A certain sum was invested at simple interest for 3 years at a certain rate of interest. Had it been invested at a rate 2% per annum higher, it would have earned Rs. 360 more as interest. What is the sum invested (in rupees)?

Introduction / Context:This question is a typical example where the change in simple interest is given when the rate changes, and we must work backward to find the principal. Because the time period is the same in both cases, the extra interest is directly proportional to the increase in rate and the principal.Given Data / Assumptions:

• Time (T) = 3 years
• Increase in rate = 2% per annum
• Extra simple interest due to the higher rate = Rs. 360
• We must find the principal (P)

Concept / Approach:Let the original rate be r%. At the higher rate, the rate becomes r + 2 percent. The extra interest only comes from the extra 2% per annum for 3 years on the same principal. The formula for extra interest is:Extra SI = (P * Increase in rate * Time) / 100We can substitute the known values and solve for P.Step-by-Step Solution:Increase in rate = 2%Time = 3 yearsExtra SI = Rs. 360Using Extra SI formula:360 = (P * 2 * 3) / 100360 = (6P) / 100Multiply both sides by 100: 36000 = 6PP = 36000 / 6 = 6000Therefore, the required principal is Rs. 6,000Verification / Alternative check:At the increased rate, extra interest per year is P * 2 / 100. For P = 6000, extra interest per year is 6000 * 2 / 100 = Rs. 120. Over 3 years, extra interest is 120 * 3 = Rs. 360, which matches the given value. This confirms that our principal is correct.Why Other Options Are Wrong:If P were Rs. 4,000, the extra interest over 3 years would be (4000 * 2 * 3) / 100 = Rs. 240. For Rs. 5,000 it would be Rs. 300, and for Rs. 9,000 it would be Rs. 540. None of these match Rs. 360, so these options are not correct.Common Pitfalls:Some students mistakenly apply the full rate instead of the increase in rate when using the formula. Others incorrectly divide by time or forget the factor of 100 in the denominator. It is important to clearly distinguish between the original interest and the extra interest caused by the rate change.Final Answer:The sum invested is Rs. 6,000.