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?

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,

`\text(Monthly saving of Aryan): 3x - 5y = 15, 000\]`

`\text(Monthly saving of Babban): 4x - 7y = 15, 000\]`The above system of equations can be written in the matrix form as follows:

LaTeX

`[[3 - 5],[ 4 -7]]``[[x],[y]]`=` [[15000 ],[15000]]`

or,
AX = B, where

A = \begin{bmatrix}3 & - 5 \\ 4 & - 7\end{bmatrix}, X =`[[x],[y]]`and B = ` [[15000 ],[15000]]`

Now,

 

|A| = \begin{vmatrix}3 & - 5 \\ 4 & - 7\end{vmatrix} = - 21 -  (- 20 )= - 1

Adj A=\[\begin{bmatrix}- 7 & - 4 \\ 5 & 3\end{bmatrix}^T = \begin{bmatrix}- 7 & 5 \\ - 4 & 3\end{bmatrix}\]

So,

\[A^{- 1} = \frac{1}{\left| A \right|}adjA = - 1\begin{bmatrix}- 7 & 5 \\ - 4 & 3\end{bmatrix} = \begin{bmatrix}7 & - 5 \\ 4 & - 3\end{bmatrix}\]

\[\therefore X = A^{- 1} B\]
⇒`[[x],[y]]`=`[[        7    -5],[4          3]]``[[15000],[15000]]`

⇒`[[x],[y]]` =`[[105000 - 75000],[60000 - 45000]]`
⇒`[[x],[y]]`= `[[30000],[15000]]`
\[ \Rightarrow x = 30, 000 \text{and  y}= 15, 000\]

Therefore,

Monthly income of Aryan =

3 × Rs 30, 000 = Rs 90, 000

Monthly income of Babban =

4 ×   Rs 30, 000 = 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.