A sum is equally invested in two different schemes at compound interest for 2 years, one at 15 percent per annum and the other at 20 percent per annum. If the interest gained from the sum invested at 20 percent is Rs. 528.75 more than the interest gained from the sum invested at 15 percent, what is the total sum invested in both schemes together?

Introduction / Context:
This question involves two equal investments placed in different compound interest schemes for the same time period. One scheme has a rate of 15 percent per annum, and the other has 20 percent per annum. The difference in interest earned between the two schemes is given. We need to find the total sum invested, which is twice the amount invested in each individual scheme. This type of problem helps strengthen understanding of compound interest growth and comparison between different rates.

Given Data / Assumptions:

• Two equal sums, each of amount P, are invested for 2 years.
• First scheme: 15 percent per annum compound interest, compounded annually.
• Second scheme: 20 percent per annum compound interest, compounded annually.
• Difference in interest between the second and the first scheme is Rs. 528.75.
• Total sum invested overall is 2P, which we need to find.

Concept / Approach:
We first compute the compound interest rate multipliers for 2 years at each rate. For 15 percent, the multiplier is (1.15)^2, and for 20 percent it is (1.20)^2. The interest for each scheme is the final amount minus the principal P. The difference between these two interests is given as Rs. 528.75, which gives us an equation in P that we can solve. Finally, we multiply P by 2 to get the total invested sum.

Step-by-Step Solution:
Let the amount invested in each scheme be P. For 15 percent scheme, amount after 2 years = P * (1.15)^2 (1.15)^2 = 1.3225, so CI at 15 percent = P * 1.3225 - P = 0.3225 * P For 20 percent scheme, amount after 2 years = P * (1.20)^2 (1.20)^2 = 1.44, so CI at 20 percent = P * 1.44 - P = 0.44 * P Difference in interest = 0.44 * P - 0.3225 * P = (0.1175) * P Given this difference equals Rs. 528.75 So 0.1175 * P = 528.75 P = 528.75 / 0.1175 = 4500 Total sum invested = 2 * P = 2 * 4500 = 9000

Verification / Alternative check:
We can verify by computing the actual interests using P = 4500. At 15 percent: CI at 15 percent = 0.3225 * 4500 = 1451.25 At 20 percent: CI at 20 percent = 0.44 * 4500 = 1980 Difference = 1980 - 1451.25 = 528.75, which matches the given difference. Therefore, the total investment of 9000 is correct.

Why Other Options Are Wrong:
If the total sum were 7000, 4500, 8200, or 6000, then the corresponding individual amount P would be 3500, 2250, 4100, or 3000 respectively. When these values are substituted into the formulas for compound interest, the difference in interest between 15 percent and 20 percent would not equal 528.75. Only when P is 4500 and the total is 9000 do we obtain the required difference.

Common Pitfalls:
A frequent mistake is to confuse the difference in interest with the difference in amounts and forget to subtract the principal in each case. Another error is to accidentally use simple interest formulas rather than compound interest growth factors. Some students also forget that the total sum invested is twice P and may stop at P instead of doubling it. Writing each step clearly and distinguishing between total amount and interest amount helps prevent these errors.

Final Answer:
The total sum invested in both schemes together is Rs. 9000.