Incomes of x and y are in the ratio 4 : 3. Their expenditures are in the ratio 12 : 7. Both save Rs 3200 at the end of the month, then the income of x is :

Options

A. Rs 6000
B. Rs8000
C. Rs2000
D. Rs4000

Answer

To solve the problem step by step, let's break it down:

1. **Define the Variables**:
- Let the incomes of X and Y be represented as \(4A\) and \(3A\) respectively, based on the given ratio of their incomes (4:3).
- Let the expenditures of X and Y be represented as \(12B\) and \(7B\) respectively, based on the given ratio of their expenditures (12:7).

2. **Set Up the Savings Equation**:
- According to the problem, both X and Y save Rs 3200 at the end of the month. Therefore, we can write the following equations based on the savings:
- For X:
\[ \text{Income of X} - \text{Expenditure of X} = \text{Savings of X} \]
\[ 4A - 12B = 3200 \quad \text{(1)} \]
- For Y:
\[ \text{Income of Y} - \text{Expenditure of Y} = \text{Savings of Y} \]
\[ 3A - 7B = 3200 \quad \text{(2)} \]

3. **Multiply the Equations**:
- To eliminate \(B\), we can multiply equation (1) by 7 and equation (2) by 12:
- From (1):
\[ 7(4A - 12B) = 7(3200) \]
\[ 28A - 84B = 22400 \quad \text{(3)} \]
- From (2):
\[ 12(3A - 7B) = 12(3200) \]
\[ 36A - 84B = 38400 \quad \text{(4)} \]

4. **Subtract the Equations**:
- Now, subtract equation (3) from equation (4):
\[ (36A - 84B) - (28A - 84B) = 38400 - 22400 \]
\[ 36A - 28A = 16000 \]
\[ 8A = 16000 \]

5. **Solve for A**:
- Divide both sides by 8:
\[ A = \frac{16000}{8} = 2000 \]

6. **Calculate the Income of X**:
- Now that we have the value of \(A\), we can find the income of X:
\[ \text{Income of X} = 4A = 4 \times 2000 = 8000 \]

Thus, the income of X is **Rs 8000**.