Question

a motorboat takes 6 hours to cover 100 KM downstream and 30 kilometre upstream if the boat goes 75 km downstream and return back to the starting point in 8 hour

Answer 1

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

Let the rate of flow of the stream be y km/h
Speed of boat upstream = (x - y) km/h.
Speed of boat downstream = (x + y)km/h.

we know time = distance/speed.

now , A to Q,

Time for 100 km downstream and 30 km upstream

100/(x + y) + 30/(x - y)


And it takes 6 hrs to cover downstream and upstream. Then

100/(x + y) + 30/(x - y) = 6



Time for 75 km downstream and returning (means 75 km upstream)
= 75/(x + y) + 75/(x - y)

Given that the time taken is 8 hours

75/(x + y) + 75/(x - y) = 8

$\bold{PUT} \left[\begin{array}{cc}\frac{1}{x + y} = p\:\: and \:\:\frac{1}{x - y}=q\end{array}\right]$

now the equation should be .

100p + 30q = 6
50p + 15q = 3------------( 1 )
75p + 75q = 8----------( 2 )


from--------( 1 ) &---------( 2 )
multiply by ( 3 ) in -----( 1 )

250p + 75q = 15
75p + 75q = 8
(–)______(–)____(–)
-------------------------------
175p = 7

p = 1/25 [ put in -------( 1 ) ]

50(1/25) + 15q = 3

2 + 15q = 3

q = 1/15 = 1/(x - y)

x - y = 15------------( 3 )

p = 1/25 = 1/(x + y)

x + y = 25---------( 4 )


From---------( 3 ) &----------( 4 )

x - y = 15
x + y = 25
---------------
2x = 40

x = 20 [ put in ------( 3 ) ]


x - y = 15

20 - y = 15

y = 20 - 15

y = 5 , x = 20

Hence, the speed of the motor boat in still water is 20 km/h and rate of flow of the stream is 5 km/h.

Answer 2

✴✴$\bf{HEY \: FRIENDS!!}$✴✴

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✴✴$\underline{Here \: is \: your \: answer↓}$⬇⏬⤵

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

↪➡ Speed upstream= ( x - y ) km/hr.

↪➡ and, speed downstream = ( x + y ) km/hr.

▶⏩ Time taken to cover 30km upstream
$\boxed{ = \frac{30}{(x - y)} hrs}$

▶⏩ Time taken to cover 24km downstream
$\boxed{ = \frac{100}{(x + y)} hrs}$

▶⏩ Total time taken = 6hrs.

$\bf{ = > \frac{30}{x - y} + \frac{100}{x + y} = 6............(1)}$

▶⏩ Again, time taken to cover 75km upstream
$\boxed{ = \frac{75}{(x - y)} hrs}$
( Returning 75km back from 75km downstream).

▶⏩ Time taken to cover 75km downstream
$\boxed{ = \frac{75}{(x + y)} hrs}$

▶⏩ Total time taken 8hrs.

$\bf{ => \frac{75}{(x - y)} + \frac{75}{(x + y)} = 8................(2)}$

$\bf \underline{putting \: \frac{1}{(x - y)} = u \: and \: \frac{1}{(x + y)} = v.}$
in equation (1) and (2), we get:-)

↪➡ 30u + 100v = 6.
↪➡ 15u + 50v = 3.....................(3)

and,
↪➡ 75u + 75v = 8.....................(4)

▶⏩ Multiply by 5 in equation (3).
we get,

↪➡ 75u + 250v = 15....................(5)

▶▶ Substract in equation (4) and (5).

75u + 250v = 15
75u + 75v = 8
(-)......(-)...........(-)
______________
=> 175v = 7.
$\boxed{ = > \: v = \frac{7}{175} = \frac{1}{25} }$

▶⏩ Put the value of ‘v’ in equation (3).

$\bf{ = > 15u + 50( \frac{1}{25} ) = 3}$

$\bf{ = > 15u + 2 = 3.}$

$\bf{ \: = > 15u = 3 - 2.}$

$\bf{ = > \: u = \frac{1}{15} .}$

$\bf{ = > \frac{1}{(x - y)} = \frac{1}{15} .}$

$\boxed{ = > \: x - y = 15...............(6)}$
and,
$\bf{ \: = > v = \frac{1}{25} .}$
$\bf{ = > \frac{1}{(x + y)} = \frac{1}{25} .}$
$\boxed{ = > \: x + y = 25..........(7)}$
▶⏩ Substract equation (6) and (7).

x - y = 15
x + y = 25
(-)...(-)....(-)
_________
=> -2y = -10
$\boxed{ \: = > y = \frac{ - 10}{ - 2} = 5.}$
▶⏩ Now, put the value of ‘y’ in equation (6).

↪➡ x - 5 = 15.

↪➡ x = 15 + 5.

$\bf \boxed{ \: = > x = 20.}$

✅✅Hence, the speed of motorboat in still water is 20 km/hr.
and , the speed of motorboat in stream is 5 km/hr.✔✔

✴✴$\boxed{THANKS}$✴✴

☺☺☺$\bf \underline{Hope \: it \: is \: helpful \: for \: you}$✌✌✌.