For a certain sum of money, what is the rate of interest per cent per annum? (I) The amount becomes double in 5 years under simple interest. (II) The difference between compound interest and simple interest on a certain amount for 2 years is Rs 400. (III) The simple interest earned per year is Rs 2000. Which information is sufficient to determine the rate of interest p.a.?

Introduction:
This is a data sufficiency style question about simple and compound interest. The goal is not to compute the exact rate numerically, but to decide which pieces of information are sufficient to uniquely determine the rate of interest per annum for a given sum.

Given Data / Assumptions:

• Statement I: An amount doubles itself in 5 years on simple interest.
• Statement II: The difference between compound interest and simple interest on a certain amount in 2 years is Rs 400.
• Statement III: Simple interest earned per annum is Rs 2000.
• The rate of interest is constant and is expressed per annum.

Concept / Approach:
For simple interest, A = P * (1 + r * t / 100). If an amount doubles in t years, then 1 + r * t / 100 = 2. For the difference between compound and simple interest over 2 years, the formula is Difference = P * (r/100)^2. Knowing SI per year also gives P * r / 100. We check which statements alone or in combination give enough equations to solve for r uniquely.

Step-by-Step Solution:
Using Statement I: 2P = P * (1 + r * 5 / 100) so 1 + 5r/100 = 2Thus 5r/100 = 1, so r = 20%. Hence I alone is sufficient.Using Statement II alone: Difference = P * (r/100)^2 = 400, which contains two unknowns P and r, so II alone is not sufficient.Using Statement III alone: SI per year = P * r / 100 = 2000, again two unknowns P and r, so III alone is not sufficient.Combining II and III: From III, P * r / 100 = 2000.From II, P * (r/100)^2 = 400.Dividing the second equation by the first, we get (P * r^2 / 10000) / (P * r / 100) = 400 / 2000.This simplifies to r / 100 = 1/5, so r = 20%. Hence II and III together are sufficient.

Verification / Alternative Check:
Both approaches, via Statement I alone and via Statements II and III together, give the same unique rate of 20% per annum. This confirms that I alone is sufficient and also that II and III, when used together, are sufficient.

Why Other Options Are Wrong:
Option I only: misses the fact that II and III together are also sufficient.Option II and III only: ignores that I alone is already enough.Option All I, II and III: overstates the requirement; you do not need all three.Option None of these: incorrect because sufficient information clearly exists in I and also in II with III.

Common Pitfalls:
Students may wrongly try to plug numbers into each statement without focusing on sufficiency. Others may treat II or III alone as sufficient, forgetting that both involve two unknowns. It is important to translate each statement into equations, count unknowns and see when you have enough independent equations to solve for the rate.

Final Answer:
The rate of interest per annum can be determined from Statement I alone or from Statements II and III together, so the correct choice is “I only or II and III only”.