A person borrows some money for 4 years at a rate of simple interest. If the ratio of the principal to the total interest earned in 4 years is 5 : 1, what is the annual simple interest rate (in percent)?

Introduction / Context:
Ratio-based interest questions are designed to test whether you can convert a relationship between principal and interest into the usual simple interest formula. In this problem, instead of giving explicit rupee values, the question states that the ratio of principal to total interest after 4 years is 5 : 1. This means that the interest is one-fifth of the principal. Using the simple interest formula and this ratio, we can determine the annual rate of interest in percent per annum.

Given Data / Assumptions:

• Time period T = 4 years.
• Ratio of principal to total interest (P : I) = 5 : 1.
• Interest is calculated using simple interest.
• Rate of interest R is constant throughout the 4 years.

Concept / Approach:
For simple interest, we use SI = P * R * T / 100. The ratio P : I = 5 : 1 implies I / P = 1 / 5. Therefore, we can write I = P / 5. But from the formula, I = P * R * T / 100. By equating P * R * T / 100 to P / 5, we can cancel P on both sides and solve for R. This gives a straightforward way to find the rate without needing actual numerical values for principal or interest.

Step-by-Step Solution:
Step 1: Let the principal be P rupees. Step 2: According to the ratio P : I = 5 : 1, the interest I = P / 5. Step 3: Simple interest formula: I = P * R * T / 100. Step 4: Substitute T = 4 years and I = P / 5 into the formula. Step 5: P / 5 = P * R * 4 / 100. Step 6: Cancel P from both sides (P is non-zero): 1 / 5 = R * 4 / 100. Step 7: Multiply both sides by 100 to clear the denominator: 100 / 5 = 4R. Step 8: 100 / 5 = 20, so 20 = 4R. Step 9: Therefore, R = 20 / 4 = 5. Step 10: The annual simple interest rate is 5% per annum.

Verification / Alternative check:
Assume a convenient principal to check. Let P = Rs. 500 (since the ratio suggests multiples of 5). If R = 5% and T = 4 years, then I = 500 * 5 * 4 / 100 = 500 * 20 / 100 = 100 rupees. Now P : I = 500 : 100, which simplifies to 5 : 1, exactly matching the given ratio. This confirms that an annual rate of 5% is consistent with the given relationship between principal and interest.

Why Other Options Are Wrong:
If R = 25%, then I = P * 25 * 4 / 100 = P. In that case, P : I would be 1 : 1, not 5 : 1. If R = 10%, then I = P * 10 * 4 / 100 = 0.4P, so P : I would be 5 : 2, not 5 : 1. If R = 20%, then I = P * 20 * 4 / 100 = 0.8P, giving a ratio of P : I = 5 : 4. Only R = 5% gives P : I = 5 : 1.

Common Pitfalls:
A frequent mistake is misinterpreting the ratio and taking I : P as 5 : 1 instead of P : I. Another error is substituting the ratio incorrectly into the formula or forgetting to cancel P from both sides. Sometimes students also plug in random values for P and I without maintaining the given ratio. Always express the ratio as a precise algebraic relationship, then solve using the standard simple interest formula.

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