An amount of $720 is invested for 36 months and earns $205.20 in simple interest; what is the corresponding yearly simple interest rate in percent per annum for this investment?

Introduction / Context:
This question gives the principal, the total simple interest earned, and the time expressed in months. The task is to determine the annual simple interest rate. It is a practical problem similar to real fixed deposit or savings account calculations, and it tests conversion of months to years and correct use of the simple interest formula for rate.

Given Data / Assumptions:
- Principal P = 720 dollars. - Simple interest SI = 205.20 dollars. - Time period = 36 months. - Rate of simple interest R percent per annum is unknown. - Interest is calculated according to the simple interest formula.

Concept / Approach:
First convert time from months to years because the rate is per annum. Since 12 months equal 1 year, 36 months correspond to 36 / 12 = 3 years. Then apply SI = (P * R * T) / 100 and rearrange to R = (SI * 100) / (P * T). Substituting the values of SI, P, and T gives the annual rate in percent per annum.

Step-by-Step Solution:
Step 1: Convert 36 months to years: T = 36 / 12 = 3 years. Step 2: Identify P = 720 dollars, SI = 205.20 dollars, T = 3 years. Step 3: Use the formula SI = (P * R * T) / 100. Step 4: Rearrange for R: R = (SI * 100) / (P * T). Step 5: Substitute SI = 205.20, P = 720, T = 3. Step 6: Compute denominator P * T = 720 * 3 = 2,160. Step 7: Compute numerator SI * 100 = 205.20 * 100 = 20,520. Step 8: Divide: R = 20,520 / 2,160. Step 9: Calculate 20,520 / 2,160 = 9.5. Step 10: Therefore, the annual simple interest rate is 9.5 percent per annum.

Verification / Alternative check:
Verify by recalculating SI with R = 9.5 percent and T = 3 years. SI = (720 * 9.5 * 3) / 100. First 720 * 9.5 = 6,840, then multiply by 3 to get 20,520, and divide by 100 to get 205.20 dollars of interest. This matches the given interest amount, confirming that the rate of 9.5 percent is correct.

Why Other Options Are Wrong:
A rate of 7.5 percent would give SI = (720 * 7.5 * 3) / 100 = 162 dollars, which is too small. At 8.5 percent, SI is 183.60 dollars, still less than 205.20. At 10.5 percent, SI becomes 226.80 dollars, which is too large. Only 9.5 percent produces exactly 205.20 dollars of interest over 3 years on a principal of 720 dollars.

Common Pitfalls:
Some learners forget to convert months to years and incorrectly set T = 36, which leads to an extremely small and unrealistic rate. Others may misplace the principal and interest in the rate formula. Carefully converting time units and writing R = SI * 100 / (P * T) helps avoid these errors.

Final Answer:
The yearly simple interest rate is 9.5% per annum.