Ratio between the monthly incomes of A and B is `9 : 8` and the ratio between their expenditures is `8:7`. If they save Rs 500 each, find 'A's monthly income.

Options

A. Rs 3,500
B. Rs 4,000
C. Rs 4,500
D. Rs 5,000

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let A's monthly income be \( 9x \) and B's monthly income be \( 8x \), where \( x \) is a common multiplier.

### Step 2: Define Expenditures
Let A's monthly expenditure be \( 8y \) and B's monthly expenditure be \( 7y \), where \( y \) is another common multiplier.

### Step 3: Set Up the Savings Equation
According to the problem, both A and B save Rs 500 each. Therefore, we can express their savings as:
- For A:
\[ \text{Savings of A} = \text{Income of A} - \text{Expenditure of A} \]
\[ 500 = 9x - 8y \quad \text{(1)} \]

- For B:
\[ \text{Savings of B} = \text{Income of B} - \text{Expenditure of B} \]
\[ 500 = 8x - 7y \quad \text{(2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 9x - 8y = 500 \) (Equation 1)
2. \( 8x - 7y = 500 \) (Equation 2)

We can solve these equations simultaneously.

#### From Equation 1:
Rearranging gives:
\[ 8y = 9x - 500 \quad \text{(3)} \]

#### Substitute Equation (3) into Equation (2):
Substituting \( y \) from Equation (3) into Equation (2):
\[ 8x - 7\left(\frac{9x - 500}{8}\right) = 500 \]

Multiply through by 8 to eliminate the fraction:
\[ 64x - 7(9x - 500) = 4000 \]
\[ 64x - 63x + 3500 = 4000 \]
\[ x + 3500 = 4000 \]
\[ x = 4000 - 3500 \]
\[ x = 500 \]

### Step 5: Calculate A's Monthly Income
Now that we have \( x \), we can find A's monthly income:
\[ A's \, Income = 9x = 9 \times 500 = 4500 \]

### Final Answer
A's monthly income is Rs 4500.

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