The simple interest on a certain sum of money for 20 years is equal to 5/4 of the principal. What is the rate of simple interest per annum (in percentage)?

Introduction / Context:
This question deals with simple interest over a long period and asks for the annual rate when the total interest is a certain fraction of the principal. It checks your understanding of the simple interest formula and how to manipulate it algebraically to solve for the rate.

Given Data / Assumptions:
- Time period = 20 years. - Let the principal (initial amount) be P. - Simple interest earned over 20 years = 5/4 of the principal = (5/4) * P. - We need to find the annual simple interest rate r in percent.

Concept / Approach:
The formula for simple interest is:
Simple interest = (P * r * t) / 100, where P is principal, r is rate per annum in percent, and t is time in years. We are told that the simple interest equals (5/4) * P over 20 years. We set the formula equal to this amount and solve for r. The principal P cancels out, leaving a simple equation in r.

Step-by-Step Solution:
Step 1: Write the simple interest formula: SI = (P * r * t) / 100. Step 2: Substitute t = 20 years, so SI = (P * r * 20) / 100. Step 3: We are given that simple interest over 20 years equals (5/4) * P. Step 4: Set the two expressions equal: (P * r * 20) / 100 = (5/4) * P. Step 5: Cancel P from both sides since P is non zero. Step 6: This gives (r * 20) / 100 = 5/4. Step 7: Simplify left side: (20 / 100) * r = (1 / 5) * r. Step 8: So we have (1 / 5) * r = 5/4. Step 9: Multiply both sides by 5: r = 5 * (5/4) = 25/4. Step 10: Convert to decimal or mixed number: 25/4 = 6.25. Step 11: Therefore, the rate of simple interest per annum is 6.25%.

Verification / Alternative check:
Take P = Rs. 100 for a simple check. If r = 6.25% and t = 20 years, then SI = (100 * 6.25 * 20) / 100 = 6.25 * 20 = 125. For P = 100, (5/4) * P = 125. Both match, confirming that the rate 6.25% is correct.

Why Other Options Are Wrong:
Option B (12.5): Would give SI = (P * 12.5 * 20) / 100 = 2.5P, which is more than (5/4) * P. Option C (2.5): Would give SI = (P * 2.5 * 20) / 100 = 0.5P, less than the required fraction. Option D (5): Would give SI = (P * 5 * 20) / 100 = P, not (5/4) * P. Option E (7.5): Would give SI = (P * 7.5 * 20) / 100 = 1.5P, again not equal to (5/4) * P.

Common Pitfalls:
Common errors include forgetting to divide by 100 in the interest formula or misinterpreting 5/4 of the principal as 4/5. Some students also incorrectly keep the principal P and complicate the algebra unnecessarily. Remember that when both sides of an equation are proportional to P, it can be cancelled safely, simplifying the solving process for r.

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