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

Introduction / Context:
This is a typical ratio based income and expenditure question. Two persons A and B have incomes in a given ratio and both save the same fixed amount every month. The ratio of their expenditures is also given. Using these relationships, we are asked to determine their actual monthly incomes in rupees. This tests the ability to translate verbal conditions into algebraic equations and then solve for the unknowns.

Given Data / Assumptions:
- Income ratio of A to B is 3 : 4.
- Each person saves Rs 100 per month.
- Expenditure ratio of A to B is 1 : 2.
- Expenditure of a person = income minus savings.
- We need the actual incomes of A and B in rupees.

Concept / Approach:
Let the common factor for incomes be k. Then the income of A is 3k and the income of B is 4k. Since both save Rs 100, the expenditures are 3k minus 100 and 4k minus 100. The ratio of these expenditures is 1 : 2. We set up the equation (3k minus 100) divided by (4k minus 100) equals 1 divided by 2 and solve for k. After determining k, we compute actual incomes 3k and 4k and match them with the options.

Step-by-Step Solution:
Step 1: Let income of A = 3k and income of B = 4k. Step 2: Since each saves Rs 100, expenditure of A = 3k - 100 and expenditure of B = 4k - 100. Step 3: Given that expenditure ratio A : B = 1 : 2, write (3k - 100) / (4k - 100) = 1 / 2. Step 4: Cross multiply to get 2(3k - 100) = 4k - 100. Step 5: Simplify 6k - 200 = 4k - 100 which leads to 2k = 100. Step 6: Solve for k to get k = 50. Step 7: Income of A = 3k = 3 * 50 = Rs 150, income of B = 4k = 4 * 50 = Rs 200.

Verification / Alternative check:
Check the savings and expenditures with these values. A saves Rs 100, so expenditure of A = 150 - 100 = Rs 50. B saves Rs 100, so expenditure of B = 200 - 100 = Rs 100. The expenditure ratio is 50 : 100 which simplifies to 1 : 2, matching the condition. The income ratio 150 : 200 simplifies to 3 : 4, which also matches. Hence, the derived incomes are fully consistent with the given data.

Why Other Options Are Wrong:
Rs 200 and Rs 400 do not satisfy the condition that both save Rs 100 and have expenditure ratio 1 : 2. Rs 100 and Rs 300 give an income ratio of 1 : 3, not 3 : 4. Rs 100 and Rs 200 lead to a ratio of 1 : 2 which again does not match. Only Rs 150 and Rs 200 give the required income ratio of 3 : 4 along with a savings of Rs 100 each and an expenditure ratio of 1 : 2.

Common Pitfalls:
Students often mix up the ratio of income with the ratio of expenditure and may try to directly impose 3 : 4 on expenditures. Another common mistake is to forget that savings are the same fixed amount for both, not in ratio. This can lead to setting up incorrect equations. Careless algebra, especially while cross multiplying and simplifying, can also cause wrong values of k and therefore wrong income figures.

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