Question

a boat covers 40 km upstream and 36 km downstream in 8 hours also it covers 48 km upstream and 72 km downstream in 12hrs . find the speed of the boat in still water and that of the stream .​

Answer 1

       $\underline{\bold{Solution:}}$

       $\text{Let the speed of the boat and stream be }x \text{ and }y \text{ respectively}$

       $\text{Upstream means against the flow of the stream}$

       $\text{Downstream means towards the flow of the stream}$

$\implies \text{Speed of boat upstream} = (x-y)$

$\implies \text{Speed of boat downstream} = (x+y)$

       $\text{According to the question:}$

       $\text{Boat covers 40 km upstream and 36 km downstream in 8 hours}$

       $\text{We know that Time} = \dfrac{\text{Distance}}{\text{Speed}}$

$\implies \dfrac{40}{x - y} + \dfrac{36}{x + y} = 8$

$\implies \dfrac{10}{x - y} + \dfrac{9}{x + y} = 2 \text{ ------ (i)}$

       $\text{Boat also covers 48 km upstream and 72 km downstream in 12 hours}$

$\implies \dfrac{48}{x - y} + \dfrac{72}{x + y} = 12$

$\implies \dfrac{4}{x - y} + \dfrac{6}{x + y} = 1 \text{ ------ (ii)}$

       $\text{Let }a = \dfrac{1}{x-y}\text{ and }b = \dfrac{1}{x + y}$

$\implies \text{The new equations are:}$

       $10a + 9b = 2 \text{ ------ (iii)}$

       $4a + 6b = 1 \text{ ------ (iv)}$

       $\text{From (iv) we get that}$

       $a = \dfrac{1-6b}{4}$

       $\text{Substituting in (iii)}$

$\implies 10(\dfrac{1-6b}{4}) + 9b = 2$

       $\text{Simplifying...}$

$\implies 5(\dfrac{1-6b}{2}) + 9b = 2$

       $\text{Simplifying...}$

$\implies \dfrac{5-30b}{2} + 9b = 2$

       $\text{Simplifying...}$

$\implies \dfrac{5-30b+ 18b}{2} = 2$

       $\text{Simplifying...}$

$\implies \dfrac{5-12b}{2} = 2$

       $\text{Transposing 2 to R.H.S and simplifying...}$

$\implies 5 - 12b = 4$

       $\text{Transposing 5 to R.H.S}$

$\implies - 12b = 4 - 5$

       $\text{Simplifying...}$

$\implies - 12b = -1$

       $\text{Transposing 12 to R.H.S}$

$\implies b = \dfrac{1}{12}$

       $\text{Substituting in (iv)}$

$\implies 4a + 6(\dfrac{1}{12})= 1$

       $\text{Simplifying...}$

$\implies 4a + \dfrac{1}{2} = 1$

       $\text{Transposing } \dfrac12\text{ to R.H.S and simplifying...}$

$\implies 4a = \dfrac12$

       $\text{Transposing } 4 \text{ to R.H.S and simplifying...}$

$\implies a = \dfrac18$

       $\text{We know that }a = \dfrac{1}{x-y}\text{ and }b = \dfrac{1}{x + y}$

$\implies \dfrac{1}{x-y} = \dfrac18$                    $\implies\dfrac{1}{x + y} = \dfrac{1}{12}$

$\implies x-y = 8 \text{ ------ (v)}$       $\implies x + y = 12 \text{ ------ (vi)}$

       $\text{Adding (v) and (vi)}$

$\implies x-y + x+ y = 8 + 12$

       $\text{Simplifying...}$

$\implies 2x = 20$

       $\text{Transposing } 2 \text{ to R.H.S and simplifying...}$

$\implies x = 10$

       $\text{Substituting in (v)}$

$\implies 10 - y = 8$

       $\text{Transposing } 10 \text{ to R.H.S and simplifying...}$

$\implies - y = -2$

       $\text{Simplifying...}$

$\implies y = 2$

 $\therefore \text{ }\text{ }\boxed{\text{Speed of boat = 10 km/h and Speed of stream = 2 km/h}}$