A certain amount is invested at compound interest. You are asked: "What is the rate of compound interest per annum?" I. The principal was invested for 4 years. II. The compound interest earned over that period was Rs 1,491. Based on these two statements, which option correctly describes whether the information is sufficient to determine the rate?

Introduction:
This is a classic data sufficiency style question about compound interest. Rather than asking you to compute a numerical value directly, it tests whether the given pieces of information are enough to determine the unknown rate of compound interest. Such problems are common in management and banking entrance exams.

Given Data / Assumptions:

• We have an unknown principal P invested at compound interest.
• We need to determine the rate of compound interest per annum.
• Statement I: The principal was invested for 4 years.
• Statement II: The total compound interest earned over the investment period was Rs 1,491.
• No information about the value of the principal P itself is provided.

Concept / Approach:
For compound interest, the amount after n years is given by:
A = P * (1 + r)^nThe compound interest is:
CI = A - PTo find the rate r, we must know both the principal P and either the final amount A or the interest CI for a specified time period n. If P is unknown, we get only one equation with two unknowns, which is not enough to solve uniquely.

Step-by-Step Analysis of Statements:
Step 1: Consider Statement I alone.I: Time n = 4 years.No information about CI or P is given. With only time, there is no way to compute r.Statement I alone is not sufficient.Step 2: Consider Statement II alone.II: CI = Rs 1,491 over the entire period.We still do not know P or the time. Without at least one of these, CI alone does not allow us to compute r.Statement II alone is not sufficient.Step 3: Combine Statements I and II.From I and II together, we know:Time n = 4 years, CI = Rs 1,491But both P and r are unknown. Using the formula:CI = P * ((1 + r)^4 - 1)This is one equation with two unknowns (P and r). Infinitely many combinations of P and r can satisfy this equation, so r cannot be determined uniquely.

Verification / Alternative check:
Assume two different principal values, say P1 and P2, and see that you can adjust r1 and r2 respectively to produce the same CI of Rs 1,491 over 4 years. This shows that the given information does not pin down one unique rate, confirming that the data is insufficient even together.

Why Other Options Are Wrong:

• Option a: Time alone without CI or P cannot provide r.
• Option b: CI alone without P and time cannot provide r.
• Option c: Even combining both statements leaves two unknowns in one equation.
• Option d: Clearly false because neither statement alone is sufficient.

Common Pitfalls:
Many students rush to apply formulas without checking how many unknowns are involved. In data sufficiency questions, always check whether you have enough independent equations to solve for all unknowns uniquely. If not, the data is insufficient, even if it feels like a lot of information is given.

Final Answer:
The correct conclusion is that even both statements I and II together are not sufficient to determine the rate of compound interest.