An amount of money was lent for 3 years. What will be the difference between the simple interest and the compound interest earned on it at the same rate? \n\nI. The rate of interest was 8% per annum. \nII. The total simple interest earned in 3 years was Rs 1,200.\n\nBased on the information in statements I and II, which option correctly describes the sufficiency of the data?

Introduction / Context:
This is another data sufficiency question related to the difference between simple interest (SI) and compound interest (CI) over 3 years. Instead of asking for a direct numerical value in the options, the question asks whether the given statements provide enough information to determine the difference between SI and CI. We must analyze the information from each statement and then from both together, focusing on whether they allow us to compute a unique value for the difference.

Given Data / Assumptions:

• A principal P is lent at an annual rate r% for 3 years.
• Interest is compounded annually for CI.
• Statement I: The rate of interest was 8% per annum.
• Statement II: The total simple interest earned in 3 years was Rs 1,200.
• We must decide whether these statements individually or jointly are sufficient to determine the numerical difference between SI and CI for 3 years.

Concept / Approach:
For time t = 3 years, simple interest is SI = P * r * t / 100, and compound interest is CI = P * [(1 + r/100)^3 - 1]. The difference between CI and SI depends on both P and r. To compute this difference numerically, we need enough information to determine both P and r or at least enough independent equations linking them. Statement I gives r, but not P. Statement II gives SI and t, and r is unknown unless we combine the statements. Data sufficiency problems require us to check whether the available information results in a fully determined value for the required quantity.

Step-by-Step Solution:
Consider statement I alone: r = 8% per annum, but P and the simple interest are unknown. We cannot compute SI or CI numerically, so the difference remains unknown. Statement I alone is not sufficient. Consider statement II alone: SI for 3 years is Rs 1,200, but P and r are both unknown. The formula SI = P * r * 3 / 100 gives one equation in two unknowns, which is not enough to determine either P or r uniquely. Statement II alone is not sufficient. Now combine statements I and II: from statement I, r = 8%. Substitute into the SI formula: SI = P * 8 * 3 / 100 = P * 24 / 100 = 0.24P. Statement II says SI = 1,200, so 0.24P = 1,200. This gives P = 1,200 / 0.24 = 5,000. Now that we know both P and r, we can compute both SI and CI and hence their difference.

Verification / Alternative check:
Let us actually compute the difference: With P = 5,000 and r = 8%, SI for 3 years is 5,000 * 8 * 3 / 100 = 1,200, matching the given value. The amount under CI after 3 years is A = 5,000 * (1.08)^3. Compute (1.08)^2 = 1.1664; (1.08)^3 = 1.259712. Thus, A = 5,000 * 1.259712 = 6,298.56, so CI ≈ 1,298.56. The difference CI - SI ≈ 98.56. Although we do not need the exact numerical difference for the data sufficiency answer, this calculation shows that with both statements we can indeed determine a unique difference.

Why Other Options Are Wrong:
Option a is wrong because knowing only the rate (8%) does not fix the principal, so we cannot obtain any numerical interest values. Option b is wrong because knowing only the total SI (1,200) without the rate does not allow us to identify P or r uniquely. Option d is wrong because neither statement alone is sufficient, so they certainly cannot both be sufficient individually. Option e is wrong because, as shown above, with both statements combined we can compute both P and r and then the exact difference between SI and CI. Only option c correctly states that both statements together are sufficient, but neither alone is sufficient.

Common Pitfalls:
Many students see a known rate in one statement and a known SI value in the other and jump to the conclusion that each is sufficient on its own. However, for compound interest problems, the difference between SI and CI depends on both principal and rate, and an incomplete set of data leaves ambiguity. Always check if there are enough independent equations to determine all the necessary unknowns. If two unknowns remain with only one equation, the data is insufficient.

Final Answer:
Both statements I and II together are sufficient to determine the difference between SI and CI, but neither statement alone is sufficient, so the correct choice is “Both statements I and II together are sufficient, but neither alone is sufficient.”