Exact-Time Simple Interest on a Short-Term Loan: A loan of $15,000 is taken at 7% per annum simple interest on April 7 and is due exactly 7 months later on November 7. Using exact time in days (April 7 to November 7 = 214 days) and a 365-day year, what is the total amount to be repaid (in $)?

Introduction / Context:
This problem involves simple interest calculated using exact time in days and a 365 day year. You must compute the interest for a specific number of days and then add it to the principal to obtain the maturity value. This type of calculation is common in financial mathematics for short term loans and notes.

Given Data / Assumptions:

• Principal P = $15,000.
• Annual simple interest rate r = 7% per annum.
• Exact time from April 7 to November 7 = 214 days.
• A 365 day year is used for the time fraction.

Concept / Approach:
When using exact time and a 365 day year, the time in years is t = number of days / 365. Simple interest is then I = (P * r * t) / 100. After finding the interest, the total amount to be repaid (maturity value) is A = P + I. Careful handling of the day count and percentage conversions is essential to get the correct answer.

Step-by-Step Solution:
Step 1: Compute the time in years: t = 214 / 365. Step 2: Use the SI formula I = (P * r * t) / 100. Step 3: Substitute P = 15,000, r = 7, and t = 214 / 365. Step 4: I = (15,000 * 7 * 214) / (100 * 365). Step 5: Numerator = 15,000 * 7 * 214 = 22,470,000. Step 6: Denominator = 100 * 365 = 36,500. Step 7: I = 22,470,000 / 36,500 ≈ $615.52 (rounded to two decimal places). Step 8: Total amount A = P + I ≈ 15,000 + 615.52 = $15,615.52. Step 9: Therefore, the amount to be repaid is approximately $15,615.52.

Verification / Alternative check:
You can estimate to check reasonableness. Seven percent of $15,000 for a full year is 0.07 * 15,000 = $1,050. The fraction of the year is 214 / 365, which is a little more than half. Half of $1,050 is about $525, and taking the exact fraction gives around $615.52, which is consistent with the precise calculation. This confirms that the computed interest and repayment amount are sensible.

Why Other Options Are Wrong:

• $13,615.52 and $14,615.52: These are less than the principal, which is impossible when positive interest is charged.
• $16,615.52: This would require about $1,615.52 in interest, far too high for 214 days at 7%.
• $15,500.00: This implies interest of only $500, lower than the accurate calculation.

Common Pitfalls:
Errors occur if you mistakenly treat 7 months as 7 / 12 of a year instead of using the given 214 days, or if you use a 360 day year instead of 365. Rounding too early or incorrectly handling the division can also cause small mismatches. Always follow the formula with the specified day count and year length.

Final Answer:
The total amount to be repaid on the due date is $15,615.52 (approximately).