Question

The monthly incomes of Aryan and Babban are in the ratio $3:4$ and their monthly expenditures are in the ratio $5:7$. If each saves Rs. $15000$ per month, find their monthly incomes using matrix method. This problem reflects which value?

Answer 1

Let the incomes of Aryan and Babban be $3x$ and $4x$ respectively.

Similarly, their expenditures would be $5y$ and $7y$ respectively.

Since each saves Rs. $15000$, we get

$3x-5y=15000...\left(1\right)$

$4x-7y=15000...\left(2\right)$

This can be written in matrix form as $\left[\begin{array}{cc}3& -5\\ 4& -7\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

$R_2\to R_2-\frac{4}{3}{R}_1$

We get $\left[\begin{array}{cc}3& -5\\ 0& \frac{-1}{3}\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ -5000\end{array}\right]$

$R_2\to R_2\times -3$

We get $\left[\begin{array}{cc}3& -5\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

$R_1\to R_1+5R_2$

We get $\left[\begin{array}{cc}3& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}90000\\ 15000\end{array}\right]$

$R_1\to R_1\times \frac{1}{3}$

We get $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}30000\\ 15000\end{array}\right]$

$\therefore x=30000$

Their incomes thus become Rs. $90,000$ and Rs. $120,000$ respectively.