Question

A boat goes 16kmper hour upstream 24 km down stream in 6 hour Also it cover 12 km upstream and 36 km down stream
Find the speed of boat

Answer 1

$\large{\underline{\underline{\mathfrak{\blue{\sf{Answer-}}}}}}$

speed of boat is 8 km/h.

$\large{\underline{\underline{\mathfrak{\blue{\sf{Explanation-}}}}}}$

$\large{\orange{\boxed{\green{\underline{\red{\mathfrak{Given-}}}}}}}$

• A boat goes 16 km upstream and 24 km down stream in 6 hour.

• Also it cover 12 km upstream and 36 km down stream.

$\large{\orange{\boxed{\green{\underline{\red{\mathfrak{To\:find-}}}}}}}$

• Speed of boat

$\large{\orange{\boxed{\green{\underline{\red{\mathfrak{Solution-}}}}}}}$

Let the speed of boat be $\bold\pink{x\:km/h}$ and speed of current be $\bold\pink{y\:km/h}$.

Downstream speed = $\bold{x+y\:km/h}$

upstream speed = $\bold{x-y\:km/h}$

We know that,

$\huge{\underline{\boxed{\purple{Time=\dfrac{Distance}{Speed}}}}}$

$\implies$ $\sf{\dfrac{24}{x+y}+\dfrac{16}{x-y}=6}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀–eq (1).

$\implies$ $\sf{\dfrac{36}{x+y}+\dfrac{12}{x-y}=6}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀–eq (2).

Now,

put $\sf{\dfrac{1}{x+y}=a}$ and $\sf{\dfrac{1}{x-y}=b}$

• $\sf{24a+16b=6}$

By taking 2 as common,

$\implies$ $\sf{12a+8b=3}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀–eq (3).

• $\sf{36a+12b=6}$

By taking 6 as common,

$\implies$ $\sf{6a+2b=1}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀–eq (4).

★ Multiply equation (4) by 4.

$\sf{\green{24a+8b=4}}$ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀–eq (5).

From equation (3) and (5), by subtracting,

$\sf{12a=1}$

$\implies$ $\sf{a=\dfrac{1}{12}}$

Now put the value of a in equation (4),

$\sf{b=4}$

_________________

$\sf{\dfrac{1}{x+y}}$ = $\sf{\dfrac{1}{12}}$ and $\sf{\dfrac{1}{x-y}}$ = $\sf{\dfrac{1}{4}}$

we get,

• $\sf{x+y=12}$ ______(A)

• $\sf{x-y=4}$ _______(B)

Solving both A and B, we get,

$\huge{\underline{\boxed{\red{x=8}}}}$

Hence, speed of boat is 8 km/h.

___________________________

#answerwithquality !!

#BAL :)

Answer 2

$\tt\huge\star{\red{\underline{\mathfrak{Correct\:Question :-}}}}$

A boat goes 16kmper hour upstream 24 km down stream in 6 hour Also it cover 12 km upstream and 36 km down stream in the same time.

Find the speed of boat in still water and the speed of the current.

$\tt\huge\star{\blue{\underline{\mathfrak{Answer :-}}}}$

The speed of the boat in still water x km/hr = 8 km/hr. and the speed of the stream be y km/hr = 4 km/hr.

$\tt\huge\star{\green{\underline{\mathfrak{Explanation :-}}}}$

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

The speed of the boat downstream = ( x + y ) km/hr. and the speed of the boat upstream = ( x - y ) km/hr.

We know that,Time =$\dfrac {Distance}{Time}$

So, Time = $\dfrac{Distance}{Speed}$

In the first case when the boat goes 16 km upstream, the time taken t1 will be = $\dfrac {16}{x-y}$

Now, the time taken t2 by the board to go downstream for about 24 kilometres will be = $\dfrac {24}{x+y}$

Now, we are given that the total time taken is 6 hrs.

$So, \:t_1 + t_2 = 6 \:hours$

$Therefore, \:\pink{\dfrac{16}{x-y}+\dfrac{24}{x+y}=6 :- equation\: 1}$

In the second case, in the same time that is 6 hours it covers 12 km upstream and 36 km downstream.

So, the equation becomes = $\pink {\dfrac{12}{x-y}+\dfrac{36}{x+y}=6 :- equation\: 2}$

Let us assume $\red {\dfrac {1 }{x-y}=u }$ and $\red {\dfrac {1 }{x+y}=v}$

$Equation\: 1 :- \green {16 u + 24 v = 6 } \:and\: Equation\: 2 :- \green {12 u + 36 v = 6}$

${Simplifying\: the \:equations\: further :-}$

$\tt \implies Equation\:1 :-\green {16 u + 24 v - 6 = 0 }\:and \:Equation \:2 :-\green {12 u + 36 v - 6 = 0}$

$\tt \implies Equation\:1 :-\green {2 (8 u + 12 v - 3)= 0 } \:and\: Equation \:2 :- \green {2 (6 u + 18 v - 3) = 0 }$

$\tt \implies Equation \:1 :-\green { 8 u + 12 v - 3= 0 }\: and\:Equation\: 2 :- \green { 3 (2 u + 6 v - 1) = 0 }$

$\tt\implies Equation \:1 :-\green { 8 u + 12 v - 3= 0 }\: and\:Equation\: 2 :- \green { 2 u + 6 v - 1 = 0 }$

Now, we will solve this equationby substitution method :-

From equation 2,

$\implies u= \dfrac{1-6v}{2}$

Substituting it in equation 1,

$\implies 8 (\dfrac{1-6v}{2})+12v=3$

$\implies \dfrac{8-48v+24v}{2}=3$

$\implies 8-48v+24v=6$

$\implies 8-24v=6$

$\implies -24v=6-8$

$\implies \cancel{-}24v=\cancel {-}2$

$\implies v=\dfrac {2}{24}$

$\purple{\huge\underline{\implies v=\dfrac{1}{12}}}$

Substituting the value of v in u to find actual value of u,

$\implies u= \dfrac{1-\cancel{6}(\dfrac{1}{\cancel{12}})}{2}$

$\implies\dfrac{\dfrac{2-1}{2}}{2}$

$\huge\purple{\underline{\implies u=\dfrac{1}{4}}}$

So, $\dfrac{1}{x-y}=\dfrac{1}{4} \:and \: \dfrac{1}{x+y}=\dfrac{1}{12}$

$\implies x-y=4\:and\:x+y=12$

By elimination method, we get,

$\implies 2x = 16$

$\huge\red{\underline{\implies x = 8}}$

Substituting the value of x in any of the equations above :-

$\implies 8+y=12$

$\red{\huge\underline{\implies y = 4}}$

So, The speed of the boat in still water x km/hr = 8 km/hr. and the speed of the stream be y km/hr = 4 km/hr.

_______________

#AnswerWithQuality

#BAL

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