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 ${\text{Rs}}.15000$per month, find their monthly incomes using matrix method. This problem reflects which value?

Answer 1

Hint: Convert the rations into a system of equations of linear equations and then solve using Gaussian Elimination method.

Let the incomes of Aryan and Babban be $3x$ and $4x$ respectively.
In the same way, their expenditures would be $5y$ and $7y$ respectively.
Since Aryan and Babban saves ${\text{Rs}}.15000$ , we get
$3x - 5y = 15000.........................................\left( 1 \right) \\\ 4x - 7y = 15000........................................\left( 2 \right) \\\$
These are the systems of linear equations in the form $AX = B$, which can be solved by using Gaussian Elimination method.
So, A = \left[ {\begin{array}{*{20}{c}} 3&{ - 5} \\\ 4&{ - 7} \end{array}} \right],B = \left[ {\begin{array}{*{20}{c}} {15000} \\\ {15000} \end{array}} \right]{\text{ and }}X = \left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right]
This can be written in matrix form as
\left[ {\begin{array}{*{20}{c}} 3&{ - 5} \\\ 4&{ - 7} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {15000} \\\ {15000} \end{array}} \right]
Applying the row operation ${R_2} \to {R_2} - \dfrac{4}{3}{R_1}$
We get \left[ {\begin{array}{*{20}{c}} 3&{ - 5} \\\ 0&{ - \dfrac{1}{3}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {15000} \\\ { - 5000} \end{array}} \right]
Applying the row operation ${R_2} \to {R_2} \times - 3$
We get \left[ {\begin{array}{*{20}{c}} 3&{ - 5} \\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {15000} \\\ {15000} \end{array}} \right]
Applying the row operation ${R_1} \to {R_1} + 5{R_2}$
We get \left[ {\begin{array}{*{20}{c}} 3&0 \\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {90000} \\\ {15000} \end{array}} \right]
Applying the row operation ${R_1} \to {R_1} \times \dfrac{1}{3}$
We get \left[ {\begin{array}{*{20}{c}} 1&0 \\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {30000} \\\ {15000} \end{array}} \right]
Converting back into system of equations we have
$x = 30000 \\\ y = 15000 \\\$
Thus, the income of Aryan is $3x = 3 \times 30000 = 90000$ and the income of Babban is $4x = 4 \times 30000 = 120000$.
Hence incomes of Aryan and Babban are ${\text{Rs}}.90,000{\text{ and Rs}}.1,20,000$ respectively.
The value of the problem is Aryan is more interested in saving money than Babban. So, one must save money, no matter how much one earns.

Note: In this problem the matrix method is used to solve the system of $n$ linear equations in $n$ unknowns. Some types of matrix methods to solve the system of linear equations are Gaussian Elimination method, Inverse matrix method and Cramer's rule. Here we have used the Gaussian Elimination method.