Incomes of A and B are in the ratio `4 : 3` and their annual expenses in the ratio `3 : 2`. If each saves Rs 60,000 at the end of the year, the annual income of A is:

Options

A. Rs 1,20,000
B. Rs 1,50,000
C. Rs 2,40,000
D. Rs 3,60,000

Answer

To solve the problem step by step, let's denote the incomes and expenses of A and B using variables based on the given ratios.

### Step 1: Define the variables
Let the incomes of A and B be:
- Income of A = 4x
- Income of B = 3x

Let the expenses of A and B be:
- Expense of A = 3y
- Expense of B = 2y

### Step 2: Set up the equations based on savings
According to the problem, both A and B save Rs 60,000 at the end of the year. The savings can be expressed as:
- Savings of A = Income of A - Expense of A
- Savings of B = Income of B - Expense of B

This gives us the following equations:
1. \( 4x - 3y = 60,000 \) (for A)
2. \( 3x - 2y = 60,000 \) (for B)

### Step 3: Solve the equations
We have two equations:
1. \( 4x - 3y = 60,000 \) (Equation 1)
2. \( 3x - 2y = 60,000 \) (Equation 2)

Now, we can solve these equations simultaneously.

#### Step 3.1: Multiply Equation 2 to eliminate y
To eliminate y, we can multiply Equation 2 by 3:
\[ 3(3x - 2y) = 3(60,000) \]
This simplifies to:
\[ 9x - 6y = 180,000 \quad \text{(Equation 3)} \]

#### Step 3.2: Multiply Equation 1 to match coefficients of y
Now, we multiply Equation 1 by 2:
\[ 2(4x - 3y) = 2(60,000) \]
This simplifies to:
\[ 8x - 6y = 120,000 \quad \text{(Equation 4)} \]

### Step 4: Subtract Equation 4 from Equation 3
Now, we can subtract Equation 4 from Equation 3:
\[ (9x - 6y) - (8x - 6y) = 180,000 - 120,000 \]
This simplifies to:
\[ x = 60,000 \]

### Step 5: Find the income of A
Now that we have the value of x, we can find the income of A:
\[ \text{Income of A} = 4x = 4(60,000) = 240,000 \]

### Final Answer
The annual income of A is Rs 240,000.

---