Income of A is Rs 140 more than income of B, and income of C is Rs 80 more than income of D. The ratio of incomes of A and C is 2 : 3 and the ratio of incomes of B and D is 1 : 2. What are the incomes of A, B, C and D respectively?

Introduction / Context:
This question involves four related incomes with both difference conditions and ratio conditions. Income of A is related to B by a fixed difference, and income of C is related to D by another fixed difference. At the same time, the ratios A : C and B : D are given. We need to solve this system of relationships to find the exact incomes of A, B, C and D. This is essentially a simultaneous equations problem with ratios.

Given Data / Assumptions:
Income of A is Rs 140 more than income of B. Income of C is Rs 80 more than income of D. Ratio of incomes A : C is 2 : 3. Ratio of incomes B : D is 1 : 2. We must determine incomes of A, B, C and D.

Concept / Approach:
We translate each word statement into an algebraic equation. We use one variable to represent income of B and express income of D in terms of B using the ratio B : D. Then we express A in terms of B using the difference condition, and C in terms of D using the other difference condition. Finally, we apply the ratio A : C = 2 : 3 to connect these expressions and solve for B. Once B is known, A, C and D follow directly from their relationships with B.

Step-by-Step Solution:
Step 1: Let income of B be B and income of D be D. Step 2: Given B : D = 1 : 2, so D = 2B. Step 3: A is Rs 140 more than B, so A = B + 140. Step 4: C is Rs 80 more than D, so C = D + 80. Step 5: Substitute D = 2B into C expression to get C = 2B + 80. Step 6: Ratio A : C is given as 2 : 3, so (A / C) = 2 / 3. Step 7: Substitute A = B + 140 and C = 2B + 80 into the ratio equation: (B + 140) / (2B + 80) = 2 / 3. Step 8: Cross multiply: 3(B + 140) = 2(2B + 80). Step 9: Expand both sides: 3B + 420 = 4B + 160. Step 10: Rearrange to solve for B: 420 - 160 = 4B - 3B gives 260 = B. Step 11: So income of B is Rs 260. Step 12: Income of D is D = 2B = 2 * 260 = Rs 520. Step 13: Income of A is A = B + 140 = 260 + 140 = Rs 400. Step 14: Income of C is C = D + 80 = 520 + 80 = Rs 600.

Verification / Alternative check:
Check each condition with A = 400, B = 260, C = 600 and D = 520. Difference A - B = 400 - 260 = 140, which matches the first condition. Difference C - D = 600 - 520 = 80, matching the second condition. Ratio A : C = 400 : 600 simplifies by dividing by 200 to 2 : 3. Ratio B : D = 260 : 520 simplifies by dividing by 260 to 1 : 2. All given relationships hold, so these values are consistent and correct.

Why Other Options Are Wrong:
Other options either fail the difference condition, the ratio condition, or both. For example, in option A, 260 and 120 do not have a difference of 140, and the ratios also fail. Similarly, in option B and option D, at least one of the given ratios or differences between incomes is incorrect. Only option C fulfils A - B = 140, C - D = 80, A : C = 2 : 3 and B : D = 1 : 2 simultaneously.

Common Pitfalls:
Some learners may try to work with multiple variables without reducing them using ratios, which makes the algebra heavier. Others may accidentally reverse the ratio B : D and set B = 2D, which changes the relationships. Careful reading of the ratios and differences and using a small number of variables with systematic substitution helps avoid errors and reach the correct incomes quickly.

Final Answer:
The incomes of A, B, C and D are Rs 400, Rs 260, Rs 600 and Rs 520 respectively.