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?

Introduction / Context:
This question asks for the rate of simple interest when the total interest over a certain time period is given as a fraction of the principal. Instead of providing a numerical interest value, the question states that simple interest for 4 years equals one fifth of the principal. This kind of relationship can be handled directly using the simple interest formula and is a common pattern in aptitude exams.

Given Data / Assumptions:
Let the principal be P rupees.
Simple interest over 4 years is equal to P divided by 5, that is P / 5.
Type of interest is simple interest, not compound interest.
Rate of interest r is constant and is to be found.
Time period t is 4 years.

Concept / Approach:
For simple interest, I = P * r * t / 100. Here I is given in terms of P as P / 5. Substituting this into the simple interest formula leads to P / 5 = P * r * 4 / 100. Because P is non zero, it cancels from both sides, leaving an equation involving only r. Solving this equation yields the annual interest rate. This method uses proportional reasoning and basic algebra.

Step-by-Step Solution:
Let principal be P rupees.Simple interest over 4 years is I = P / 5.From the simple interest formula, I = P * r * 4 / 100.Set these equal: P / 5 = P * r * 4 / 100.Cancel P from both sides to get 1 / 5 = r * 4 / 100.Multiply both sides by 100 to remove the denominator: 100 / 5 = 4r.Compute 100 / 5 = 20, so 20 = 4r.Divide both sides by 4 to find r: r = 20 / 4 = 5 percent per annum.Hence, the rate of simple interest per annum is 5 percent.

Verification / Alternative check:
Take a convenient principal value, such as P = 100 rupees, to check. One fifth of 100 is 20, so simple interest for 4 years should be 20 rupees. Using r = 5 percent, interest for 4 years is I = 100 * 5 * 4 / 100 = 100 * 20 / 100 = 20 rupees, which matches P / 5. This confirms that a rate of 5 percent per annum fits the given condition.

Why Other Options Are Wrong:
At 4 percent per annum, simple interest for 4 years would be P * 4 * 4 / 100 = 0.16P, which is 16 percent of the principal, not one fifth.
At 6 percent, the interest would be 24 percent of the principal, more than one fifth.
Inadequate data is incorrect because the given relation is enough to uniquely determine the rate.
At 8 percent, interest would be 32 percent of the principal, again not matching one fifth. Only 5 percent gives exactly one fifth of P after 4 years.

Common Pitfalls:
Some learners mishandle the fraction P / 5 and forget to cancel P on both sides of the equation. Others incorrectly interpret one fifth of principal as 5 percent directly, which would only be correct if the time were one year. It is important to use the full simple interest formula with the correct time and then solve for the rate systematically.

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