The ratio of the incomes of A and B is 5:4 and the ratio of their expenditures is 3:2. If at the end of the year, each saves Rs. 1600, then the income of A is :

Options

A. Rs3400
B. Rs. 3600
C. Rs. 4000
D. Rs. 4400

Answer

To solve the problem step by step, we will use the information given about the incomes and expenditures of A and B, as well as their savings.

### Step 1: Define the Ratios
Let the income of A be \(5x\) and the income of B be \(4x\), where \(x\) is a common multiplier.
Let the expenditure of A be \(3y\) and the expenditure of B be \(2y\), where \(y\) is another common multiplier.

### Step 2: Set Up the Savings Equation
According to the problem, both A and B save Rs. 1600 at the end of the year.
The savings can be expressed as:
- For A: \( \text{Income of A} - \text{Expenditure of A} = 1600 \)
\[ 5x - 3y = 1600 \quad \text{(1)} \]
- For B: \( \text{Income of B} - \text{Expenditure of B} = 1600 \)
\[ 4x - 2y = 1600 \quad \text{(2)} \]

### Step 3: Solve the Equations
Now we have a system of equations (1) and (2). We can solve these equations simultaneously.

From equation (2):
\[ 4x - 2y = 1600 \]
We can express \(y\) in terms of \(x\):
\[ 2y = 4x - 1600 \implies y = 2x - 800 \quad \text{(3)} \]

### Step 4: Substitute Back
Now substitute equation (3) into equation (1):
\[ 5x - 3(2x - 800) = 1600 \]
Expanding this gives:
\[ 5x - 6x + 2400 = 1600 \]
Combining like terms:
\[ -x + 2400 = 1600 \]
Rearranging gives:
\[ -x = 1600 - 2400 \implies -x = -800 \implies x = 800 \]

### Step 5: Find the Income of A
Now that we have \(x\), we can find the income of A:
\[ \text{Income of A} = 5x = 5 \times 800 = 4000 \]

### Final Answer
The income of A is Rs. 4000.

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