The simple interest accrued on a sum of money after 4 years is equal to one fifth of its principal. Assuming simple interest at a constant rate, what is the rate of interest per annum in percentage terms?

Introduction / Context:
This question asks for the rate of simple interest when the simple interest over a fixed time is expressed as a fraction of the principal. The simple interest for 4 years is given to be one fifth of the principal. Under simple interest, interest is directly proportional to both rate and time. By equating the expression for simple interest to one fifth of the principal, we can solve for the rate. This type of question is common in conceptual tests of simple interest where fractional relationships are used instead of actual rupee values.

Given Data / Assumptions:

• Simple interest for 4 years = one fifth of the principal.
• Let principal be P rupees.
• Time T = 4 years.
• Rate of interest per annum = R% (unknown).
• Interest is calculated using simple interest only.

Concept / Approach:
The simple interest formula is:
SI = (P * R * T) / 100 The problem states:
SI = P / 5 Substituting the simple interest formula for SI and equating to P / 5, we can cancel P from both sides and solve the resulting equation for R. Because the principal cancels out, the answer does not depend on the actual rupee amount invested, only on the ratio of interest to principal and the time period.

Step-by-Step Solution:
Let the principal be P rupees. Given that simple interest for 4 years is P / 5. Using SI formula: SI = (P * R * 4) / 100. Set this equal to P / 5: (P * R * 4) / 100 = P / 5. Cancel P from both sides, since P is nonzero. (R * 4) / 100 = 1 / 5. 4R / 100 = 1 / 5. Cross multiply: 4R * 5 = 100 * 1. 20R = 100. R = 100 / 20 = 5. Therefore, the rate of interest is 5% per annum.

Verification / Alternative check:
Assume any convenient principal, for example P = Rs 500. One fifth of P is 500 / 5 = Rs 100. At 5% per annum for 4 years, simple interest SI = (500 * 5 * 4) / 100 = 100, which matches one fifth of the principal. Trying any other principal will also show that 5% satisfies the given condition consistently, confirming the answer.

Why Other Options Are Wrong:
At 4% per annum, simple interest for 4 years would be (P * 4 * 4) / 100 = 16P / 100 = 4P / 25, which is less than P / 5.
At 6% per annum, simple interest becomes 24P / 100 = 6P / 25, which is larger than P / 5.
At 8% per annum, simple interest for 4 years is 32P / 100 = 8P / 25, again not equal to P / 5.
Thus the data is adequate, and 5% is the unique correct rate, making the option “Inadequate data” incorrect.

Common Pitfalls:
Some students forget to cancel the principal and try to assume a numerical value without simplifying the equation, which can lead to arithmetic errors. Others misinterpret “one fifth of its principal” and incorrectly set up the equation. It is also possible to miscalculate cross multiplication if steps are rushed. Carefully writing the equation and canceling common factors simplifies the problem greatly.

Final Answer:
The rate of interest per annum is 5%.