The compound interest on a certain principal at a rate of 16 2/3% per annum for 3 years is Rs 1270. What is the simple interest on the same principal at the same rate of 16 2/3% per annum for the same period of 3 years?

Introduction / Context:
This question compares compound interest and simple interest on the same principal, at the same rate of interest, and for the same time period. The rate given, 16 2/3%, is a fractional percentage that is often used in exam problems because it simplifies to a neat fraction. We are given the compound interest and must work backwards to find the principal and then compute the simple interest for 3 years.

Given Data / Assumptions:

• Rate of interest r = 16 2/3% per annum, which is equal to 1 / 6 as a fraction.
• Time period t = 3 years.
• Compound interest for 3 years = Rs 1270.
• Interest is compounded annually.
• We must find simple interest for 3 years on the same principal and at the same rate.

Concept / Approach:
For compound interest, the amount after t years is:
A = P * (1 + r / 100)^t For r = 16 2/3%, we write r / 100 = 1 / 6, so the factor is (7 / 6). For t = 3 years:
A = P * (7 / 6)^3 = P * (343 / 216) The compound interest CI is A - P. Once we know P from CI, the simple interest SI is:
SI = (P * r * t) / 100

Step-by-Step Solution:
Let P be the principal. Amount after 3 years: A = P * (7 / 6)^3 = P * (343 / 216). Compound interest CI = A - P = P * (343 / 216 - 1) = P * (127 / 216). We are given CI = 1270, so P * (127 / 216) = 1270. Therefore P = 1270 * 216 / 127 = 10 * 216 = Rs 2160. Now compute simple interest for 3 years at 16 2/3%. SI = P * r * t / 100 = 2160 * (50 / 3) * 3 / 100. Here 16 2/3% = 50 / 3 percent. SI = 2160 * 50 / 100 = 2160 * 1 / 2 = Rs 1080.

Verification / Alternative Check:
Note that at 16 2/3% for 3 years, simple interest is (r * t) / 100 = 50 percent of P. Since P = 2160, 50 percent is indeed 1080, which matches our computed SI.

Why Other Options Are Wrong:
1090 and 1109 are close but do not match the exact percentage calculation. 1180 is higher than the correct 50 percent of 2160, so it cannot be right.

Common Pitfalls:
Misreading 16 2/3% as 16.23% leads to completely wrong answers. Some students attempt to find simple interest directly from compound interest without first determining the principal. Errors can also arise from incorrect fraction arithmetic when using (7 / 6)^3.

Final Answer:
The correct simple interest for 3 years at 16 2/3% per annum is 1080.