Question

$need \: solution \: wit h \: the \: method \: of \: subsitution \: solve \: it \: by \: full \: method \: fast$
Note ... solve it by full method with right ans​

Answer 1

Question

Roohi travels 300 km to home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and thr remaining by bus .If she travels 100km by train and the remaining by bus , she takes 10 minutes longer. Find the speed of the train and the bus separately .

Given that

• Roohi travels 300 km to home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and thr remaining by bus .If she travels 100km by train and the remaining by bus , she takes 10 minutes longer.

We need to Find

• Find the speed of the train and the bus separately .

Formula

• $\longrightarrow \boxed{ \ \frak{ \: \underline{Speed = \frac{ Distance }{Time}}}}$

• $\longrightarrow \boxed{ \underline{Time = \frac{ Distance}{ Speed} }}$

Solution

• In order to find the speed of bus and train, first we will assume the speed of train and bus as variables, then we will make two equations with two variables with the help of a given statement and by simplifying it we will proceed further to get the perfect answer.

Step-by-step explanation

Let the speed of the train be x km/hr and the speed of the bus is y km/hr

Given that

Roohi travels 300km to her home partly by train and partly by bus .She takes 4 hours if She travels 60km by train & the remaining by bus

As we know formula of time

$\blue{ \large \qquad \qquad \qquad \sf \:Train}$

❒ Distance = 60km

❒Speed = x

❒ Time = ?

$\sf \: we \: know \: that \: \boxed{ \red{time = \dfrac{distance}{speed} }}$

substituting values , we get

$\tt: \longrightarrow \: time \: = \dfrac{60}{x}$

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$\blue{ \large \qquad \qquad \qquad \sf \:bus}$

❒ Distance = 240km

❒Speed = y

❒ Time = ?

$\sf \: we \: know \: that \: \boxed{ \red{time = \dfrac{distance}{speed} }}$

substituting values

$\tt: \longrightarrow \: time \: = \dfrac{240}{y}$

Now

Total Time = 4 hours

$\tt: \longrightarrow \: time \: = \dfrac{60}{x} + \dfrac{240}{y}=4--(1)$

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Also,

She travels 100km by train and remaining by bus, she takes 4 hours 10 minutes

missing solution in attachment

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Now ,

Total Time taken = 4 hours 10 minutes

$\sf:\longmapsto \: \dfrac{100}{x} + \dfrac{200}{y} = 4 + \dfrac{10}{60} \: hours \: \\ \\ \\ \sf:\longmapsto \: \frac{100}{x} + \frac{200}{y} = 4 + \frac{1 \cancel0}{6 \cancel0} \: hours \\ \\ \\ \sf:\longmapsto \: \frac{100}{x} + \frac{200}{y} = 4 + \frac{1}{6} \\ \\ \\ \sf:\longmapsto \frac{100}{x} + \frac{200}{y} = \frac{4(6) + 1}{6} \\ \\ \\ \sf:\longmapsto \frac{100}{x} + \frac{200}{y} = \frac{24 + 1}{6} \\ \\ \\ \pink{ \sf:\longmapsto \frac{100}{x} + \frac{200}{y} = \frac{25}{6} - - - (2)}$

From (3)

$\sf:\longmapsto \: 60u \: + 240v = 4 \\ \\ \\ \sf:\longmapsto60u = 4 - 240v \\ \\ \\ \pink{\sf:\longmapsto \: u = \frac{4 - 240v}{60} }$

putting value of u in (4)

$\sf:\longmapsto24 + 48v = 1 \\ \\ \\ \sf:\longmapsto \cancel{24} \bigg( \frac{4 - 240v}{ \cancel{60}} \bigg) + 48 = 1 \\ \\ \\ \sf:\longmapsto \: 2 \bigg( \frac{4 - 240v}{5} \bigg) + 48v = 1$

Multiplying both sides by 5

$\sf:\longmapsto \: 5 \times 2 \bigg( \dfrac{4 - 240v}{5} \bigg) + 5 \times 48v = 5 \times 1$

$\sf:\longmapsto \: \cancel 5 \times 2 \bigg( \dfrac{4 - 240v}{ \cancel5} \bigg) + 5 \times 48v = 5 \times 1$

$\sf:\longmapsto2(4 - 240v) + 240 = 5 \\ \\ \\ \sf:\longmapsto8 - 480v + 240v = 5 \\ \\ \\ \sf:\longmapsto - 480v + 240v = 5 - 8 \\ \\ \\ \sf:\longmapsto - 240v = - 3 \\ \\ \\ \sf:\longmapsto \: v = \cancel\frac{ \cancel - 3}{ \cancel - 240} \\ \\ \\ \pink{ \large\sf:\longmapsto \: \: v = \frac{1}{180} }$

Putting v = 1/180 in (3)

$\sf:\longmapsto \: 60u + 240v = 4 \\ \\ \\ \sf:\longmapsto60u + 240 \bigg( \frac{1}{80} \bigg) = 4 \\ \\ \\ \sf:\longmapsto60u + \cancel{240} \times \frac{1}{ \cancel{80}} \\ \\ \\ \sf:\longmapsto60u + 3 = 4 \\ \\ \\ \sf:\longmapsto60u = 4 - 3 \\ \\ \\ \sf:\longmapsto60u = 1 \\ \\ \\ \large \pink{\sf:\longmapsto \: u = \frac{1}{60} }$

$\large \bf \: Hence, \: u \: = \dfrac{1}{60} \: \: \: , \: \: \: v = \dfrac{1}{80}$

But we need to Find values of x and y

we know that

$\qquad \: \bf \: \qquad \: u \: = \frac{1}{x} \\ \\ \frac{1}{60} = \frac{1}{x} \\ \\ \large \pink{x = 60}$

and

$\qquad \qquad \bf \: v = \frac{1}{y} \\ \\ \frac{1}{80} = \frac{1}{y} \\ \\ \large \pink{y = 80}$

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So , x = 60 , y = 80 is the solution of the given equation

Hence

$\sf \qquad \qquad \: the \: speed \: of \: the \: train \: is \: \: \red{60km/hr}$

And

$\sf \qquad \qquad \: the \: speed \: of \: the \: bus \: \: is \: \: \red{80km/hr}$