In a simple interest setup on the same principal amount: If increasing the rate of simple interest by 2 percentage points increases the interest by ₹108, and increasing the time period by 2 years increases the interest by ₹180, which of the following statements about the principal (sum) is correct?

Introduction:
This question checks whether the given changes in rate and time provide enough information to uniquely determine the principal under simple interest. Simple interest depends on three variables: principal P, rate r, and time t. If the problem only gives information about how interest changes when r or t changes, we must see if those constraints uniquely identify P, or if multiple combinations of P, r, and t can satisfy them.

Given Data / Assumptions:

• Original principal = P
• Original rate = r% per annum
• Original time = t years
• Increase in rate by 2% increases interest by ₹108
• Increase in time by 2 years increases interest by ₹180
• Formula: SI = (P * r * t) / 100

Concept / Approach:
Translate each “increase” into an equation. A 2% increase in rate changes SI by (P * 2 * t)/100. A 2-year increase in time changes SI by (P * r * 2)/100. These produce relationships involving P*t and P*r, but do not directly give P alone unless we can also relate r and t uniquely.

Step-by-Step Solution:
Rate increase condition: (P * 2 * t) / 100 = 108 So P * t = 108 * 100 / 2 = 5400 Time increase condition: (P * r * 2) / 100 = 180 So P * r = 180 * 100 / 2 = 9000 We now know P*t and P*r, but we do not know r or t individually. Many pairs (r, t) can satisfy these while giving different P values.

Verification / Alternative check:
Example 1: If t = 6 years, then P = 5400/6 = 900, and r = 9000/900 = 10%. Example 2: If t = 9 years, then P = 5400/9 = 600, and r = 9000/600 = 15%. Both satisfy the given increases, but P is different. Hence P is not uniquely determined.

Why Other Options Are Wrong:
₹540, ₹415, ₹404, and ₹450 are specific values, but the information allows multiple valid principals, so no single number can be confirmed as the only answer.

Common Pitfalls:
Assuming the original rate or time is implicitly known, or trying to compute SI without enough independent equations. Another mistake is to treat “increase by 2%” as “multiply by 1.02” (that would be a different interpretation than 2 percentage points).

Final Answer:
Data is not sufficient to uniquely determine the principal.