Question

The boat goes$30km$ upstream and $44km$ downstream in $10hours$. In $13hours$, it can go $40km$ upstream and $55km$ downstream. Determine the speed of the stream and that of the boating till water.

Answer 1

Let the speed of the stream be $=ukm/h$

Let the speed of the boat be$=vkm/h$

Then

Speed of the upstream will be $=\left(v–u\right)km/h$

Speed of the downstream will be $=\left(v+u\right)km/h$

We know that,

$$\begin{gathered} speed=\frac{dis\tan ce}{time} \\ \Rightarrow time=\frac{dis\tan ce}{speed} \end{gathered}$$

So,

Time is taken to cover$30km$ upstream $=\frac{30}{\left(v-u\right)}hr$

Time is taken to cover$44km$ downstream $=\frac{44}{\left(v+u\right)}hr$

As per the first condition,

$\frac{44}{\left(v+u\right)}+\frac{30}{\left(v-u\right)}=10\_\_\_\_\_\_\_\_\_\_\_\_\left(1\right)$

Time is taken to cover $40km$ upstream $=\frac{40}{\left(v-u\right)}hr$

Time is taken to cover$55km$ downstream $=\frac{55}{\left(v+u\right)}hr$

As per the second condition,

$\frac{55}{\left(v+u\right)}+\frac{40}{\left(v-u\right)}=13\_\_\_\_\_\_\_\_\_\_\_\_\left(2\right)$

Let

$$\begin{gathered} \frac{1}{v+u}=x \\ \frac{1}{v-u}=y \end{gathered}$$

Equation ($1$) becomes

$44x+30y=10\_\_\_\_\_\_\_\_\_\left(3\right)$

Equation ($2$) becomes

$55x+40y=13\_\_\_\_\_\_\_\_\left(4\right)$

To find the value of $x$and $y$

Multiply an equation $\left(3\right)$ by $5$and equation $\left(4\right)$ by $4$and subtract them, we get

$$\begin{gathered} \left(220x+150y\right)-\left(220x+160y\right)=50-52 \\ \Rightarrow -10y=-2 \\ \Rightarrow y=\frac{2}{10}=\frac{1}{5} \end{gathered}$$

Substitute the value of $y$ in equation$\left(3\right)$ we get

$$\begin{gathered} 44x+30\times \frac{1}{5}=10 \\ \Rightarrow 44x+6=10 \\ \Rightarrow 44x=10-6 \\ \Rightarrow x=\frac{4}{44}=\frac{1}{11} \end{gathered}$$

Thus,

$$\begin{gathered} \frac{1}{v+u}=x=\frac{1}{11} \\ \Rightarrow v+u=11 \\ \frac{1}{v-u}=y=\frac{1}{5} \\ \Rightarrow v-u=5 \end{gathered}$$

So our two equations,

$$\begin{gathered} v–u=5\_\_\_\_\_\_\_\_\_\left(5\right) \\ v+u=11\_\_\_\_\_\_\_\left(6\right) \end{gathered}$$

Adding equations$\left(5\right)$ and$\left(6\right)$ , we get

$$\begin{gathered} \left(v–u\right)+\left(v+u\right)=5+11 \\ \Rightarrow 2v=16 \\ \Rightarrow v=\frac{16}{2} \\ \Rightarrow v=8 \end{gathered}$$

Putting the value of $v$in equation $\left(6\right)$we get

$$\begin{gathered} v+u=11 \\ \Rightarrow 8+u=11 \\ \Rightarrow u=11–8 \\ \Rightarrow u=3 \end{gathered}$$

Hence,

Speed of the stream $=3km/h$

Speed of the boat $=8km/h$