If the rate of simple interest increases by 2%, the interest on a certain sum of money increases by Rs. 108. If instead the time period is increased by 2 years, the simple interest on the same sum increases by Rs. 180. What is the sum of money involved?

Introduction / Context:
This question is about understanding how changes in rate and time affect simple interest. It asks you to determine the original sum of money, given how the interest changes when the rate is increased by 2% and when the time is increased by 2 years. However, it is also a data sufficiency style problem, and you must check whether the information provided is enough to uniquely determine the sum.

Given Data / Assumptions:

• Original principal = P (unknown).
• Original rate = R% per annum (unknown).
• Original time = T years (unknown).
• If the rate increases by 2%, the additional interest is Rs. 108.
• If the time increases by 2 years, the additional interest is Rs. 180.
• Simple interest formula SI = (P * R * T) / 100 is used.

Concept / Approach:
Let the original simple interest be SI = (P * R * T) / 100. A change in rate affects interest as proportional to P * T, while a change in time affects interest as proportional to P * R. We can create two equations from the changes and see whether the principal P can be uniquely determined from them.

Step-by-Step Solution:
Step 1: When rate increases by 2%, new interest increment = (P * 2 * T) / 100 = 108. Step 2: So (2 * P * T) / 100 = 108, which simplifies to P * T = 108 * 100 / 2 = 5400. Step 3: When time increases by 2 years, the additional interest = (P * R * 2) / 100 = 180. Step 4: So (2 * P * R) / 100 = 180, which simplifies to P * R = 180 * 100 / 2 = 9000. Step 5: Now you have two equations: P * T = 5400 and P * R = 9000. Step 6: There are three unknowns P, R, and T, but only two equations, so there is an infinite family of solutions. You cannot uniquely determine P.

Verification / Alternative check:
You can try to express R in terms of T: from P * T = 5400, P = 5400 / T. Substitute this into P * R = 9000 to get (5400 / T) * R = 9000, giving R = 9000 * T / 5400 = (5 * T) / 3. This shows that many combinations of R and T are possible, each leading to a different P. Hence, there is no unique principal.

Why Other Options Are Wrong:
Rs. 540, Rs. 415, and Rs. 404 are specific values, but no unique principal can be derived from the equations. Each of these values would correspond to some arbitrary choice of R and T that is not uniquely justified by the given conditions.

Common Pitfalls:
Many students mistakenly assume that because there are two pieces of information, they must be able to find a unique answer. Others try to arbitrarily fix either R or T without any justification. The correct reasoning is to count unknowns and independent equations and to check whether the system is solvable uniquely.

Final Answer:
The given information does not uniquely determine the sum of money, so the correct response is Data is not sufficient.