The annual income of A and B are in the ratio `4 : 3` and the ratio of their expenditures is `3:2`. If each of them saves Rs 600 in the year, the annual income of A is:

Options

A. Rs 4800
B. Rs 1800
C. Rs 1200
D. Rs 2400

Answer

To find the annual income of A, we can follow these steps:

### Step 1: Define the Variables
Let the annual income of A be \( 4x \) and the annual income of B be \( 3x \), where \( x \) is a common multiplier.

### Step 2: Define Expenditures
Let the annual expenditure of A be \( 3y \) and the annual expenditure of B be \( 2y \), where \( y \) is another common multiplier.

### Step 3: Set Up the Savings Equation
According to the problem, both A and B save Rs 600 each year. The savings can be expressed as:
- For A: \( \text{Income of A} - \text{Expenditure of A} = 600 \)

Thus, we can write:
\[ 4x - 3y = 600 \quad \text{(1)} \]

- For B: \( \text{Income of B} - \text{Expenditure of B} = 600 \)

Thus, we can write:
\[ 3x - 2y = 600 \quad \text{(2)} \]

### Step 4: Solve the Equations
Now we have a system of equations:
1. \( 4x - 3y = 600 \)
2. \( 3x - 2y = 600 \)

We can solve these equations simultaneously.

### Step 5: Multiply the Equations to Eliminate Variables
To eliminate \( y \), we can multiply equation (1) by 2 and equation (2) by 3:
- From equation (1):
\[ 2(4x - 3y) = 2(600) \implies 8x - 6y = 1200 \quad \text{(3)} \]
- From equation (2):
\[ 3(3x - 2y) = 3(600) \implies 9x - 6y = 1800 \quad \text{(4)} \]

### Step 6: Subtract the Equations
Now we can subtract equation (3) from equation (4):
\[ (9x - 6y) - (8x - 6y) = 1800 - 1200 \]
This simplifies to:
\[ x = 600 \]

### Step 7: Substitute Back to Find y
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). Let's use equation (1):
\[ 4(600) - 3y = 600 \]
This simplifies to:
\[ 2400 - 3y = 600 \]
\[ 3y = 2400 - 600 \]
\[ 3y = 1800 \implies y = 600 \]

### Step 8: Calculate the Income of A
Now we can find the income of A:
\[ \text{Income of A} = 4x = 4(600) = 2400 \]

### Final Answer
The annual income of A is Rs 2400.

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