The ratio of monthly income of Deepak and Raj is 4 : 9 respectively and the ratio of their expenditures is 1 : 3 respectively. If each saves Rs 9000 per month, then what will be the monthly income (in Rs) of Raj?

Options

A. a)54000
B. b)33000
C. c)24000
D. d)42000

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the monthly income of Deepak be \( 4x \) and the monthly income of Raj be \( 9x \).

**Hint:** Use the ratio given in the problem to express the incomes in terms of a common variable.

### Step 2: Define the Expenditures
Let the expenditures of Deepak and Raj be \( y \) and \( 3y \) respectively, according to the ratio of their expenditures.

**Hint:** Use the ratio of expenditures similarly to how you defined the incomes.

### Step 3: Set Up the Savings Equation for Deepak
According to the problem, Deepak saves Rs 9000 per month. Therefore, we can write the equation:
\[ 4x - y = 9000 \]

**Hint:** Savings can be calculated as income minus expenditure.

### Step 4: Set Up the Savings Equation for Raj
Similarly, for Raj, who also saves Rs 9000, we can write:
\[ 9x - 3y = 9000 \]

**Hint:** Again, use the same formula for savings to set up the equation.

### Step 5: Solve the Two Equations
Now we have two equations:
1. \( 4x - y = 9000 \) (Equation 1)
2. \( 9x - 3y = 9000 \) (Equation 2)

From Equation 1, we can express \( y \):
\[ y = 4x - 9000 \]

**Hint:** Rearranging the first equation will help substitute \( y \) in the second equation.

### Step 6: Substitute \( y \) in Equation 2
Substituting \( y \) in Equation 2 gives:
\[ 9x - 3(4x - 9000) = 9000 \]
Expanding this:
\[ 9x - 12x + 27000 = 9000 \]
Combining like terms:
\[ -3x + 27000 = 9000 \]
Now, isolate \( x \):
\[ -3x = 9000 - 27000 \]
\[ -3x = -18000 \]
\[ x = 6000 \]

**Hint:** Isolate the variable to find its value.

### Step 7: Calculate Raj's Monthly Income
Now that we have \( x \), we can find Raj's monthly income:
\[ \text{Income of Raj} = 9x = 9 \times 6000 = 54000 \]

**Hint:** Substitute the value of \( x \) back into the expression for Raj's income.

### Final Answer
The monthly income of Raj is Rs 54,000.

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