Question

The ratio of incomes of two persons is \(9:7\) and the ratio of their expenditures is \(4:3\). If each of them saves Rs. \(5000\) per month, find their monthly incomes.

Hint:

In ratio problems involving income, expenditure, and savings, first assume the quantities in terms of variables according to the given ratios, then form equations using Income - Expenditure = Savings.

Answer 1

Step 1: Assume the incomes and expenditures.}
Let the monthly incomes of the two persons be \(9x\) and \(7x\), since their income ratio is \(9:7\).
Let their monthly expenditures be \(4y\) and \(3y\), since their expenditure ratio is \(4:3\).

Step 2: Use the condition of savings.}
It is given that each person saves Rs. \(5000\) per month.
So, for the first person:
\[ 9x - 4y = 5000 \] For the second person:
\[ 7x - 3y = 5000 \]
Step 3: Solve the pair of equations.}
We have:
\[ 9x - 4y = 5000 \qquad \text{...(1)} \] \[ 7x - 3y = 5000 \qquad \text{...(2)} \] Multiply equation (1) by \(3\):
\[ 27x - 12y = 15000 \] Multiply equation (2) by \(4\):
\[ 28x - 12y = 20000 \] Now subtract the first from the second:
\[ (28x - 12y) - (27x - 12y) = 20000 - 15000 \] \[ x = 5000 \]
Step 4: Find the monthly incomes.}
Now substitute \(x = 5000\):
First person's income:
\[ 9x = 9 \times 5000 = 45000 \] Second person's income:
\[ 7x = 7 \times 5000 = 35000 \]
Step 5: Verify the result.}
From equation (2):
\[ 7(5000) - 3y = 5000 \] \[ 35000 - 3y = 5000 \] \[ 3y = 30000 \] \[ y = 10000 \] So expenditures are \(4y = 40000\) and \(3y = 30000\).
Savings are:
\[ 45000 - 40000 = 5000 \] \[ 35000 - 30000 = 5000 \] Hence, the answer is correct.

Step 6: State the final answer.}
Therefore, the monthly incomes of the two persons are:
\[ \boxed{\text{Rs. }45000 \text{ and Rs. }35000} \]