If the compound interest on a certain sum at a rate of 16⅔% per annum (that is, 16 2/3 percent per year) for 3 years is Rs. 1,270, then what will be the simple interest on the same sum at the same rate for the same period of 3 years?

Introduction:
Here we are given compound interest for 3 years at a fractional rate of 16⅔% per annum and are asked to find the simple interest for the same principal, same rate, and same time. The problem tests skills in handling fractional percentages and in moving between compound interest and simple interest calculations.

Given Data / Assumptions:
Compound interest CI for 3 years = Rs. 1,270. Rate r = 16⅔% per annum = 16.666...%. Time t = 3 years. The same rate and time apply for computing simple interest.

Concept / Approach:
First, convert 16⅔% to a fraction: 16⅔% = 50/3 percent, which is 1/6 as a yearly fraction. For compound interest, the amount after 3 years is: A = P * (1 + 1/6)^3 = P * (7/6)^3 = P * 343/216. So CI = A − P = P * (343/216 − 1) = P * (127/216). Given CI, we can solve for P. Then, simple interest for 3 years at rate 1/6 per year equals P * (1/6) * 3 = P/2.

Step-by-Step Solution:
Step 1: Use CI expression. CI = P * 127/216 = 1,270. P = 1270 * 216 / 127. 1270 / 127 = 10, so P = 10 * 216 = Rs. 2,160. Step 2: Compute simple interest for 3 years. Rate as fraction per year = 1/6. SI = P * (1/6) * 3 = P * 3/6 = P/2. SI = 2160 / 2 = Rs. 1,080.

Verification / Alternative check:
We can quickly check the CI expression with P = 2,160: CI = 2160 * 127 / 216 = 10 * 127 = Rs. 1,270, which matches the given CI. This confirms that the principal is correct, and therefore the simple interest calculation P/2 is also reliable.

Why Other Options Are Wrong:
Rs. 1,090, Rs. 1,180, Rs. 1,109, and Rs. 980 all arise from computational errors such as misreading the fractional rate, using an incorrect exponent, or not simplifying P * 3/6 properly. None of them equal P/2 with P = 2,160.

Common Pitfalls:
Students often misinterpret 16⅔% or treat it as 16.23%. Others may forget that the given Rs. 1,270 is compound interest rather than the amount. Careful fraction handling and stepwise substitution help avoid these mistakes.

Final Answer:
The simple interest on the same sum for 3 years at the same rate is Rs. 1,080.