Data Sufficiency – Finding Principal from Simple Interest: What is the principal sum (in ₹) that earned the interest? Statement I: The total simple interest is ₹7,000 after 7 years. Statement II: The total of principal and simple interest becomes double of the principal after 5 years at the same simple interest rate.

Introduction / Context:
This is a data sufficiency question on simple interest. Instead of asking you to compute a numerical value directly, it asks whether the given statements provide enough information to determine the principal. You must analyse each statement independently and then together, using the simple interest formula, without necessarily computing the final numeric principal.

Given Data / Assumptions:

• We are dealing with simple interest at some annual rate r%.
• Principal = P (unknown).
• Statement I: Total simple interest is ₹7,000 after 7 years.
• Statement II: Amount (principal + interest) becomes double of the principal after 5 years at the same rate.
• The rate r is constant over time.

Concept / Approach:
For simple interest, I = (P * r * t) / 100, where I is interest, P is principal, r is rate, and t is time in years. Data sufficiency problems require deciding whether a statement or combination of statements is enough to uniquely determine P. We are not required to calculate P explicitly, but we can do it to verify understanding.

Step-by-Step Solution:
Step 1: Analyse Statement I alone. It gives I = 7,000 for t = 7 years, so 7,000 = (P * r * 7) / 100. Step 2: Rearranging, P * r = (7,000 * 100) / 7 = 100,000. This relation involves two unknowns P and r, so P cannot be uniquely found. Step 3: Therefore, Statement I alone is not sufficient. Step 4: Analyse Statement II alone. The amount doubles in 5 years, so P + I = 2P, hence I = P. Step 5: Under SI, I for 5 years is (P * r * 5) / 100, so (P * r * 5) / 100 = P, giving r = 100 / 5 = 20%. Step 6: Statement II alone gives r = 20% but no information about the actual value of P, so P is still unknown. Step 7: Combine Statements I and II. From I: P * r = 100,000. From II: r = 20%. So P = 100,000 / 20 = ₹5,000. Step 8: Only by using both statements together do we get a unique principal P.

Verification / Alternative check:
With P = ₹5,000 and r = 20%, SI for 7 years would be (5,000 * 20 * 7) / 100 = 5,000 * 1.4 = ₹7,000, matching Statement I. Amount after 5 years would be P + I = 5,000 + (5,000 * 20 * 5) / 100 = 5,000 + 5,000 = ₹10,000, which is double the principal, matching Statement II. Thus both statements are consistent and jointly sufficient.

Why Other Options Are Wrong:

• Statement I alone is sufficient: Incorrect, because it leaves two unknowns (P and r) in one equation.
• Statement II alone is sufficient: Incorrect, as it only determines r, not P.
• Either statement alone is sufficient: Incorrect for the reasons above.
• Both statements not sufficient: Incorrect, because together they clearly determine P.

Common Pitfalls:
Many test takers mistakenly stop when they find r from Statement II and assume this is enough to answer the question, forgetting that the question asks for the principal. Others try to compute P from Statement I alone and ignore that r is unknown. In data sufficiency, always check whether the variables of interest can be uniquely determined from the information provided.

Final Answer:
The correct choice is Both Statement I and Statement II are necessary to determine the principal.