Question

A boat goes $16$ km upstream and $24$ km downstream in $6$ hrs. Also it covers $12$ km upstream and $36$ km downstream in the same time. Find the speed of the boat upstream and downstream.

Answer 1

Let speed of the boat in still water $=x$ km/hr, and

Speed of the current $=y$ km/hr

Downstream speed $=(x+y)$ km/hr

Upstream speed $=(x-y)$ km/hr

$T=\frac{D}{S}$

$\frac{24}{x+y}+\frac{16}{x-y}=6$ .......$\left(1\right)$

$\frac{36}{x+y}+\frac{12}{x-y}=6$ .......$\left(2\right)$

Put $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ the above equation becomes,

$24u+16v=6$

Or, $12u+8v=3$ ... $\left(3\right)$

$36u+12v=6$

Or, $6u+2v=1$ ... $\left(4\right)$

Multiplying $\left(4\right)$ by $4$, we get,

$24u+8v=4$v … $\left(5\right)$

Subtracting $\left(3\right)$ by $\left(5\right)$, we get,

$12u=1$

$\Rightarrow u=\frac{1}{12}$

Putting the value of $u$ in $\left(4\right)$, we get, $v=\frac{1}{4}$

$\Rightarrow \frac{1}{x+y}=\frac{1}{12}$ and $\frac{1}{x-y}=\frac{1}{4}$

$\Rightarrow x+y=12$ and $x-y=4$

Thus, speed of the boat upstream $=4$ km/hr

Speed of the boat downstream $=12$ km/hr