The simple interest on a sum of money for 20 years is equal to five fourths of the principal. What is the per annum rate of interest in percentage?

Introduction / Context:
This problem involves the basic formula for simple interest in quantitative aptitude. You are told that over 20 years, the simple interest earned on a principal amount is equal to five fourths of the principal itself. The task is to find the annual rate of interest in percentage terms. This type of question helps reinforce the relationship between principal, rate, time and interest in financial mathematics.

Given Data / Assumptions:
- Let the principal amount be P rupees.
- Time period t = 20 years.
- Simple interest over 20 years is equal to five fourths of the principal, that is SI = (5 / 4) * P.
- We need to find the rate of interest per annum r in percent.

Concept / Approach:
The simple interest formula is SI = (P * r * t) / 100, where P is principal, r is the annual rate in percent and t is the time in years. The problem directly gives SI as (5 / 4) * P. By equating the formula expression to this given value and simplifying, we can solve for r. Since P appears on both sides of the equation, it cancels out, making the calculation straightforward.

Step-by-Step Solution:
Step 1: Write the formula for simple interest: SI = (P * r * t) / 100.Step 2: Substitute t = 20 years into the formula: SI = (P * r * 20) / 100.Step 3: According to the question, SI is also equal to (5 / 4) * P.Step 4: Set the two expressions equal: (P * r * 20) / 100 = (5 / 4) * P.Step 5: Cancel P from both sides, assuming P is not zero.Step 6: This gives (r * 20) / 100 = 5 / 4.Step 7: Simplify left side: r * 20 / 100 = r / 5.Step 8: So r / 5 = 5 / 4.Step 9: Multiply both sides by 5: r = (5 / 4) * 5 = 25 / 4 = 6.25.Step 10: Therefore, the rate of interest per annum is 6.25 percent.

Verification / Alternative check:
Take a simple principal, for example P = 4 rupees for easy fractional calculation. If r = 6.25 percent and t = 20 years, then SI = (4 * 6.25 * 20) / 100 = (4 * 125) / 100 = 500 / 100 = 5 rupees. Five rupees is equal to five fourths of 4 rupees, so SI = (5 / 4) * P is satisfied. This numerical check confirms that the derived rate of 6.25 percent is consistent with the problem statement.

Why Other Options Are Wrong:
If r were 12.5, 2.5 or 5 percent, substituting into the formula SI = (P * r * 20) / 100 would not yield (5 / 4) * P. For example, with r = 5 percent, SI over 20 years would be (P * 5 * 20) / 100 = P, which is only equal to the principal, not five fourths of it. Only 6.25 percent produces the required interest to principal ratio.

Common Pitfalls:
Some students confuse simple interest with compound interest and attempt more complex formulas. Others misinterpret five fourths of the principal as a rate instead of an amount. Always read carefully whether the statement is about the interest amount or the rate. Here the interest amount over 20 years is compared directly to the principal, which fits perfectly into the simple interest formula.

Final Answer:
The rate of interest is 6.25 percent per annum.