Question

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

Answer 1

• 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

As we know that,

`\bigstar \: \: \sf Time = (Distance)/(Speed) \\ \\`

`\bigstar\:\underline{\boldsymbol{According\: to \:the\: Question\:now :}} \\[tex]</p><p></p><p></p><p>[tex]:\implies \sf (30)/(x - y) + (44)/(x + y) = 10 \\ \\ \\`

`:\implies \sf (40)/(x - y) + (55)/(x + y) = 13 \\ \\ \\`

`\sf Let \: (1)/(x - y) = \textsf{\textbf{m}} \sf \: \: and \: \: \sf{(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 \\ \\ </p><p>\sf 120m + 165n = 39 \\ \\`

`\sf \: \: ( - ) \:\:\: \: \: ( - ) \: \: \: \: \: \: \: ( - )`

_____________________

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

`\: \: \: \: \: \qquad\sf n= (1)/(11) \\ \\`

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

`\dashrightarrow\:\:\sf 30m + 44 * (1)/(11) = 10 \\ \\ \\`

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

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

`\dashrightarrow\:\:\sf m= (6)/(30) \\ \\ \\`

`\dashrightarrow\:\:\sf m= (1)/(5) \\ \\ \\`

____________________....

`\dashrightarrow\:\:\sf (1)/(x - y) = m\\ \\ \\`

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

`\dashrightarrow\:\:\sf (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}}\\`

Answer 2

Average speed of boat = 3.67 km/h

Average speed of stream = 0.63 km/h

Explanation:

30 km upstream in 10 hours gives a speed of 30/10 = 3 km/h relative speed of boat and stream

Same way, 44 km in 10 hrs = 4.4 km/h relative speed of boat and stream.

Let the speed of the stream be y

Let the speed of both be x, then

x - y = 3.... (1)

x + y = 4.4.... (2)

Subtract 1 from 2

2y = 1.4

y = 1.4/2 = 0.7 km/h

substitute value of y in 1

x - 0.7 = 3

x = 3.7 km/h

Also, 40 km upstream in 13 hrs gives speed of 3.08 km/h

55 km in 13 hrs gives 4.2 km/h

Let speed of boat be x and that of stream be y

x - y = 3.08.... (1)

x + y = 4.2..... (2)

Subtract 1 from 2

2y = 1. 12

y = 0.56 km/h

Substitute value of y in 1

x - 0.56 = 3.08

x = 3.64 km/h

Average speed of boat = (3.7 + 3.64)/2

= 3.67 km/h.

Average speed of stream = (0.7 + 0.56)/2 = 0.63 km/h