A person borrows money for 8 years at a simple interest rate. If the ratio of the principal to the total interest earned in 8 years is 5 : 8, what is the annual simple interest rate (in % per annum)?

Introduction:
This question tests converting a principal-to-interest ratio into a simple interest rate. Under simple interest, total interest over t years is SI = (P * r * t)/100, so the fraction SI/P equals (r*t)/100. A given ratio between P and SI immediately fixes SI/P, and therefore fixes r when time is known.

Given Data / Assumptions:

• Principal : total interest in 8 years = 5 : 8
• Time t = 8 years
• Simple interest formula: SI = (P * r * t) / 100
• Thus SI/P = (r * t) / 100

Concept / Approach:
From P:SI = 5:8, compute SI/P = 8/5. Then set (r*8)/100 = 8/5 and solve for r. Since 8 appears on both sides, it cancels neatly, making the computation quick and reliable.

Step-by-Step Solution:
Given P : SI = 5 : 8 So SI/P = 8/5 But SI/P = (r * t) / 100 = (r * 8) / 100 (r * 8) / 100 = 8/5 Cancel 8: r / 100 = 1/5 r = 100/5 = 20

Verification / Alternative check:
If r = 20% for 8 years, then SI/P = (20*8)/100 = 160/100 = 1.6 = 8/5, which matches the ratio exactly. So the annual rate is consistent.

Why Other Options Are Wrong:
10% gives SI/P = 0.8 (ratio 5:4). 25% gives SI/P = 2 (ratio 5:10). 30% gives 2.4 (ratio 5:12). Only 20% gives SI/P = 1.6 (ratio 5:8).

Common Pitfalls:
Inverting the ratio, using yearly interest instead of total 8-year interest, or forgetting the division by 100 when converting percent to a fraction.

Final Answer:
The annual simple interest rate is 20% per annum.