Question

A motor boat can travel $30km$ upstream and $28km$ downstream in $7$ hours. It can also travel $21Km$ upstream and return in $5$ hours. Calculate the speed of the boat when it is in still water in $km/h$.

Answer 1

Step 1: Given data.

1. A motor boat can travel $30km$ upstream and $28km$ downstream in $7$ hours.
2. The same motorboat can travel $21km$ upstream and return in $5$ hours.

Step 2: Formula to be used.

1. The formula for speed $S$ when the time $T$ and distance $D$ is given can be given by $S=\frac{D}{T}$.
2. The formula for time $T$ when the speed $S$ and distance $D$ is given can be given by $T=\frac{D}{S}$.
3. The formula for distance $D$ when the speed $S$ and time $T$ is given can be given by $D=S·T$.

Step 3: Find the speed of the boat in still water.

Assume that, the speed of the boat in still water is $xkm/h$ and the speed of the water steam is $ykm/h$.

Therefore, the speed of the boat in upstream is $x-ykm/h$ and the speed of the boat in downstream is $x+ykm/h$.

So, According to the first situation given in the question.

$\frac{30}{x-y}+\frac{28}{x+y}=7...1$

According to the second situation given in the question.

$\frac{21}{x-y}+\frac{21}{x+y}=5...2$

Assume that, $\frac{1}{x-y}=p$ and $\frac{1}{x+y}=q$.

So, equation $1$ becomes:

$30p+28q=7...3$

And equation $2$ becomes:

$21p+21q=5...4$

Now, multiply equation $3$ by $21$ and $4$ by $28$ and subtract.

$$\begin{gathered} 630p+588q-588p-588q=147-140 \\ \Rightarrow 42p=7 \\ \Rightarrow p=\frac{1}{6} \end{gathered}$$

Now, substitute the value of $p$ in equation $4$.

$$\begin{gathered} 21p+21q=5 \\ \Rightarrow p+q=\frac{5}{21} \\ \Rightarrow \frac{1}{6}+q=\frac{5}{21} \\ \Rightarrow q=\frac{5}{21}-\frac{1}{6} \\ \Rightarrow q=\frac{30-21}{126} \\ \Rightarrow q=\frac{9}{126} \\ \Rightarrow q=\frac{1}{14} \end{gathered}$$

Since, $\frac{1}{x-y}=p$ and $\frac{1}{x+y}=q$.

Therefore, the value of $x+y=14...5$ and $x-y=6...6$.

Add equation $5$ and equation $6$.

$$\begin{gathered} 2x=14+6 \\ \Rightarrow 2x=20 \\ \Rightarrow x=10 \end{gathered}$$

Therefore, the speed of the boat when it is in still water is $10km/h$.