The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves ₹ 6,000 at the end of the month, their monthly incomes respectively are (in ₹)

The correct option is 24,000 and 18,000.

Let's solve this step-by-step to understand why this is the correct answer.

Step 1: Set up the variables for Income
The monthly incomes of Peter and Paul are in the ratio of 4 : 3.
Let the monthly income of Peter be $4x$ and the monthly income of Paul be $3x$, where $x$ is a common multiplier.

Step 2: Set up the variables for Expenses
Their monthly expenses are in the ratio of 3 : 2.
Let the monthly expense of Peter be $3y$ and the monthly expense of Paul be $2y$, where $y$ is a common multiplier.

Step 3: Relate Income, Expense, and Savings
We know the basic financial formula:
Income - Expense = Savings
Since both Peter and Paul save ₹ 6,000 at the end of the month, we can write two equations:

For Peter:
$4x-3y=6000$ --- (Equation 1)

For Paul:
$3x-2y=6000$ --- (Equation 2)

Step 4: Solve the system of equations
To find the incomes, we need to solve for $x$. We can eliminate $y$ by multiplying Equation 1 by 2 and Equation 2 by 3:
Multiplying Equation 1 by 2 gives:
$8x-6y=12000$ --- (Equation 3)
Multiplying Equation 2 by 3 gives:
$9x-6y=18000$ --- (Equation 4)

Subtracting Equation 3 from Equation 4 to eliminate $y$:
$(9x-6y)-(8x-6y)=18000-12000$
$x=6000$

Step 5: Calculate the monthly incomes
Now we substitute the value of $x$ back into our income representations:
Peter's monthly income = $4x=4\times 6000=24000$
Paul's monthly income = $3x=3\times 6000=18000$

Thus, their monthly incomes are ₹ 24,000 and ₹ 18,000 respectively.