Question

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

Answer 1

Let the speed of the stream $=x$ km/hr

Let the speed of the boat in still water $=y$ km/hr

Upstream speed $=y-x$ km/hr

Downstream speed $=y+x$ km/hr

$time=\frac{distance}{speed}$

The boat goes $30$ km upstream and $44$ km downstream in $10$ hours.

Time taken $=\frac{30}{y-x}+\frac{44}{y+x}$

$10=\frac{30}{y-x}+\frac{44}{y+x}$ ................. (1)

The boat goes $40$ km upstream and $55$ km downstream in $13$ hours.

Time taken $=\frac{40}{y-x}+\frac{55}{y+x}$

$13=\frac{40}{y-x}+\frac{55}{y+x}$.................. (2)

Let $\frac{1}{y-x}=u$

and $\frac{1}{y+x}=v$

From (1) and (2),

$30u+44v=10$ ...................(3)

$40u+55v=13$ ...................(4)

Multiply equation (3) with 4 and equation (4) with 3,

$120u+176v=40$ ........... (5)

$120u+165v=39$ ........... (6)

subtract equation (6) from (5),

$176v-165v=40-39$

$11v=1$

$v=\frac{1}{11}$

$\Rightarrow \frac{1}{y+x}=\frac{1}{11}$

$y+x=11$ .......................... (7)

From equation (3),

$30u=10-44v$

$30u=10-44\times \frac{1}{11}$

$30u=10-4=6$

$u=\frac{1}{5}$

$\frac{1}{y-x}=\frac{1}{5}$

$\Rightarrow y-x=5$ .................. (8)

Adding (7) and (8), we get,

$2y=16$

$y=8$

From equation (7),

$x=11-y$

$x=11-8=3$