Question

Formulate the following problem as a pair of equations and then find the solution.

Rahim travels $600 km$ to his home partly by train and partly by car. He takes $8 hours$ if he travels $120 km$ by train and rest by car. He takes $20 minutes$ more if he travels $200 km$ by train and rest by car. Find the speed of the train and the car.

Accepted Answer

The total distance is $600 km$ .

Distance covered by train is $120 km$ .

Distance covered by car is $600 km - 120 km = 480 km$

Total time is taken by train and car $= 8 h$

Let the speed of the train be $x km / h$ and the speed of the car be $y km / h$ .

We know that $T i m e = \frac{D i s \tan c e}{S p e e d}$

So, $\frac{120}{x} + \frac{480}{y} = 8$

$\Rightarrow \frac{15}{x} + \frac{60}{y} = 1$ ............. $(i)$

Again, the distance covered by train is $200 km$

Distance covered by car is $600 km - 200 km = 400 km$

Total time is taken by train and car $= 8 h 20 \min = 8 \frac{20}{60} h = \frac{25}{3} h$

$\frac{200}{x} + \frac{400}{y} = \frac{25}{3}$

$\Rightarrow \frac{8}{x} + \frac{16}{y} = \frac{1}{3}$ ............. $(ii)$

Let $\frac{1}{x} = u$ and $\frac{1}{y} = v$

On substituting these values in the equation $(i) & (ii)$ , we get

$15 u + 60 v = 1$ ............. $(iii)$

and $8 u + 16 v = \frac{1}{3}$

$\Rightarrow 24 u + 48 v = 1$ ............. $(iv)$

Coefficients of $u$ are $15 & 24$ and their LCM is $120$ . Using the elimination method:

Equation $(iii) \times 8$ and Equation $(iv) \times 5$ , we get

$\begin{matrix} 120 u & + 480 v & = & 8 \\ 120 u & + 240 v & = & 5 \end{matrix}$

We see that the variable $u$ have the same sign, we subtracting the above equation

$\begin{matrix} 120 u & + 480 v & = & 8 \\ 120 u & + 240 v & = & 5 \\ (-) & (-) & (-) \\ 240 v & = & 3 \end{matrix}$

$v = \frac{3}{240} = \frac{1}{80}$

On putting the value of $v$ in eq. $(iii)$

$15 u + 60 \times \frac{1}{80} = 1$

$\Rightarrow 15 u + \frac{3}{4} = 1$

$\Rightarrow 15 u = 1 - \frac{3}{4}$

$\Rightarrow 15 u = \frac{4 - 3}{4} = \frac{1}{4}$

$\Rightarrow u = \frac{1}{4 \times 15} = \frac{1}{60}$

Now, $\frac{1}{x} = u = \frac{1}{60}$

$\Rightarrow x = 60$

and $\frac{1}{y} = v = \frac{1}{80}$

$\Rightarrow y = 80$

Hence, the speed of train is $60 km / h$ and speed of car is $80 km / h$ .

Important Questions on Pair of Linear Equations in Two Variables

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