Question

The ratio of incomes of two persons is $9:7$. The ratio of their expenses is $4:3$. Every person saves ₹ 200 , find the income of each.

Answer 1

Let the income of first person be $₹x$ and that of second person be $₹y$.
According to the first condition,
the ratio of their incomes is $9:7$.
$$\begin{array}{cc}& \therefore \frac{x}{y}=\frac{9}{7}\\ & \therefore 7x=9y\\ & \therefore 7x-9y=0\dots \dots \text{(i)}\end{array}$$
Each person saves ₹ 200 .
Expenses of first person $=$ Income $-$ Saving $=x-200$
Expenses of second person $=y-200$
According to the second condition,
the ratio of their expenses is $4:3$
$$\begin{array}{cc}& \therefore 3(x-200)=4(y-200)\\ & \therefore 3x-600=4y-800\\ & \therefore 3x-4y=-800+600\\ & \therefore 3x-4y=-200\dots \text{(ii)}\end{array}$$
Multiplying equation (i) by 4 ,
$$28x-36y=0\text{...(iii)}$$
Multiplying equation (ii) by 9 ,
$$27x-36y=-1800\text{...(iv)}$$
Subtracting equation (iv) from (iii),
Subtracting equation (iv) from (iii),
$$\begin{array}{cc}28x-36y& =0\\ 27x-36y=& -1800\\ -+& +\\ x& =1800\end{array}$$
Substituting $x=1800$ in equation (i),
$$\begin{array}{cc}& 7x-9y=0\\ & \therefore 7\left(1800\right)-9y=0\\ & \therefore 9y=7\times 1800\\ & \therefore y=\frac{7\times 1800}{9}\\ & y=7\times 200\\ & \therefore y=1400\end{array}$$
$\therefore$ The income of first person is $₹1800$ and that of second person is $₹1400$.