Data Sufficiency – Principal Sum from Simple Interest and Doubling: What is the principal sum (in ₹) that earned the interest? Statement A: The total simple interest is ₹9,000 after 9 years. Statement B: The total of principal and simple interest becomes double of the principal after 6 years at the same simple interest rate.

Introduction / Context:
This data sufficiency question is similar in structure to earlier ones: you must decide whether each statement alone or both together are sufficient to determine the principal sum on which simple interest is calculated. The problem combines information about the total simple interest over 9 years with an amount doubling condition over 6 years at the same rate.

Given Data / Assumptions:

• Simple interest at a constant annual rate r%.
• Principal = P (unknown).
• Statement A: Total SI = ₹9,000 after 9 years.
• Statement B: After 6 years, the amount (principal + SI) becomes double of the principal.

Concept / Approach:
Simple interest follows I = (P * r * t) / 100. From Statement A we get one equation linking P and r. From Statement B we get another equation for the same P and r based on the doubling condition. The key is to see whether these two equations together are sufficient to solve uniquely for P, while each alone is not.

Step-by-Step Solution:
Step 1: Analyse Statement A. It gives I = 9,000 after t = 9 years. Step 2: Apply SI formula: 9,000 = (P * r * 9) / 100. Step 3: Rearranging, P * r = (9,000 * 100) / 9 = 100,000. Step 4: This is one equation with two unknowns (P and r), so P cannot be determined uniquely from Statement A alone. Step 5: Analyse Statement B. Amount doubles in 6 years, so P + I = 2P, which gives I = P. Step 6: Under SI for 6 years, I = (P * r * 6) / 100 = P. Step 7: Cancelling P, r * 6 / 100 = 1, so r = 100 / 6 ≈ 16.67% per annum. Step 8: Statement B alone determines r but not P, so P is still unknown. Step 9: Combine both statements. From A, P * r = 100,000, and from B, r = 100 / 6. Step 10: Substitute r into P * r = 100,000: P * (100 / 6) = 100,000. Step 11: Therefore, P = 100,000 * (6 / 100) = 6,000. Step 12: Thus, only by using both Statement A and Statement B together can we uniquely find the principal.

Verification / Alternative check:
With P = ₹6,000 and r = 100 / 6% per annum, check Statement A: SI in 9 years = (6,000 * (100 / 6) * 9) / 100 = 6,000 * (9 / 6) = 6,000 * 1.5 = ₹9,000, which matches. For Statement B, SI in 6 years = (6,000 * (100 / 6) * 6) / 100 = 6,000, so amount becomes P + I = 6,000 + 6,000 = 12,000 = 2P, so doubling holds. Both statements are consistent with these values.

Why Other Options Are Wrong:

• Only Statement A is sufficient: Incorrect, as A alone leaves two unknowns (P and r).
• Only Statement B is sufficient: Incorrect, as B alone gives r but not P.
• Either statement alone is sufficient: Incorrect; each alone is insufficient.
• Neither statement is sufficient: Incorrect, because together they yield a unique P.

Common Pitfalls:
Test takers sometimes stop after finding r from Statement B and incorrectly claim B is sufficient, forgetting that the question asks for the principal. Others misinterpret the doubling condition and apply it to 9 years instead of 6. Always connect each statement correctly to the formula and check whether all required variables can be uniquely found.

Final Answer:
The correct choice is Both Statement A and Statement B are sufficient together to determine the principal sum.