The incomes of two persons A and B are in the ratio 3 : 4. Each of them saves Rs. 100 per month, and the ratio of their expenditures is 1 : 2. What are their monthly incomes?

Introduction / Context:
This question combines ratios of income and expenditure with a fixed saving amount. We know that the incomes of A and B stand in a given ratio, their expenditures are in a different ratio, and both save the same fixed amount each month. Using this data, we need to calculate their actual monthly incomes. This tests algebraic skill and the ability to convert verbal relationships into equations.

Given Data / Assumptions:
• Income ratio A : B = 3 : 4. • Expenditure ratio A : B = 1 : 2. • Each saves Rs. 100 per month. • Savings = Income − Expenditure for each person.

Concept / Approach:
Let the incomes of A and B be 3x and 4x respectively. Let their expenditures be y and 2y respectively, consistent with 1 : 2. Since each person saves Rs. 100, we get two equations: 3x − y = 100 and 4x − 2y = 100. Solving these simultaneously gives x and y. Then we compute 3x and 4x to obtain the actual incomes.

Step-by-Step Solution:
Step 1: Let A's income = 3x and B's income = 4x. Step 2: Let A's expenditure = y and B's expenditure = 2y. Step 3: Savings of A: 3x − y = 100. Step 4: Savings of B: 4x − 2y = 100. Step 5: From 3x − y = 100, express y = 3x − 100. Step 6: Substitute into 4x − 2y = 100: 4x − 2(3x − 100) = 100. Step 7: Simplify: 4x − 6x + 200 = 100 ⇒ −2x + 200 = 100. Step 8: Solve for x: −2x = −100 ⇒ x = 50. Step 9: Then A's income = 3x = 3 * 50 = 150, and B's income = 4x = 4 * 50 = 200.

Verification / Alternative check:
Compute expenditures: y = 3x − 100 = 150 − 100 = 50, so A spends Rs. 50 and saves Rs. 100. B's expenditure is 2y = 2 * 50 = 100, so B spends Rs. 100 and saves Rs. 100 since 200 − 100 = 100. The expenditure ratio is 50 : 100 = 1 : 2, and the income ratio is 150 : 200 = 3 : 4. All conditions match the question.

Why Other Options Are Wrong:
• The other options would break either the income ratio, the expenditure ratio, or the equality of savings of Rs. 100 each.

Common Pitfalls:
Some students incorrectly assume that equal savings imply equal incomes or expenditures. Others may mis-handle the algebraic substitution. Always express one variable in terms of another and substitute carefully to avoid sign errors.

Final Answer:
The monthly incomes of A and B are Rs. 150 and Rs. 200 respectively.