The ratio of income of two persons A and B is 8 : 5 and ratio of their…

Concept: When income and expenditure are given only as ratios (not amounts), assign one unknown multiplier to the income ratio and a different unknown multiplier to the expenditure ratio. Since Savings = Income − Expenditure, each person's savings gives one linear equation in the two multipliers; solving the pair by elimination gives the actual income.

Step-by-step:

1. Let income of A = 8x and income of B = 5x, since the income ratio A : B = 8 : 5.

2. Let expenditure of A = 5y and expenditure of B = 3y, since the expenditure ratio A : B = 5 : 3.

3. Using Savings = Income − Expenditure for A: 8x − 5y = 2,400.

4. Using Savings = Income − Expenditure for B: 5x − 3y = 2,000.

5. Multiply the first equation by 3 and the second by 5 to make the y-coefficients equal: 24x − 15y = 7,200 and 25x − 15y = 10,000.

6. Subtracting the first from the second eliminates y: x = 10,000 − 7,200 = 2,800.

7. Income of A = 8x = 8 × 2,800 = ₹22,400.

Cross-check: Substituting x = 2,800 into 5x − 3y = 2,000 gives y = 4,000. Checking A's equation: 8(2,800) − 5(4,000) = 22,400 − 20,000 = 2,400, which matches the given savings, confirming income of A = ₹22,400.