Question

A motor boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km. downstream. Determine speed of boat in still water and speed of stream. ​

Answer 1

Answer:

$\bigstar{\bold{Speed\:of\:the\:stream=3\:km/hr}}$

$\bigstar{\bold{Speed\:of\:the\:boat=8\:km/hr}}$

Step-by-step explanation:

$\Large{\underline{\underline{\rm{Given:}}}}$

• The boat goes 30 km upstream and 44 km downstream in 10 hours
• The boat goes 40 km upstream and 55 km downstream in 13 hours

$\Large{\underline{\underline{\rm{To\:Find:}}}}$

• Speed of the boatt in still water
• Speed of the stream

$\Large{\underline{\underline{\rm{Solution:}}}}$

➝ Let the speed of the boat be x km/hr

➝ Let the speed of the stream be y km/hr

➝ Hence,

    Speed while going upstream = x - y km/hr

    Speed while goind downstream = x + y km/hr

➝ We know that,

    Time = Distance/Speed

➝ Hence by first case,

    $\sf{\dfrac{30}{x-y}+\dfrac{44}{x+y} =10-----(1)}$

➝ By second case,

   $\sf{\dfrac{40}{x-y} +\dfrac{55}{x+y} =13----(2)}$

➝ Let 1/x -y = p, 1/x + y = q

➝ Substituting the data in equation 1 and 2

    30p + 44q = 10-------(3)

    40p + 55q = 13-----(4)

➝ Multiply equation 3 by 4 and equation 4 by 3

    120 p + 176q = 40

    120p + 165 q = 39

➝ Solving by elimination method,

                 11q = 1

                    q = 1/11

➝ Substitute the value of q in equation 3

    30p + 44  × 1/11 = 10

    30p + 4 = 10

    30p = 6

         p = 6/30 = 1/5

➝ But we know that,

    1/x - y = p = 1/5

    x - y = 5

    x = 5 + y ------(5)

➝ 1/x + y = q = 1/11

    x + y = 11

➝ Substitute value of x from equation 5

    5 + y + y = 11

    2y = 6

      y = 3

➝ Hence the speed of the stream is 3 km/hr

    $\boxed{\bold{Speed\:of\:the\:stream=3\:km/hr}}$

➝ Now substitute the value of y in equation 5

   x = 5 + 3

   x = 8 km/hr

➝ Hence the speed of the boat is 8 km/hr

    $\boxed{\bold{Speed\:of\:the\:boat=8\:km/hr}}$

$\Large{\underline{\underline{\rm{Notes:}}}}$

➝ A linear equation in two variables can be solved by,

• Substitution method
• Elimination method
• Cross multiplication method

   

Answer 2

Answer:

• Let the speed of boat in stream be x km/hr.

• And the speed of boat in still water be y km/hr.

• For upstream = x - y

• For downstream = x + y

We know,

$\bigstar \: \: \sf Time = \dfrac{Distance}{Speed}$

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

$:\implies \sf \dfrac{30}{x - y} + \dfrac{44}{x + y} = 10$

$:\implies \sf \dfrac{40}{x - y} + \dfrac{55}{x + y} = 13$

$\sf Let \: \dfrac{1}{x - y} = \textsf{\textbf{m}} \sf \: \: and \: \: \sf{\dfrac{1}{x + y} = \textsf{\textbf{n} }}$

$:\implies \sf 30m + 44n = 10\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (i)}}\Bigg\rgroup$

$:\implies \sf 40m + 55n = 13\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (ii)}}\Bigg\rgroup$

$\qquad\tiny \underline{\frak{ Multiply \: equation \: (ii) \: by \: 3 \: and \: equation \: (i) \: by \: 4 :}}$

$:\implies \sf 120m + 176n = 40\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (iii)}}\Bigg\rgroup$

$:\implies \sf 120m + 165n = 39\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (iv)}}\Bigg\rgroup$

$\qquad\tiny \underline{\frak{ Substracting \: equation \: (iii) \: from \: equation \: (iv) \: we \: get :}}$

$\sf 120m + 176n = 40$

$\sf 120m + 165n = 39$

$\sf \: \: \: ( - ) \: \: \: ( - ) \: \: \: \: \: \: \: ( - )$

_______________________

$\: \: \: \: \qquad\sf 11n= 1$

$\: \: \: \: \qquad\sf n= \dfrac{1}{11}$

$\qquad\tiny {\frak{ Put\: n = \dfrac{1}{11}\:in\:equation \: (i) \: we \: get :}}$

$\dashrightarrow\:\:\sf 30m + 44 \times \dfrac{1}{11} = 10$

$\dashrightarrow\:\:\sf 30m= 10 - 4$

$\dashrightarrow\:\:\sf 30m= 6$

$\dashrightarrow\:\:\sf m= \dfrac{6}{30}$

$\dashrightarrow\:\:\sf m= \dfrac{1}{5}$

____________________....

$\dashrightarrow\:\:\sf \dfrac{1}{x - y} = m$

$\dashrightarrow\:\:\sf x - y= 5\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (v)}}\Bigg\rgroup$

$\dashrightarrow\:\:\sf \dfrac{1}{x + y} = n$

$\dashrightarrow\:\:\sf x + y= 11\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (vi)}}\Bigg\rgroup$

$\qquad\tiny {\frak{Adding\:equation \: (v) \: and \: equation \: (vi) \: we \: get :}}$

$:\implies \sf 2x = 16$

$:\implies \underline{ \boxed{ \textsf {\textbf{x = 8 km/hr}}}}$

$\qquad\tiny {\frak{Putting\:x = 8\: in \: equation \: (v) \: we \: get :}}$

$:\implies \sf 8 - y = 5$

$:\implies \sf y = 8 - 5$

$:\implies \underline{ \boxed{ \textsf {\textbf{y = 3 km/hr}}}}$

_________________....

$\bigstar\:\underline{\sf{Therefore\: speed\: of \:boat\: in\: still \:water\: and\: speed\: of\: stream:}}$

$\bullet\:\:\textsf{Speed of boat in stream = \textbf{8 km/hr}}$

$\bullet\:\:\textsf{Speed of boat in still water = \textbf{3 km/hr}}$