Question

Roohi travels 200 km to her home partly by train and partly by bus. She takes 4 hours if she travels 80 km by train and the remaining by bus. If she travels 110 km by train and the remaining by bus, she takes 15 minutes longer. Find the speed of the train and the bus separately.

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

• Speed of the train be x km/hr

• Speed of the bus be y km/hr

According to first condition

Roohi travels 200 km to her home partly by train and partly by bus. She takes 4 hours if she travels 80 km by train and the remaining by bus.

So,

↝ Distance travelled by train = 80 km

↝ Distance travelled by bus = 120 km

So,

↝ Time taken to travel 200 km is 4 hours.

$\rm :\longmapsto\:\boxed{ \tt{ \: \frac{80}{x} + \frac{120}{y} = 4 \: }} - - - - (1)$

According to second condition

If she travels 110 km by train and the remaining by bus, she takes 15 minutes longer.

So,

↝ Distance travelled by train = 110 km

↝ Distance travelled by bus = 90 km

So,

↝ Time taken to travel 200 km is 4 hour 15 minutes

$\rm :\longmapsto\:\dfrac{110}{x} + \dfrac{90}{y} = 4 \dfrac{15}{60}$

$\rm :\longmapsto\:\dfrac{110}{x} + \dfrac{90}{y} = 4 \dfrac{1}{4}$

$\rm :\longmapsto\:\dfrac{110}{x} + \dfrac{90}{y} = \dfrac{17}{4}$

$\rm :\longmapsto\:\boxed{ \tt{ \: \dfrac{440}{x} + \dfrac{360}{y} = 17}} - - - - (2)$

Now, we have two equations

$\rm :\longmapsto\:\dfrac{80}{x} + \dfrac{120}{y} = 4 - - - - (1)$

and

$\rm :\longmapsto\:\dfrac{440}{x} + \dfrac{360}{y} = 17 - - - - (2)$

Now, we use method of Eliminations to get the values of x and y.

So, multiply equation (1) by 3, we get

$\rm :\longmapsto\:\dfrac{240}{x} + \dfrac{360}{y} = 12 - - - - (3)$

On Subtracting equation (3) from equation (2), we get

$\rm :\longmapsto\:\dfrac{200}{x} = 5$

$\rm :\longmapsto\:\dfrac{200}{5} = x$

$\rm \implies\:\boxed{ \tt{ \: x = 40 \: }} - - - - - (4)$

On substituting x = 40, in equation (1), we get

$\rm :\longmapsto\:\dfrac{80}{40} + \dfrac{120}{y} = 4$

$\rm :\longmapsto\:2 + \dfrac{120}{y} = 4$

$\rm :\longmapsto\: \dfrac{120}{y} = 4 - 2$

$\rm :\longmapsto\: \dfrac{120}{y} = 2$

$\rm :\longmapsto\: \dfrac{120}{2} = y$

$\rm \implies\:\boxed{ \tt{ \: y = 60 \: }} - - - - - (5)$

Hence,

• Speed of train = 40 km/ hr

• Speed of bus = 60 km/ hr

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Concept Used :-

There are 4 methods to solve this type of pair of

equations.

1. Method of Substitution

2. Method of Eliminations

3. Method of Cross Multiplication

4. Graphical Method

We prefer here Method of Eliminations :-

To solve systems using elimination, follow this procedure:

The Elimination Method

Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient.

Step 2: Subtract the second equation from the first to eliminate one variable

Step 3: Solve this new equation for other variable.

Step 4: Substitute the value of variable thus evaluated into either Equation 1 or Equation 2 and get the value other variable.