The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 16 percent per annum is Rs. 320. For this investment, what is the principal sum in rupees?

Introduction / Context:
This problem is yet another application of the shortcut formula for the difference between compound interest (CI) and simple interest (SI) over 2 years at a given rate. The question supplies the rate, the time, and the difference, and asks you to find the principal amount. Efficiently using this formula helps save time in quantitative aptitude sections that involve many interest-related questions.

Given Data / Assumptions:

• Annual rate of interest r = 16 percent.
• Time period t = 2 years.
• Difference between CI and SI over 2 years, D₂ = Rs. 320.
• Interest for CI is compounded annually.
• Principal P is unknown.

Concept / Approach:
For a principal P invested at rate r percent per annum for 2 years, the difference between CI and SI is:
D₂ = P * (r/100)^2.
This result is derived by comparing the compound amount P * (1 + r/100)^2 with the simple interest amount P * (1 + 2r/100). Once D₂ and r are known, we substitute and solve for P directly. This method avoids long calculations of CI and SI separately.

Step-by-Step Solution:
Step 1: Use the formula D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 320 and r = 16: 320 = P * (16/100)^2. Step 3: Compute (16/100)^2 = 256/10000 = 0.0256. Step 4: So 320 = P * 0.0256. Step 5: Rearranging for P, we get P = 320 / 0.0256. Step 6: Divide: 320 / 0.0256 = 12500. Step 7: Therefore, the principal sum is Rs. 12500.

Verification / Alternative check:
We can verify by explicitly computing CI and SI for P = 12500 at r = 16 percent over 2 years. Simple interest SI = (12500 * 16 * 2) / 100 = 4000. Compound amount A = 12500 * (1.16)^2. Compute (1.16)^2 = 1.3456, so A = 12500 * 1.3456 = 16820. Compound interest CI = A − P = 16820 − 12500 = 4320. The difference CI − SI = 4320 − 4000 = 320, matching the given difference. This confirms that P = 12500 is correct.

Why Other Options Are Wrong:
If P were Rs. 25000, then D₂ would be 25000 * 0.0256 = 640, which is double the required 320. For P = 50000, the difference becomes 1280, and for P = 37500 it is 960. At P = 10000, the difference is 256, smaller than 320. Only P = 12500 yields the correct difference of Rs. 320, making the other options inconsistent with the data and formula.

Common Pitfalls:
Common sources of error include miscalculating (r/100)^2, forgetting to convert the percentage to a fraction before squaring, or incorrectly rearranging the equation to solve for P. Some test-takers also mistakenly apply the simple interest formula instead of using the CI–SI difference shortcut. Carefully following the formula and checking the arithmetic helps avoid these mistakes.

Final Answer:
The principal sum on which the difference between simple interest and compound interest over 2 years at 16 percent per annum is Rs. 320 is Rs. 12500.