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 ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Answer 1

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

Suppose their monthly expenditures are 5y and 7y, respectively.

Since each saves Rs 15,000 per month,

$$\begin{gathered} \mathrm{Monthly}\mathrm{saving}\mathrm{of}\mathrm{Aryan}:3x-5y=15,000 \\ \mathrm{Monthly}\mathrm{saving}\mathrm{of}\mathrm{Babban}:4x-7y=15,000 \end{gathered}$$

The above system of equations can be written in the matrix form as follows:

$\left[\begin{array}{cc}3& -5\\ 4& -7\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

or,
AX = B, where $\mathrm{A}=\left[\begin{array}{cc}3& -5\\ 4& -7\end{array}\right],\mathrm{X}=\left[\begin{array}{c}x\\ y\end{array}\right]\mathrm{and}\mathrm{B}=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

Now,

$\left|\mathrm{A}\right|=\left|\begin{array}{cc}3& -5\\ 4& -7\end{array}\right|=-21-\left(-20\right)=-1$

Adj A=${\left[\begin{array}{cc}-7& -4\\ 5& 3\end{array}\right]}^T=\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]$

So, $A^{-1}=\frac{1}{\left|A\right|}adjA=-1\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]$

$$\begin{gathered} \therefore \mathrm{X}={\mathrm{A}}^{-1}\mathrm{B} \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]\left[\begin{array}{c}15000\\ 15000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}105000-75000\\ 60000-45000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}30000\\ 15000\end{array}\right] \\ \Rightarrow x=30,000\mathrm{and}y=15,000 \end{gathered}$$

Therefore,

Monthly income of Aryan = $3\times \mathrm{Rs}30,000=\mathrm{Rs}90,000$

Monthly income of Babban = $4\times \mathrm{Rs}30,000=\mathrm{Rs}1,20,000$

From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future.