Question

Example 19: A boat goes 30 km upstream and 44 km downstream in10 hours. In 13 hours, it can go40 km upstream and 55 kmdown-stream. Determine the speedof the stream and that of the boat in still water​

Answer 1

Given

• A boat goes 30 km upstream and 44 km downstream in10 hours.

• In 13 hours, it can go40 km upstream and 55 kmdown-stream.

To Find

• speed of the stream and that of the boat in still water

Concept used

• Elimination method

Solution

Let the speed of the boat in still water be xkm/h

and speed of the stream be ykm/h

While going downstream

• speed of boat=(x+y)km/h
• Distance=44km
• $\bf \: time = \dfrac{distance}{speed} \\ \\ \bf \implies \boxed{ \bf t_{1} = \dfrac{44}{x + y} }$

While going upstream

• speed of boat=(x-y)km/h
• Distance=30
• $\boxed{ \bf t_{2} = \dfrac{30}{x - y} }$

and t1+t2=10hr

$\bf \: t_{1} + t_{2} = \dfrac{44}{x + y} + \dfrac{30}{x - y} \\ \\ \underline{ \boxed{\bf \: \dfrac{44}{x + y} + \dfrac{30}{x - y} = 10}}....1$

In the second case,In 13 hours, it can go40 km upstream and 55 km down-stream.

$\underline{ \boxed{\bf \dfrac{55}{x + y} + \dfrac{40}{x - y} = 13}} ...2$

$\bf \: let \: \dfrac{1}{x + y} \: be \: u \: and \: \dfrac{1}{ x - y } \: be \: v$

On substituting these values in equation 1 and 2 we get

$\bf \: 44u + 30v = 10 \\ \bf 55u + 40u = 13$

By elimination method

$\bf \: \: \: \: 176u + 120v = 40\\ \\ \bf \boxed{ - }165u + 120v = 39 \\ \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \bf \: 11u = 1 \\ \\ \underline{ \boxed{ \bf \: u = \frac{1}{11} \: and \: v = \frac{1}{5} }}$

Now putting values of u and v

$\bf \: \dfrac{1}{x + y} = \frac{1}{11} \: and \: \dfrac{1}{x - y} = \frac{1}{5} \\ \\ \implies \bf \: x + y = 11 \: and \: x - y = 5$

Now elimination method in these equations

$\bf \: x + y = 11 \\ \bf \: x - y = 5 \: \\ \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \bf \: x = 8 \: and \: y = 3$

$\bf \therefore \: the \: speed \: of \: the \: boat \: in \: still \: water \: is \: 8km \: per \: hr \\ \bf and \: \: speed \: of \: stream \: is \: 3km \: per \: hr$