Question

The ratio of incomes of two persons A and B is $3:4$ and the ratio of their expenditures is $5:7$. If their savings are $\text{Rs}.15,000$ annually find their annual incomes. What value will be promoted if the expenditure is under control?

Answer 1

Step 1: Obtain the system of linear equations:

Given the ratio of incomes$=3:4$
The ratio of their expenditures$=5:7$
Savings of each person annually$\text{=Rs}.15,000$
Let the incomes of the two persons be $3x$and$4x$ , and the expenditures be $5y$ and$7y$ respectively.

$\begin{array}{rcl}3x-5y& =& 15000...\left(1\right)\\ 4x-7y& =& 15000...\left(2\right)\end{array}$

Hence, the above equations represent a system of linear equations.

Step 2: Solve the above system of linear equations using the cross-multiplication method

$\begin{array}{rcl}3x-5y& =& 15000...\left(1\right)\\ 4x-7y& =& 15000...\left(2\right)\end{array}$

Now we get:

$\begin{array}{rcl}& \Rightarrow & \frac{x}{\left|\begin{array}{cc}-5& -15000\\ -7& -15000\end{array}\right|}=\frac{y}{\left|\begin{array}{cc}-15000& 3\\ -15000& 4\end{array}\right|}=\frac{1}{\left|\begin{array}{cc}3& -5\\ 4& -7\end{array}\right|}\\ & \Rightarrow & \frac{x}{75000-105000}=\frac{y}{-60000+45000}=\frac{1}{-21+20}\\ & \Rightarrow & \frac{x}{-30000}=\frac{y}{-15000}=\frac{1}{-1}\end{array}$

So, we get $x=\frac{-30000}{-1}=30000$ and $y=\frac{-15000}{-1}=15000$.

$\therefore$The annual income of both persons are $3x=3\left(30000\right)=90000$ and $4x=4\left(30000\right)=120000$ respectively.

Hence, the annual income of both persons are $\text{Rs}.90,000\text{and Rs}.1,20,000$ respectively.

Also, the values to be promoted if the expenditure is under control are saving attitude and economic value.