Question

A race-boat covers a distance of 60 km downstream in one and a half hour. It covers this distance upstream in 2 hours. The speed of the race-boat in stillwater is 35 km/hr. Find the speed of the stream.​

Answer 1

Given,

A race-boat covers a distance of 60 km downstream in one and a half hour. It covers this distance upstream in 2 hours. The speed of the race-boat in still water is 35 km/hr.

To find :

Find the speed of the stream.​

Solution :

At first, speed of boat in still water = 35 km/h

Let the speed of stream be n km/h

∴ Speed of boat in downstream = (35 + n) km/h

∴ Speed of boat in upstream = (35 - n) km/h

Now distance covered in both streams = 60 km/h

Time in downstream = 1 and a half hour = 1 + 1/2 = 3/2 h

Time in upstream = 2 h

∴ Distance covered in downstream = (35 + n) * (3/2)

∴ Distance covered in upstream = (35 - n) * 2

Now atq,

⇒ (35 + n) * (3/2) = (35 - n) * 2

⇒ (105 + 3n)/2 = 70 - 2n

⇒ 105 + 3n = 140 - 4n

⇒ 140 - 105 = 4n + 3n

⇒ 35 = 7n

⇒ n = 35/7

⇒ n = 5 km/h

Therefore,

Speed of the stream = 5 km/h

Answer 2

Answer :-

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

Here the concept of Linear Equations and Distance - Speed Relationship has been used. Here we are give to find an unknown quantity, which we can make as variable and find its value. Also, we know that half (the speed in downstream - speed in upstream) is the speed of the stream. Let's do it :-

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★ Formula Used :-

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

$\: \\ \large{\sf{Speed\: of \: Stream \: \: = \: \: \bf{\dfrac{1}{2} \: \times \: (Speed \: in \: Downstream, \:S_{1}\: - \: Speed \: in \: Upstream, \: S_{2})}}}$

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★ Question :-

A race-boat covers a distance of 60 km downstream in one and a half hour. It covers this distance upstream in 2 hours. The speed of the race-boat in stillwater is 35 km/hr. Find the speed of the stream.

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★ Solution :-

Given,

» Distance covered in Downstream = 60 Km

» Time taken for downstream distance = 1.5 hr

» Distance covered in upstream = 60 Km» Time taken for upstream distance = 2 hr

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~ For the Speed in Downstream :-

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed \: \; = \: \; \bf{\dfrac{Distance}{Time}}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed_{(in \: downstream)}, \: S_{1} \: \; = \: \; \bf{\dfrac{Distance_{(in \: downstream)}}{Time_{(while\:downstream)}}}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed_{(in \: downstream)}, \: S_{1} \: \; = \: \; \bf{\dfrac{\cancel{60} \: km}{\cancel{1.5} \: hr} \: \: = \: \: \underline{\underline{40 \: \: Km\:hr^{-1}}}}}}$

$\: \\ \: \large{\boxed{\boxed{\sf{Speed \; in \; Downstream \; = \; \bf{40 \: \; Km\:hr^{-1}}}}}}$

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~ For the Speed in Upstream :

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed \: \; = \: \; \bf{\dfrac{Distance}{Time}}}}$

$\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed_{(in \: upstream)}, \: S_{2} \: \; = \: \bf{\dfrac{Distance_{(in \: upstream)}}{Time_{(while\:upstream)}}}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \: \: Speed_{(in \: upstream)}, \: S_{2} \: \; = \: \; \bf{\dfrac{\cancel{60} \: km}{\cancel{2} \: hr} \: \: = \: \: \underline{\underline{30 \: \: Km\:hr^{-1}}}}}}$

$\: \\ \: \large{\boxed{\boxed{\sf{Speed \; in \; Upstream \; = \; \bf{30 \: \; Km\:hr^{-1}}}}}}$

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~ For the Speed of the Stream :-

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: Speed\: of \: Stream \: \: = \: \: \bf{\dfrac{1}{2} \: \times \: (Speed \: in \: Downstream, \:S_{1}\: - \: Speed \: in \: Upstream, \: S_{2})}}}$

$\: \\ \qquad \large{\sf{:\Longrightarrow \: \: \: Speed\: of \: Stream \: \: = \: \: \bf{\dfrac{1}{2} \: \times \: (40 \:\: Km\:hr^{-1} \: - \: 30 \:\: Km\:hr^{-1})}}}$

$\large{\sf{:\Longrightarrow \: \: \: Speed\: of \: Stream \: \: = \: \: \bf{\: Speed\: of \: Stream \: \: = \: \: \bf{\dfrac{1}{\cancel{2}} \: \times \: (\cancel{10} \:\: Km\:hr^{-1})} \: \: = \: \: \underline{\underline{5 \: \: Km\:hr^{-1}}}}}}$

$\: \\ \large{\underline{\underline{\rm{\mapsto \: \: \: Thus, \; the \; speed \; of \; the \; stream \; is \;\; \boxed{\bf{5 \:\; Km\:hr^{-1}}}}}}}$

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$\: \large{\underline{\sf{\leadsto \: \: \: Confused? \:\; Don't \: worry \: let's \: verify \: it \: :-}}}$

For verification , we need to simply apply the values we got into the equations.

~ Case I :-

=> Speed in downstream = 40 Km/hr

=> Speed of stream + Speed of boat = 40Km/hr

=> 5 Km/hr + 35 Km/hr = 40 Km/hr

=> 40 Km/hr = 40 Km/hr

Clearly, LHS = RHS

~ Case II :-

=> Speed in upstream = 30 Km/hr

=> Speed of boat - speed of stream = 30 Km/hr

=> 35 Km/hr - 5 Km/hr = 30 Km/hr

=> 30 Km/hr = 30

Clearly, LHS = RHS. Here both the conditions are satisfied. So our answer is correct. Hence, Verified.