The ratio of monthly incomes of A, B is `6 : 5` and their monthly expenditures are In the ratio `4 : 3`. If each of them saves Rs 400 per month, find the sum of their monthly incomes.

Options

A. 2300
B. 2400
C. 2200
D. 2500

Answer

To solve the problem, we will follow these steps:

1. **Define the Variables for Incomes and Expenditures**:
- Let the monthly income of A be \( 6x \) and the monthly income of B be \( 5x \).
- Let the monthly expenditure of A be \( 4y \) and the monthly expenditure of B be \( 3y \).

2. **Set Up the Savings Equations**:
- According to the problem, both A and B save Rs 400 per month.
- For A:
\[ \text{Income of A} - \text{Expenditure of A} = \text{Savings of A} \]
\[ 6x - 4y = 400 \quad \text{(1)} \]
- For B:
\[ \text{Income of B} - \text{Expenditure of B} = \text{Savings of B} \]
\[ 5x - 3y = 400 \quad \text{(2)} \]

3. **Solve the Equations**:
- We have two equations:
\[ 6x - 4y = 400 \quad \text{(1)} \]
\[ 5x - 3y = 400 \quad \text{(2)} \]
- We can solve these equations simultaneously. First, let's manipulate equation (1) to express \( y \) in terms of \( x \):
\[ 4y = 6x - 400 \implies y = \frac{6x - 400}{4} \quad \text{(3)} \]
- Now substitute equation (3) into equation (2):
\[ 5x - 3\left(\frac{6x - 400}{4}\right) = 400 \]
- Simplifying this:
\[ 5x - \frac{18x - 1200}{4} = 400 \]
\[ 5x - \frac{18x}{4} + 300 = 400 \]
\[ 5x - 4.5x + 300 = 400 \]
\[ 0.5x + 300 = 400 \]
\[ 0.5x = 100 \implies x = 200 \]

4. **Find the Value of \( y \)**:
- Substitute \( x = 200 \) back into equation (3):
\[ y = \frac{6(200) - 400}{4} = \frac{1200 - 400}{4} = \frac{800}{4} = 200 \]

5. **Calculate Monthly Incomes**:
- Now we can find the monthly incomes of A and B:
\[ \text{Income of A} = 6x = 6(200) = 1200 \]
\[ \text{Income of B} = 5x = 5(200) = 1000 \]

6. **Find the Sum of Monthly Incomes**:
- The total monthly income is:
\[ \text{Total Income} = \text{Income of A} + \text{Income of B} = 1200 + 1000 = 2200 \]

Thus, the sum of their monthly incomes is **Rs 2200**.