For a certain investment at compound interest, what is the annual rate of interest? \n\nI. The principal amount was invested for 4 years. \nII. The total interest earned in 4 years was Rs 1,491.\n\nBased on the information in statements I and II, which option correctly describes the sufficiency of the data?

Introduction / Context:
This is a data sufficiency question based on compound interest. Instead of asking directly for a numerical value, it asks whether the given statements provide enough information to determine the annual rate of interest. We are told that a principal is invested at compound interest for 4 years and that the total interest earned in 4 years is Rs 1,491. We must reason carefully about whether this information identifies a unique rate of interest or whether multiple rates and principals could generate the same total interest, making the rate indeterminate.

Given Data / Assumptions:

• We consider a principal P invested at an unknown annual compound interest rate r%.
• Statement I: The principal is invested for 4 years.
• Statement II: The total interest earned over 4 years is Rs 1,491.
• We do not know the value of P from either statement.
• Interest is understood to be compounded annually unless stated otherwise.

Concept / Approach:
For compound interest, the amount after 4 years is A = P * (1 + r/100)^4 and the interest earned is I = A - P. Therefore, I = P * [(1 + r/100)^4 - 1]. To determine r, we must know the relationship between I and P. If both P and I were known, we could set up an equation in r. However, we only know I = 1,491 and the time 4 years; we do not know P. The data sufficiency task is to see whether knowing only time and total interest, without principal or final amount, is enough to uniquely determine r.

Step-by-Step Solution:
Consider statement I alone: It only tells us that t = 4 years. There is no information about P, r, or the amount, so we cannot find r. Statement I alone is not sufficient. Consider statement II alone: It gives the total interest I = 1,491, but neither P nor t nor the final amount is specified. Without P, we cannot form a complete equation linking I and r. Therefore, statement II alone is not sufficient. Combine statements I and II: Now we know t = 4 years and I = 1,491, but we still do not know P. The governing equation is I = P * [(1 + r/100)^4 - 1] = 1,491. This equation involves two unknowns, P and r, but only one equation; hence, infinitely many pairs (P, r) can satisfy it. Since we cannot isolate a unique value of r from this single equation, even using both statements together is not sufficient.

Verification / Alternative check:
To see this concretely, suppose we guess r = 5%. Then the factor [(1 + 0.05)^4 - 1] is approximately 0.215506. For I = 1,491, we would get P ≈ 1,491 / 0.215506, some positive principal. If we instead guess r = 6%, we would have a different factor and another positive solution for P. Because we can always adjust P to satisfy the same total interest for different r values, the rate r cannot be unique given only I and t. Therefore, the data is insufficient to determine r uniquely.

Why Other Options Are Wrong:
Option a is wrong because statement I alone provides only the time, not P or I. Option b is wrong because knowing only the total interest without principal or time does not fix the rate. Option c is wrong as both statements together still leave us with two unknowns and one equation. Option d is clearly wrong because neither statement alone is sufficient, so they certainly cannot both be sufficient individually. Only option e correctly captures that even combined, statements I and II do not provide enough information to find a unique rate.

Common Pitfalls:
A frequent mistake is to assume that knowing the time and the total interest is always sufficient. This is true for some simple interest problems where principal might be assumed or indirectly known, but under compound interest with an unknown principal, the situation is different. Another error is to misread the question and think that the principal is given when it is not. Data sufficiency questions require strict attention to what is actually provided and whether it leads to a unique solution, not just some equation involving the unknowns.

Final Answer:
Even when considered together, statements I and II are not sufficient to determine the rate of compound interest, so the correct choice is “Even both statements I and II together are not sufficient to answer the question.”