Question

The incomes of A and B are in the ratio $3:4$and their expenditures are in the ratio $5:7$. If each of them saves $Rs.1000$per month, Find their monthly income?

Answer 1

Hint:
With the help of $income = savings + expenditure$, form an equation with the help of ratios given, solve for the values of $x\,and\,y$, and find the values of saving and expenditure individually.

Complete step by step solution:
Given, Incomes of A and B are in the ratio $3:4$
Their expenditures are given in the ratio$5:7$
Savings of both are $Rs.1000$per month.
Incomes of A and B are in the ratio$3:4$, so the incomes of A and B are $3x\,and\,4x$
Expenditures of A and B are in the ratio$5:7$, so the expenditures of A and B are $5y\,and\,7y$
As we know that, $income = savings + expenditure$, form an equation separately for A and B.
So, we have the equations
$3x = 5y + 1000\,and\,4x = 7y + 1000$
Solve the equation for the value of $x$
$
   \Rightarrow 3x = 5y + 1000\, \\
  \,\,\,\,\,\,4x = 7y + 1000 \\
   \Rightarrow 3x - 5y = 1000........(1) \\
  \,\,\,\,\,\,4x - 7y = 1000........(2) \\
 $
Multiply 4 with equation (1) and multiply 3 with equation (2)
$
   \Rightarrow 12x - 20y = 4000 \\
  \,\,\,\,\,\,12x - 21y = 3000 \\
 $
On solving the equation value of $y = 1000$
Substitute the value of $y = 1000$ in equation (1),
$
   \Rightarrow 3x - 5y = 1000 \\
   \Rightarrow 3x - 5(1000) = 1000 \\
   \Rightarrow 3x = 1000 + 5000 \\
   \Rightarrow 3x = 6000 \\
   \Rightarrow x = 2000 \\
 $
So, we have the value of $x$, substitute the value of $x$in the incomes of A and B.
Income of A is $3x = 3(2000) = Rs.\,6000$
Income of B is $4x = 4(2000) = Rs.\,8000$

So, the incomes of A and B are $Rs.\,6000\,and\,Rs.\,8000$.

Note:
Whenever questions based on incomes and saving appear, remember the formula $income = savings + expenditure$, and assume the savings of particulars in form$\dfrac{{ax}}{{bx}}$if their savings are given in the ratio $a:b$.