Data Sufficiency – Finding the Simple Interest Rate: What is the rate of simple interest per annum (in %)? Statement I: The total interest earned was ₹4,000. Statement II: The sum was invested for 4 years. Assume simple interest and that no other information is available.

Introduction / Context:
This data sufficiency question tests whether you can determine the simple interest rate when given limited information. You must decide if knowing the total interest or the time period, separately or together, is enough to determine the annual rate of interest. The key is to understand how many variables must be known in the simple interest formula.

Given Data / Assumptions:

• Simple interest is used.
• Principal = P (unknown).
• Rate per annum = r% (unknown).
• Time = t years.
• Statement I: Total interest earned I = ₹4,000.
• Statement II: Time t = 4 years.

Concept / Approach:
The simple interest formula is I = (P * r * t) / 100. There are three key variables P, r, and t, plus the interest I. To find r, we need enough independent equations to eliminate the other unknowns. Data sufficiency problems focus on whether the given statements provide enough equations, rather than on doing calculations.

Step-by-Step Solution:
Step 1: Analyse Statement I alone. It gives I = ₹4,000 but says nothing about P, r, or t. Step 2: With only I known and three other unknowns, we cannot determine r. Hence, Statement I alone is not sufficient. Step 3: Analyse Statement II alone. It gives t = 4 years. Step 4: Without knowing P, r, or I, Statement II alone is clearly not sufficient to determine r. Step 5: Combine Statements I and II. We now know I = 4,000 and t = 4 years, but P and r are still both unknown. Step 6: The SI formula becomes 4,000 = (P * r * 4) / 100, or P * r = 100,000. Step 7: This is a single equation in two unknowns (P and r), so r cannot be uniquely determined because many combinations of P and r satisfy P * r = 100,000. Step 8: Therefore, even together, Statements I and II are not sufficient to find the rate.

Verification / Alternative check:
To see why r is not unique, suppose r = 10%. Then P must be 10,000 to make P * r = 100,000. If r = 20%, P must be 5,000. Using both pairs, you still get I = 4,000 over 4 years, but the rate is different in each case. Hence, the information is insufficient to determine a unique r.

Why Other Options Are Wrong:

• Statement I alone is sufficient: Incorrect, as it does not give P, r, or t individually.
• Statement II alone is sufficient: Incorrect, as knowing only time cannot give the rate.
• Either statement alone is sufficient: Incorrect because neither alone works.
• Both statements are necessary: Even together they are not enough, so this is wrong.

Common Pitfalls:
Many learners see I = 4,000 and t = 4 and mistakenly assume they can find r, forgetting that P is unknown. In data sufficiency, you must always check how many unknowns remain after combining all statements. If more than one unknown remains and only one equation is available, the data is not sufficient.

Final Answer:
The correct choice is Both Statement I and Statement II are not sufficient to determine the rate of interest.