The simple interest on a sum of money is Rs. 50. What is the sum? I. The interest rate is 10% per annum. II. The sum earns this simple interest in 10 years.

Introduction / Context:
This is a data sufficiency question involving simple interest. The simple interest is given, and you must decide what combination of additional information is necessary and sufficient to determine the principal sum. Instead of computing a numerical answer directly, the focus is on understanding what data is required.

Given Data / Assumptions:

• Simple interest (SI) = Rs. 50.
• Statement I: The interest rate is 10% per annum.
• Statement II: The sum earns this simple interest in 10 years.
• The principal (P) is constant and earns simple interest.
• Time is expressed in years and interest is simple, not compound.

Concept / Approach:
The simple interest formula is SI = (P * R * T) / 100. To find P uniquely, you need to know both R (rate) and T (time) along with SI. The question asks which statements, alone or together, provide enough information to compute P without ambiguity.

Step-by-Step Solution:
Step 1: Start from SI = (P * R * T) / 100 with SI = 50. Step 2: Consider Statement I alone: R = 10%, but T is unknown. Step 3: With SI = 50 and R = 10, the formula becomes 50 = (P * 10 * T) / 100 = (P * T) / 10. Step 4: This gives P * T = 500, which has infinitely many combinations of P and T, so P is not uniquely determined with I alone. Step 5: Consider Statement II alone: T = 10 years, but R is unknown. Step 6: With SI = 50 and T = 10, the formula becomes 50 = (P * R * 10) / 100 = (P * R) / 10. Step 7: This implies P * R = 500, again allowing infinitely many combinations of P and R, so P is not uniquely determined with II alone. Step 8: Consider I and II together: R = 10% and T = 10 years, with SI = 50. Step 9: Substitute into SI formula: 50 = (P * 10 * 10) / 100 = (P * 100) / 100 = P. Step 10: Thus P = 50, and the principal sum is uniquely determined when both statements are used together.

Verification / Alternative check:
Check the calculation: With principal Rs. 50, rate 10% per annum, and time 10 years, SI = (50 * 10 * 10) / 100 = 50, which matches the given interest. This confirms that both statements together are sufficient and necessary.

Why Other Options Are Wrong:
Statement I alone does not fix T, so P cannot be uniquely found. Statement II alone does not fix R, so P cannot be uniquely found. Either I or II alone being sufficient is incorrect because each leaves one essential variable undetermined.

Common Pitfalls:
Students sometimes assume a default value for time or rate without being given, which leads to incorrect conclusions about sufficiency. Others mistakenly think that knowing SI and either R or T is enough. For simple interest, both R and T must be known, along with SI, to determine P uniquely.

Final Answer:
Both statements I and II are needed to determine the sum, so the correct choice is Both I and II are necessary to answer.