Question

Swati can row her boat at a speed of $5 km/h$ in still water. It takes her $1$ an hour more to row the boat $5.25 km/h$ upstream than to return downstream. Find the speed of the stream.

Answer 1

Hint: Let us first understand the concept of speed of an object in upstream and downstream and still water
In still water means that the water is not moving and its speed is $0$, when the direction of boat is along the stream is called downstream as both are travelling in the same direction their speed gets added. In case of upstream the direction of boat is against the direction of stream so their speed gets subtracted so if we assume the speed of stream as $x$ km/hr

Formula used: Upstream= (u-v) km/hr where u is the speed of boat in still water and v is the speed of stream
Downstream= (u+v) km/hr where u is the speed of boat in still water and v is the speed of stream

Complete step-by-step answer:
Firstly it is given the speed of boat in still water is $5$ km/hr now let us assume the speed of water be $x$
On applying the formula of upstream i.e (speed of boat - speed of stream)
$ \Rightarrow (5 - x)$
Speed of boat in downstream will be (speed of boat + speed of stream)
$ \Rightarrow (5 + x)$
As the distance travelled by boat in upstream = distance travelled by boat in downstream i.e. $5.25$ Km
As the difference in time in covering the distance to upstream and downstream is $1$
On using the formula of
Time = distance $ \div $ speed
Time taken to travel upstream is
$ \Rightarrow \dfrac{{5.25}}{{5 - x}}$
Time taken to travel downstream
$ \Rightarrow \dfrac{{5.25}}{{5 + x}}$
On equating that the difference between the time taken to travel upstream and downstream is equal to $1$
$ \Rightarrow $ $\dfrac{{5.25}}{{5 - x}} - \dfrac{{5.25}}{{5 + x}} = 1$
Taking $5.25$ as common
$ \Rightarrow $ $5.25\left[ {\dfrac{1}{{5 - x}} - \dfrac{1}{{5 + x}}} \right] = 1$
And taking the L.C.M of $(5 - x)$ and $(5 + x)$
$ \Rightarrow 5.25\left[ {\dfrac{{5 + x - 5 + x}}{{(5 - x)(5 + x)}}} \right] = 1$
On adding and subtracting like terms and multiplying the terms in denominator of equation
$ \Rightarrow 5.25\left[ {\dfrac{{2x}}{{5(5 + x) - x(5 + x)}}} \right] = 1$
Solving the expression in denominator
$ \Rightarrow 5.25\left[ {\dfrac{{2x}}{{25 + 5x - 5x - {x^2}}}} \right] = 1$
$ \Rightarrow 5.25\left[ {\dfrac{{2x}}{{25 - {x^2}}}} \right] = 1$
Taking the term in denominator to the right side and multiplying it by $1$
$10.50x = 25 - {x^2}$
Rearranging the Equation
Equation formed is
${x^2} + 10.50x - 25$
To solve this quadratic equation and do the factors of $25$ and arrange those factors in such a way that we can get $10.5$ as their difference or we can say by the method of splitting the middle term
Factors are $12.5x$ and $2x$
${x^2} + 12.5x - 2x - 25$
On further solving
$ \Rightarrow $ $x(x + 12.5) - 2(x + 12.5)$
$ \Rightarrow $ $(x - 2)(x + 12.5)$
On equating each term to zero we get
$x = 2$ and $x = -12.5$
Speed can’t be negative so speed of speed stream will be $2$ km/hr
Hence the speed of stream is $2$ km/hr

Note: In this question students mainly get confused they mainly forgot in which case they have to add the speed of boat and speed of stream and in which they have to subtract and sometimes they use to do the opposite to remember that in upstream boat goes opposite to direction of flow of stream so we subtract the speeds and in downstream boat goes in same direction so we add them