Correct Option (2)
Let the monthly income of X be 4a and the monthly income of Y be 3a, based on their given ratio of 4:3.
Similarly, let the monthly expenses of X be 3b and the monthly expenses of Y be 2b, based on their given ratio of 3:2.
Since each saves Rs 6,000 per month, we can formulate two linear equations:
- For X: Income - Expenses = Savings ⇒ 4a - 3b = 6000
- For Y: Income - Expenses = Savings ⇒ 3a - 2b = 6000
To solve this system of equations, multiply equation (1) by 2 and equation (2) by 3 to equate the coefficients of 'b':
- (4a - 3b = 6000) × 2 ⇒ 8a - 6b = 12000
- (3a - 2b = 6000) × 3 ⇒ 9a - 6b = 18000
Subtract the first modified equation from the second modified equation:
(9a - 6b) - (8a - 6b) = 18000 - 12000
a = 6000
The total monthly income of X and Y is the sum of their individual incomes: 4a + 3a = 7a.
Substitute the value of 'a':
Total monthly income = 7 × 6000 = Rs 42,000.
Incorrect Options:
Options (1) Rs 28,000, (3) Rs 56,000, and (4) Rs 84,000 are incorrect. The calculations based on the given ratios of incomes and expenses, combined with the fixed monthly savings, uniquely determine the total monthly income to be Rs 42,000. These alternative values do not satisfy the conditions established by the problem's parameters.