A sum of money is equally invested in two different schemes at compound interest for 2 years, one at 15% per annum and the other at 20% per annum. The interest earned from the amount invested at 20% is Rs. 528.75 more than the interest earned from the amount invested at 15%. What is the total sum invested in both schemes together?

Introduction / Context:
This question involves comparing compound interest on the same principal invested in two different schemes with different interest rates. The principal is split equally into the two schemes for the same time period. The difference in interest amounts is given, and you must determine the total sum invested. This tests algebraic handling of compound interest rates and differences.

Given Data / Assumptions:
- Let the amount invested in each scheme be P rupees (equal investments).
- Scheme 1: Rate = 15% per annum, compound interest, time = 2 years.
- Scheme 2: Rate = 20% per annum, compound interest, time = 2 years.
- Interest from Scheme 2 is Rs. 528.75 more than from Scheme 1.
- We must find the total sum invested, which is 2P rupees.

Concept / Approach:
Write expressions for the compound interest in both schemes over 2 years. For a rate r, the amount after 2 years is P * (1 + r/100)^2, and the interest is this amount minus P. The difference between the two interest values equals Rs. 528.75. This gives an equation in P. Solving that equation yields P and hence the total invested amount 2P.

Step-by-Step Solution:
Step 1: Interest at 15% for 2 years: A1 = P * (1.15)^2 = P * 1.3225.Step 2: Compound interest from Scheme 1: I1 = A1 - P = P * (1.3225 - 1) = 0.3225P.Step 3: Interest at 20% for 2 years: A2 = P * (1.20)^2 = P * 1.44.Step 4: Compound interest from Scheme 2: I2 = A2 - P = P * (1.44 - 1) = 0.44P.Step 5: Given that I2 - I1 = 528.75.Step 6: So (0.44P - 0.3225P) = 0.1175P = 528.75.Step 7: Solve for P: P = 528.75 / 0.1175 = 4500.Step 8: Total sum invested = 2P = 2 * 4500 = Rs. 9,000.

Verification / Alternative check:
Check actual interests. For P = 4500: Scheme 1 interest = 4500 * 0.3225 = 1451.25. Scheme 2 interest = 4500 * 0.44 = 1980. Difference = 1980 - 1451.25 = 528.75, exactly as given. Hence the total sum, 9000, is correct.

Why Other Options Are Wrong:
Rs. 7,000, Rs. 4,500, Rs. 8,200 and Rs. 10,500 lead to interest differences that are not equal to Rs. 528.75 when the same formulas are applied. For example, 4,500 is only the single-scheme principal P, not the total sum; using it as the total would halve the interest difference.

Common Pitfalls:
Some students forget that the sum is equally invested and mistakenly treat the given difference as arising from different principals. Others compute simple interest instead of compound interest over 2 years, or they miscalculate squares like (1.15)^2. Always convert the condition on interest differences into an algebraic equation and solve systematically.

Final Answer:
The total sum invested in the two schemes together is Rs. 9,000.