Question

A boat goes 30 km upstream and 44 km downstream in 10 hrs. In 13 hrs, 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 boat in still water=x

$\frac{km}{hr} and \: the \: speed \: of stream = y \frac{km}{hr}$

Speed of boat at downstream

$= (x + y) \frac{km}{hr}$

Speed of boat at upstream

$= (x - y) \frac{km}{hr}$

■ $time = \frac{distance}{speed}$

Time taken to cover 30 km upstream

$= \frac{30}{x - y}$

Time taken to cover 44 km downstream

$= \frac{44}{x + y}$

According to the first condition,

$⇒ \frac{30}{x - y} = \frac{44}{x + y} = 10$

Time taken to cover 40 km upstream 

$⇒ \frac{40}{x - y}$

Time taken to cover 55 km downstream 

$⇒ \frac{55}{ x + y} </strong></p><p><strong>[tex]⇒ \frac{55}{ x + y}$

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Answer 2

$\boxed{ \bf \: SOLUTION}$

Let be the speed of boat in still water be x km/hr.

and let the speed of current (stream) be y km/hr.

Speed downstream = x + y

Speed upstream = x – y

According to the question,

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

$\rm \: and \: \frac{40}{x - y} + \frac{55}{x + y} = 13 \: \: \: \: \: \: \: ...(ii)$

$\rm \: Let \: \frac{1}{x - y} = u \: and \: \frac{1}{x + y} = \: v$

$\therefore \: \rm \: From \: Eqs. \: (i) \: and \: (ii), \: we \: get$

$\rm \: 30u + 44v = 10 \: \: \: \: \: \: \: ...(iii)$

$\rm \: 40u + 55v = 13 \: \: \: \: \: \: \: ...(iv)$

From Eq. (iii),

$\rm \: 30u = 10 - 44v$

$\rm \: u = \: \frac{10 - 44v}{30}$

Putting value of u in Eq. (iv), we get

$\rm \: 40 \bigg( \frac{10 - 44v}{30} \bigg) + 55v = 13$

$\implies \: \rm \: 4(10 - 44v) + 165v = 39$

$\implies \: \rm \: 40 - 176v + 165v = 39$

$\implies \: \rm \: - 176v + 165v = 39 - 40$

$\implies \: \rm \: - 11v = - 1$

$\implies \: \rm \: v = \frac{1}{11}$

$\rm \: Putting \: v = \frac{1}{11} \: in \: value \: of \: u, \: w e\: get$

$\rm \: u = \frac{10 - 44v}{30} = \frac{10 - 44 \bigg( \frac{1}{11} \bigg) }{30} = \frac{10 - 4}{30} = \frac{6}{30}$

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

$\rm \: Now, \: u = \frac{1}{x - y}$

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

$\implies \: \rm \: x - y = 5 \: \: \: \: \: \: \: ...(v)$

$\rm \: Now, \: v = \frac{1}{x + y}$

$\rm \: \frac{1}{11} = \frac{1}{x + y}$

$\implies \: \rm \: x + y = 11 \: \: \: \: \: \: \: ...(vi)$

From Eqs. (v) and (vi), we get

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \tt{ \red {x = 8 \: y = 3}}}$

Hence, speed of boat in still water = x = 8 km/hr

Speed of stream = y = 3 km/hr.