Question

Formulate the following problems as a pair of equations, and hence find their solutions:

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

Answer 1

Step 1: Formulate the equations

Let us assume that

Speed of the train $=xkm/h$

Speed of the bus $=ykm/h$

According to the given conditions

$\frac{60}{x}+\frac{240}{y}=4\dots \dots \dots \dots \dots \dots \dots \left(1\right)$

$\frac{100}{x}+\frac{200}{y}=\frac{25}{6}\dots \dots \dots \dots \dots .\left(2\right)$

Step 2: Find their solution

Let us substitute $m=\frac{1}{x}$ and $n=\frac{1}{y}$, in the above equations;

$60m+240n=4\dots \dots \dots \dots \dots \dots \dots \dots ..\left(3\right)$

$100m+200n=\frac{25}{6}$

$600m+1200n=25\dots \dots \dots \dots \dots \dots \dots .\left(4\right)$

Multiply equation $\left(3\right)$ by $10$, to get,

$600m+2400n=40\dots \dots \dots \dots \dots \dots \dots \dots \left(5\right)$

On subtracting equation $\left(4\right)$ from $\left(5\right)$, we get,

$1200n=15$

$\Rightarrow$ $n=\frac{1}{80}$

On substituting this in equation $\left(3\right)$ we get

$60m+3=4$

$\Rightarrow$ $m=\frac{1}{60}$

$\therefore x=\frac{1}{m}=60$

$\therefore y=\frac{1}{n}=80$

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