A person borrows some money for 8 years at a rate of simple interest. 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 percentage)?

Introduction / Context:
This question again uses a ratio-based relationship between principal and interest, but here the interest exceeds the principal, because the ratio P : I is 5 : 8. That means the interest is greater than the principal itself, which happens when the product of rate and time is more than 100. The problem asks you to translate this ratio into the annual simple interest rate over 8 years by correctly applying the simple interest formula and solving a simple algebraic equation.

Given Data / Assumptions:

• Time T = 8 years.
• Ratio of principal to total interest P : I = 5 : 8.
• Interest is calculated using simple interest.
• Rate of interest is constant over the entire 8-year period.

Concept / Approach:
Simple interest is given by I = P * R * T / 100. From the ratio P : I = 5 : 8, we can write I / P = 8 / 5 = 1.6. That means for every rupee of principal, the interest over 8 years is 1.6 rupees. Substituting I / P = R * T / 100 into this ratio equation allows us to solve for R. Because T is known to be 8 years, we get R * 8 / 100 = 1.6 and therefore can directly compute the annual rate R.

Step-by-Step Solution:
Step 1: Let principal be P rupees. Step 2: From the ratio P : I = 5 : 8, we have I / P = 8 / 5. Step 3: For simple interest, I = P * R * T / 100, so I / P = R * T / 100. Step 4: Therefore R * T / 100 = 8 / 5. Step 5: Substitute T = 8 years: R * 8 / 100 = 8 / 5. Step 6: Multiply both sides by 100: 8R = 100 * 8 / 5. Step 7: Compute the right side: 100 * 8 / 5 = 800 / 5 = 160. Step 8: Thus 8R = 160, so R = 160 / 8 = 20. Step 9: The annual rate of simple interest is 20% per annum.

Verification / Alternative check:
To verify, choose a convenient principal such as P = Rs. 500. With R = 20% and T = 8 years, total interest I = 500 * 20 * 8 / 100 = 500 * 160 / 100 = 800. The ratio P : I becomes 500 : 800, which simplifies to 5 : 8, exactly as given in the question. This confirms that an annual rate of 20% is consistent with the described relationship between principal and interest.

Why Other Options Are Wrong:
If R = 10%, then I / P = 10 * 8 / 100 = 0.8, giving a ratio P : I of 5 : 4, not 5 : 8. If R = 25%, then I / P = 25 * 8 / 100 = 2, which implies P : I = 1 : 2. If R = 30%, then I / P = 30 * 8 / 100 = 2.4, giving P : I = 5 : 12, all of which differ from 5 : 8. Only 20% yields the correct ratio.

Common Pitfalls:
Students sometimes invert the ratio or confuse P : I with I : P. Another mistake is to forget to divide by 100 when using the simple interest formula, leading to very large or very small values for R. Some may also mis-handle the fraction 8 / 5, approximating it incorrectly. To avoid such errors, always express the ratio precisely, substitute carefully into I / P = R * T / 100, and solve systematically.

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