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

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Solution

## 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, $\mathrm{Monthly}\mathrm{saving}\mathrm{of}\mathrm{Aryan}:3x-5y=15,000\phantom{\rule{0ex}{0ex}}\mathrm{Monthly}\mathrm{saving}\mathrm{of}\mathrm{Babban}:4x-7y=15,000$ 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, ${A}^{-1}=\frac{1}{\left|A\right|}adjA=-1\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]$ $\therefore \mathrm{X}={\mathrm{A}}^{-1}\mathrm{B}\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]\left[\begin{array}{c}15000\\ 15000\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}105000-75000\\ 60000-45000\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}30000\\ 15000\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒x=30,000\mathrm{and}y=15,000\phantom{\rule{0ex}{0ex}}$ Therefore, Monthly income of Aryan = $3×\mathrm{Rs}30,000=\mathrm{Rs}90,000$ Monthly income of Babban = $4×\mathrm{Rs}30,000=\mathrm{Rs}1,20,000$ From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future.

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