Question

Q2. A steamer goes downstream and covers the distance
between two ports in 9 hours, while it covers the same
distance upstream in 10 hours. If the speed of the stream is 1
km/hr, find the speed of the steamer in still water and the
distance between the two ports.​

Answer 1

Given,

A steamer goes downstream and covers the distance  between two ports in 9 hours, while it covers the same  distance upstream in 10 hours. Speed of the stream is 1  km/hr.

To find :

Find the speed of the steamer in still water and the  distance between the two ports.​

Solution :

Let the speed of streamer in still water be n km/h

Speed of stream = 1 km/h

∴ Speed of streamer in downstream = (n + 1) km/h

∴ Distance covered in downstream = 9(n + 1) km

Now, speed of streamer in upstream = (n - 1) km/h

∴ Speed of streamer in upstream = 10(n - 1) km

Now atq,

⇒ 9(n + 1) = 10(n - 1)

⇒ 9n + 9 = 10n - 10

⇒ 10n - 9n = 10 + 9

⇒ n = 19 km/h

∴ Speed of streamer in still water = 19 km/h

Now distance covered in upstream :

⇒ Distance b/w two ports = 10(19 - 1) km

⇒ Distance b/w two ports = 10 * 18

⇒ Distance b/w two ports = 180 km

∴ Distance between two ports = 180 km

Answer 2

Answer :-

$\: \\ \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Linear Equations In Two Variables has been used. We see that we are given here two unknown quantities that is Distance and Speed od Streamer. We will take them as variable quantities and then find their values using constants.

Let's do it  !!

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★ Formula Used :-

$\: \\ \large{\boxed{\boxed{\sf{\odot \: \: \: Distance \: \: = \: \: \bf{Speed \; \times \; Time}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{\odot \: \: \: Speed \: in \: Downstream \: = \: \bf{Speed \: of \: Streamer \: + \: Speed \: of \: Stream}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{\odot \: \: \: Speed \: in \: Upstream \: = \: \bf{Speed \: of \: Streamer \: - \: Speed \: of \: Stream}}}}}$

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★ Question :-

A steamer goes downstream and covers the distance between two ports in 9 hours, while it covers the same distance upstream in 10 hours. If the speed of the stream is 1 km/hr, find the speed of the steamer in still water and the distance between the two ports.

_______________________________________________

★ Solution :-

Given,

» Speed of Stream =    1  Km/hr

» Time taken while Upstream = 10 hrs

» Time taken while Downstream = 9 hrs

• Let the speed of the stream be 'x' Km/hr.

• Let the distance between the two ports be 'y' Km.

Then using these variables :-

$\: \\ \sf{\rightarrow \: \: \: Speed \: in \: Downstream \: = \: Speed \: of \: Streamer \: + \: Speed \: of \: Stream \: \: = \: \bf{(x \: + \: 1) \: \; Km\:hr^{-1}}}$

$\: \\ \sf{\rightarrow \: \: \: Speed \: in \: Upstream \: = \: Speed \: of \: Streamer \: - \: Speed \: of \: Stream \: \: = \: \bf{(x \: - \: 1) \: \; Km\:hr^{-1}}}$

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~ For the condition of Downstream :-

$\: \\ \large{\sf{\Longrightarrow \: \: \: Distance, \;_{(y)} \: \: = \: \: \bf{Speed_{(while \: downstream )} \; \times \; Time_{(in \: downstream)}}}}$

$\: \\ \large{\sf{\Longrightarrow \: \: \: y \: \: = \: \: \tt{(x \; + \; 1) \; \: Km\:\:\cancel{hr^{-1}} \;\; \times \;\: 9 \;\: \: \cancel{hr^{1}} \: \: = \: \: \underline{\bf{9(x \; + \; 1) \; \: Km}}}}}$

Let this be equation (i).

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~ For the condition of Upstream :-

$\: \\ \large{\sf{\Longrightarrow \: \: \: Distance, \;_{(y)} \: \: = \: \: \bf{Speed_{(while \: upstream )} \; \times \; Time_{(in \: upstream)}}}}$

$\: \\ \large{\sf{\Longrightarrow \: \: \: y \: \: = \: \: \tt{(x \; - \; 1) \; \: Km\:\: \cancel{hr^{-1}} \;\; \times \;\: 10 \;\: \: \cancel{hr^{1}} \: \: = \: \: \underline{\bf{10(x \; - \; 1) \; \: Km}}}}}$

Let this be equation (ii).

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From above both equations, we get,

➣ 10(x - 1) Km = 9(x + 1) Km

Cancelling off Km from both sides, we get,

➣ 10x - 10 = 9x + 9

➣ 10x - 9x = 9 + 10

➣ x = 19 Km/hr

$\: \\ \: \large{\boxed{\boxed{\sf{Hence,\;speed\; of \; streamer\; in\; still \; water, \; x \; = \; \bf{19 \: \; Kmhr^{-1}}}}}}$

By using the value of x and equation (ii), we get

➣ y =  9(x + 1)

➣ y = 9(19) + 9

➣ y = 171 + 9

➣ y = 180 Km

$\: \\ \: \large{\boxed{\boxed{\sf{Hence,\;distance\; between \; two\; ports\; is, \;y \; = \; \bf{180 \: \; Km}}}}}$

$\:\: \large{\rm{Thus,\;speed\;of\;streamer\;is\;in\;still\;water\;is\;\bf{19\;Kmhr^{-1}}\rm{\;and\;distance\;between\;ports\;is\;}\;\bf{180\;Km}}}$

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$\: \\ \large{\underbrace{\underbrace{\sf{More \: \: to \: \: know \: \: :-}}}}$

• Polynomials are the mathematical expressions that are formed using constant and variable terms but variable terms can be of many degrees.

They are :-

• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial
• Bi - Quadratic Polynomial

• Linear Equations are the equations formed using constant and variable terms but the variable terms are of single degrees.

They are :-

• Linear Equation in One Variable
• Linear Equation in Two Variable
• Linear Equation in Three Variable