Question

A Boat goes 16 km upstream and 24 km downstream in 6 hours .also it covers 12 km upstream and 36 downstream at the same time .find the speed of the boat in still water and that of the stream.

Answer 1

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a boat goes 16 km upstream and 24 km downstream in 6 hours.
it also covers 12 km upstream and 36 km downstream in same time.
find the rate of speed still in water and stream.
:
Let s = rate of speed in still water
let c = rate of the current
then
(s-c) = effective speed upstream
and
(s+c) = effective speed downstream
:
Write a time equation for each trip, time = dist/speed
:
 +  = 6
and
 +  = 6
Simplify both equations, divide the 1st by 2, the 2nd by 3 and we have
 +  = 3
 +  = 2
----------------------------------------Subtraction eliminates (s+c)
 + 0 = 1
4 = s - c
(c+4) = s
Using the first original equation replace s with (c+4)
 +  = 6
 +  = 6
4 +  = 6
 = 6 - 4
 = 2
24 = 2(2c + 4)
24 = 4c + 8
24 - 8 = 4c
c = 16/4
c = 4 mph is the rate of the stream
and
s = c+4
s = 4 + 4
s = 8 mph is the rate in still water 
:
:
See if that checks out in the 2nd equation
 +  = 6
3 + 3 = 6

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Answer 2

$\tt\huge{\red{\underline{\mathfrak{Answer :-}}}}$

____________________________________

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.

$\tt\huge{\blue{\underline{\mathfrak{Explanation :-}}}}$

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

• 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,

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

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

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

$\implies 8-24v=6$

$\implies -24v=6-8$

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

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

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

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

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

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

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

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,

$\implies 2x = 16$

$\implies{x = 8}$

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

$\implies 8+y=12$

$\implies{y = 4}$

Therefore,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.