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# 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: 15th Jul 2024
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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 $x = - 54$.