What is the principal sum of money if the following simple interest information is given: I. The sum amounts to Rs. 690 in 3 years at simple interest. II. The sum amounts to Rs. 750 in 5 years at simple interest. III. The rate of interest is 5% per annum?

Introduction / Context:
This is a data sufficiency type question involving simple interest. Instead of calculating a direct numerical answer, you must decide which combinations of the given statements are sufficient to determine the principal sum of money uniquely.

Given Data / Assumptions:

• The interest is simple interest in all statements.
• I: The sum amounts to Rs. 690 in 3 years.
• II: The sum amounts to Rs. 750 in 5 years.
• III: The rate of interest is 5% per annum.
• The principal sum is the same in all statements.

Concept / Approach:
For simple interest, the amount is given by A = P * (1 + R * T / 100), where P is principal, R is annual rate, and T is time in years. You must check, for each combination of statements, whether P can be uniquely determined. If more than one combination works, examine the pattern to see what the question is really asking.

Step-by-Step Solution:
Step 1: Using I and III: From I, 690 = P * (1 + R * 3 / 100). From III, R = 5. Step 2: Substitute R = 5. So 690 = P * (1 + 15 / 100) = P * 1.15, giving P = 690 / 1.15 = Rs. 600. Step 3: Using II and III: From II, 750 = P * (1 + R * 5 / 100). With R = 5, 750 = P * (1 + 25 / 100) = P * 1.25. Step 4: Solve for P: P = 750 / 1.25 = Rs. 600, again giving a unique principal. Step 5: Using I and II only, without III: From I, 690 = P * (1 + 3R / 100); from II, 750 = P * (1 + 5R / 100). Step 6: These are two equations in two unknowns P and R. Solving them simultaneously will give a unique value for P and a unique value for R, so I and II together are also sufficient.

Verification / Alternative check:
If you actually solve the system from I and II, you find that the yearly increase in amount is 750 minus 690 divided by 2, so the annual interest is Rs. 30. That implies P = 600, and from that you can compute R. This confirms that any pair of statements allows you to determine the principal uniquely.

Why Other Options Are Wrong:
I and III only is sufficient but not the only sufficient pair, so it does not capture the full situation. II and III only is also sufficient alone, but again it ignores the fact that I and II together are enough. I and II only is sufficient, yet the question asks which description correctly summarizes the sufficiency pattern of all three statements.

Common Pitfalls:
Students sometimes think that the rate must always be given explicitly, so they discard combinations that do not include the rate statement. Others misread the question and try to compute the actual principal instead of focusing on which information set is sufficient. It is important to recognize that two independent equations in two unknowns are enough to determine those unknowns uniquely.

Final Answer:
Any two of the three statements are sufficient to determine the principal, so the correct option is Any two of the three.