The incomes of A and B are in the ratio 3 : 2 and their ex penditures are in the ratio 5: 3. If each saves Rs. 1000, then A's income is

Options

A. Rs. 6000
B. Rs 4000
C. Rs. 2000
D. Rs. 5000

Answer

To solve the problem, we will follow these steps:

1. **Define Variables**:
- Let the income of A be \(3x\) and the income of B be \(2x\).
- Let the expenditure of A be \(5y\) and the expenditure of B be \(3y\).

2. **Set Up Savings Equation**:
- According to the problem, both A and B save Rs. 1000.
- For A:
\[ \text{Income of A} - \text{Expenditure of A} = \text{Savings of A} \]
\[ 3x - 5y = 1000 \quad \text{(1)} \]
- For B:
\[ \text{Income of B} - \text{Expenditure of B} = \text{Savings of B} \]
\[ 2x - 3y = 1000 \quad \text{(2)} \]

3. **Solve the System of Equations**:
- We have the two equations:
\[ 3x - 5y = 1000 \quad \text{(1)} \]
\[ 2x - 3y = 1000 \quad \text{(2)} \]
- We can multiply equation (2) by 5 and equation (1) by 3 to eliminate \(y\):
\[ 5(2x - 3y) = 5(1000) \implies 10x - 15y = 5000 \quad \text{(3)} \]
\[ 3(3x - 5y) = 3(1000) \implies 9x - 15y = 3000 \quad \text{(4)} \]

4. **Subtract the Equations**:
- Now we subtract equation (4) from equation (3):
\[ (10x - 15y) - (9x - 15y) = 5000 - 3000 \]
\[ 10x - 9x = 2000 \implies x = 2000 \]

5. **Find the Value of y**:
- Substitute \(x = 2000\) back into equation (2):
\[ 2(2000) - 3y = 1000 \]
\[ 4000 - 3y = 1000 \]
\[ -3y = 1000 - 4000 \implies -3y = -3000 \implies y = 1000 \]

6. **Calculate A's Income**:
- Now that we have \(x\), we can find A's income:
\[ \text{Income of A} = 3x = 3(2000) = 6000 \]

Thus, A's income is Rs. 6000.