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 $15000$ per month, then the monthly income of Babban is $k\left(10000\right)$ where $k$ is

Answer 1

Let us represent the situation through a matrix.
We will make two matrices: Income and Expenditure Matrices.

We know that Saving = Income - Expenditure.
Let the income of Aryan and Babban be $3x$ and $4x$ respectively and the expenditures be $5y$ and $7y$ respectively.

Income Matrix$=\left[\begin{array}{c}3x\\ 4x\end{array}\right]$
Expenditure Matrix $=\left[\begin{array}{c}5y\\ 7y\end{array}\right]$
Now, Saving $=\left[\begin{array}{c}3x\\ 4x\end{array}\right]-\left[\begin{array}{c}5y\\ 7y\end{array}\right]$
Given : Saving $=15000$ each
Therefore, we have,
$\left[\begin{array}{c}15000\\ 15000\end{array}\right]=\left[\begin{array}{c}3x\\ 4x\end{array}\right]-\left[\begin{array}{c}5y\\ 7y\end{array}\right]$
So,
$3x-5y=15000\cdots \left(1\right)$
$4x-7y=15000\cdots \left(2\right)$
Solving equations $\left(1\right)$ and $\left(2\right),$ we get,
$\Rightarrow y=15000\text{and}x=30000$
There monthly incomes are, $3x=3\times 30000=90000$ and $4x=4\times 30000=120000$
$\therefore$ Monthly income of Babban $=120000=12\left(10000\right)$
Hence $k=12$