The incomes of persons C and D are in the ratio 3 : 2, and the incomes of D and E are in the ratio 5 : 4. One third of the income of C is Rs 4000 more than one half of the income of E. What is the monthly income of D in rupees?

Introduction / Context:
This question tests combined use of ratios and linear equations in the context of monthly incomes. We are given two different ratios involving three people and an extra condition relating parts of two incomes. The task is to convert ratio information into algebraic expressions and then use the given difference condition to find the actual numerical values of the incomes, especially that of person D.

Given Data / Assumptions:

• Income of C : income of D = 3 : 2.
• Income of D : income of E = 5 : 4.
• One third of income of C is Rs 4000 more than one half of income of E.
• All incomes are assumed to be positive real numbers, expressed in rupees per month.

Concept / Approach:
When ratios involve three related quantities, we often express them in terms of a common base variable. First we combine the two ratios into a single three way ratio C : D : E. Then we convert the statement about one third of C and one half of E into an equation. Solving this equation gives us the common multiplier for the ratio and thus the exact incomes of C, D and E. This approach uses proportional reasoning along with basic algebra.

Step-by-Step Solution:
From C : D = 3 : 2, let C = 3k and D = 2k. From D : E = 5 : 4, let D = 5m and E = 4m. Since D is the same person, 2k = 5m, so k = (5m) / 2. Then C = 3k = 3 * (5m / 2) = (15m) / 2 and E = 4m. Multiply all by 2 to clear the denominator: C : D : E = 15m : 10m : 8m, so the combined ratio is 15 : 10 : 8. Let C = 15u, D = 10u and E = 8u for some u. Given: one third of C is 4000 more than one half of E. So (1 / 3) * 15u = (1 / 2) * 8u + 4000. This gives 5u = 4u + 4000, so u = 4000. Therefore, D = 10u = 10 * 4000 = Rs 40000.

Verification / Alternative check:
Compute C and E to confirm the condition. C = 15u = 60000 and E = 8u = 32000. One third of C is 60000 / 3 = 20000. One half of E is 32000 / 2 = 16000. The difference is 20000 − 16000 = 4000, which matches the problem statement. This confirms that u = 4000 and therefore D = 40000 is consistent.

Why Other Options Are Wrong:
43000, 50000, 60000 and 36000 do not satisfy the combined ratios 3 : 2 and 5 : 4 when checked against the given difference condition between one third of C and one half of E. Any other value for D would require a different u and then the numerical difference would not be exactly Rs 4000.

Common Pitfalls:
One common mistake is to treat the two given ratios independently without merging them into a single ratio for C, D and E. Another typical error is to misinterpret “one third of C is Rs 4000 more than one half of E” and reverse the direction of the inequality. Also, candidates sometimes directly assign numbers to C and E without checking consistency with both ratios, which leads to contradictions. Systematically forming and solving the equation prevents these errors.

Final Answer:
The monthly income of D is Rs 40000.