the speed of the stream is $6km/hr$.

Note: In this type of problem the value of the speed of the stream cannot be negative. Hence, we neglect $x = - 54$.

Given that,

Distance covered in upstream and in downstream is $ = 24km$

Speed of the boat in still water $ = 18km/hr$

Let the speed of the stream $ = xkm/hr$

Now,

Speed of the boat in upstream $ = $ speed of the boa in still water $ - $ speed of the stream

$

= 18km/hr - x km/hr \\

= (18 - x)km/hr \\

$

Speed of the boat in downstream $ = $ speed of the boat in still water $ + $ speed of the stream

$

= 18km/hr + x km/hr \\

= \left( {18 + x} \right)km/hr \\

$

We know that,

Time taken for the upstream $ = $ Time taken to cover downstream \[ + 1\]$

\dfrac{{{\text{Distance of upstream}}}}{{{\text{Speed of upstream}}}} = \dfrac{{{\text{Distance of downstream}}}}{{{\text{Speed of downstream}}}} + 1 \\

\\

\dfrac{{24}}{{18 - x}} = \dfrac{{24}}{{18 + x}} + 1 \\

\\

24\left( {18 + x} \right) = 24\left( {18 - x} \right) + \left( {18 - x} \right)\left( {18 + x} \right) \\

\\

432 + 24x = 432 - 24x + 324 - {x^2} \\

\\

24x + 24x = 324 - {x^2} \\

\\

{x^2} + 48x - 324 = 0 \\

$

By solving the quadratic equation, we get

$

{x^2} + 48x - 324 = 0 \\

{x^2} + 54x - 6x - 324 = 0 \\

\left( {x + 54} \right)\left( {x - 6} \right) = 0 \\

$

Thus, we have two values of $x$ i.e. $x = 6, - 54$

Therefore, the speed of the stream is $6km/hr$.

Note: In this type of problem the value of the speed of the stream cannot be negative. Hence, we neglect $x = - 54$.

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