Annual Incomes of Amit and Veer are in the ratio `3 : 2`, while the ratio of their expenditures is `5 : 3`. If at the end of the year each saves Rs 1000, what Is the annual Income of Amit?

Options

A. Rs. 9000
B. Rs. 8000
C. Rs 7000
D. Rs 6000

Answer

To solve the problem step by step, we will use the information given about the ratios of incomes and expenditures of Amit and Veer, along with their savings.

### Step 1: Define Variables
Let the annual incomes of Amit and Veer be represented as:
- Amit's income = 3x
- Veer's income = 2x

Here, x is a common multiplier.

### Step 2: Define Expenditures
Let the annual expenditures of Amit and Veer be represented as:
- Amit's expenditure = 5y
- Veer's expenditure = 3y

Here, y is another common multiplier.

### Step 3: Set Up the Savings Equation
According to the problem, both Amit and Veer save Rs 1000 at the end of the year. Therefore, we can set up the following equations based on their incomes and expenditures:

For Amit:
\[ \text{Income} - \text{Expenditure} = \text{Savings} \]
\[ 3x - 5y = 1000 \] (1)

For Veer:
\[ 2x - 3y = 1000 \] (2)

### Step 4: Solve the Equations
We have two equations (1) and (2). We can solve these equations simultaneously.

From equation (1):
\[ 3x - 5y = 1000 \]
Rearranging gives:
\[ 3x = 1000 + 5y \]
\[ x = \frac{1000 + 5y}{3} \] (3)

Now substitute (3) into equation (2):
\[ 2\left(\frac{1000 + 5y}{3}\right) - 3y = 1000 \]
Multiplying through by 3 to eliminate the fraction:
\[ 2(1000 + 5y) - 9y = 3000 \]
\[ 2000 + 10y - 9y = 3000 \]
\[ 2000 + y = 3000 \]
\[ y = 1000 \]

### Step 5: Substitute Back to Find x
Now substitute the value of y back into equation (3):
\[ x = \frac{1000 + 5(1000)}{3} \]
\[ x = \frac{1000 + 5000}{3} \]
\[ x = \frac{6000}{3} \]
\[ x = 2000 \]

### Step 6: Calculate Amit's Income
Now that we have the value of x, we can find Amit's income:
\[ \text{Amit's Income} = 3x = 3(2000) = 6000 \]

### Conclusion
Thus, the annual income of Amit is Rs 6000.

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