Question

A boat takes 9 hours to travel 30 km upstream and 40 down stream but it takes 12hours to travel 42 km upstream and 50 km downstream find the speed of the boat in still water a
nd the speed of the stream

Answer 1

Answer:

Speed of boat in still water= 8km/h, speed of stream=2km/h

Step-by-step explanation:

Let speed of boat in still water = x and speed of stream = y

Effective speed in downstream=x+y

Effective speed in upstream=x-y

Given time to travel 30km upstream+40km downstream=9hours

=>30/(x-y)+40/(x+y)=9__EQ1

Given time to travel 42km upstream+50km downstream=12hours

=>42/(x-y)+50/(x+y)=12__EQ2

30/(x-y)+40/(x+y)=9__EQ1

42/(x-y)+50/(x+y)=12__EQ2

Let 1/(x-y)=A and 1/(x+y)=B

30A+40B=9__EQ3

42A+50B=12___EQ4

From EQ3 and EQ4,

A=1/6  ,B=1/10

But A= 1/(x-y) and B=1/(x+y)

=> 1/6=1/(x-y)=>x-y=6__EQ5

& 1/10=1/(x+y)=>x+y=10__EQ6

From EQ5 and EQ 6

x=8, y=2

Answer 2

AnswEr :

Let's say , that the speed of boat in still water be x km/hr and Speed of boat in stream be y km/hr .

Therefore ,

• The speed of boat in downstream : ( x + y ) km/hr &
• The speed of boat in upstream : ( x – y ) km/hr .

$\\\dag\underline {\frak { As \:We \: know\:that \:\::\:}}$

$\qquad \bigstar \:\pmb{\underline {\boxed {\sf \:Time\:=\: \dfrac{ Distance}{Speed}\:\:}}}\:$

⠀⠀⠀(I) A boat takes 9 hours to travel 30 km upstream and 40 km down stream .

$\\:\implies \sf Total\:Time\:=\:\Big\{ Time_{(\:Upstream \:)}\:\Big\}\:+\: \Big\{ Time_{(\:Downstream \:)}\:\Big\}\:$

$:\implies \sf 9 \:=\:\bigg\{\: \dfrac{30}{x - y} \bigg\}\:+\: \bigg\{ \dfrac{40}{ x + y } \:\bigg\}\:$

$:\implies \sf 9 \:=\:\bigg\{\: \dfrac{30}{x - y} \bigg\}\:+\: \bigg\{ \dfrac{40}{ x + y } \:\bigg\}\:\qquad \bigg\lgroup eq^n\:(1)\:\bigg\rgroup$

⠀⠀⠀(II) It takes 12 hours to travel 42 km upstream and 50 km downstream

$\\:\implies \sf Total\:Time\:=\:\Big\{ Time_{(\:Upstream \:)}\:\Big\}\:+\: \Big\{ Time_{(\:Downstream \:)}\:\Big\}\:$

$:\implies \sf 12 \:=\:\bigg\{\: \dfrac{42}{x - y} \bigg\}\:+\: \bigg\{ \dfrac{50}{ x + y } \:\bigg\}\:$

$:\implies \sf 12 \:=\:\bigg\{\: \dfrac{42}{x - y} \bigg\}\:+\: \bigg\{ \dfrac{50}{ x + y } \:\bigg\} \:\qquad \bigg\lgroup eq^n\:(2)\:\bigg\rgroup$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀ ☯︎ Let's Consider , that ' 1/x + y = a ' and ' 1/x - y = b ' .

$\\\longrightarrow \sf 9 = 30b + 40a \:\:\qquad \bigg\lgroup eq^n\:(3)\:\bigg\rgroup$

$\longrightarrow \sf 12 = 42b + 50a \:\:\qquad \bigg\lgroup eq^n\:(4)\:\bigg\rgroup$

⠀⠀⠀ Multiplying eq.(3) by 5 and eq.(4) by 4 , we get —

$\\\longrightarrow \sf 45 = 150b + 200a \:\:\qquad$

$\longrightarrow \sf 48 = 168b + 200a \:\:\qquad$

$\qquad \bigstar \:\underline {\sf By \: Eliminating \:\pmb{\frak{ a }}\: from \:both\:Eq^n \:,\;we\:get\::\:}$

$\longrightarrow \sf 18b = 3$

$\longrightarrow \sf b \:=\:\dfrac{3}{18}$

$\longrightarrow \sf \pmb{ b \:=\:\dfrac{1}{6}}$

⠀By Using eq.(3) , we get –

$\\\dashrightarrow \sf 9 = 30b + 40a$

$\dashrightarrow \sf 9 = 30\times \dfrac{1}{6} + 40a$

$\dashrightarrow \sf 9 = 5 + 40a$

$\dashrightarrow \sf 40a = 9 - 5$

$\dashrightarrow \sf 40a = 4$

$\longrightarrow \sf a \:=\:\dfrac{4}{40}$

$\longrightarrow \sf \pmb{ a \:=\:\dfrac{1}{10}}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

Now ,

⠀⠀As We have assumed , ' 1/x + y = a ' and ' 1/x - y = b ' .

$\\\dashrightarrow \sf \dfrac{1}{x + y} = a \:\& \:\dfrac{1}{x - y} = b\:$

$\dashrightarrow \sf \dfrac{1}{x + y} = \dfrac{1}{6} \:\& \:\dfrac{1}{x - y} = \dfrac{1}{10}\:$

$\dashrightarrow \sf x + y = 6 \:\& \:x - y = 10\:$

⠀ By , Solving both Equation we get —

$\\\dashrightarrow \sf 2x = 16$

$\dashrightarrow \pmb{\sf x = 8 }$

⠀By Using , The above Equation , we get —

$\\\dashrightarrow \sf x + y = 6$

$\dashrightarrow \sf 8 + y = 6$

$\dashrightarrow \sf y = 8 - 6$

$\dashrightarrow \pmb{\sf y = 2 }$

Therefore,

• The speed of boat in still water , x = 8 km/hr &
• The Speed of boat is stream , y = 2 km/hr