Max Wholesaler borrowed $12,500 on a 9%, 120-day note. After 45 days, Max paid $4,375 on the note. Thirty days later, Max paid an additional $3,750. Use ordinary interest.
Required:
a. Determine the total interest using the U.S. Rule.
Note: Round your intermediate balances and interest amounts to the nearest cent. Round your final answer to the nearest cent.
Total interest amount
b. Determine the ending balance due using the U.S. Rule.
Note: Round your intermediate balances and interest amounts to the nearest cent. Round your final answer to the nearest cent.
Ending balance due.
Answer (1)
Calculate the interest for the first 45 days: I 1 = 12500 × 0.09 × 360 45 = $140.63 .
Calculate the balance after the first 45 days: B 1 = 12500 + 140.63 = $12 , 640.63 , and subtract the first payment: P 1 = 12640.63 − 4375 = $8 , 265.63 .
Calculate the interest for the next 30 days: I 2 = 8265.63 × 0.09 × 360 30 = $61.99 , calculate the balance: B 2 = 8265.63 + 61.99 = $8 , 327.62 , and subtract the second payment: P 2 = 8327.62 − 3750 = $4 , 577.62 .
Calculate the interest for the remaining 45 days: I 3 = 4577.62 × 0.09 × 360 45 = $51.50 , and the ending balance is B 3 = 4577.62 + 51.50 = $4 , 629.12 . The total interest is 4375 + 3750 − ( 12500 − 4629.12 ) = $254.12 .
Explanation
Calculate Interest for the First 45 Days First, we need to calculate the interest accrued during the first 45 days. The formula for simple interest is I = P × R × T , where I is the interest, P is the principal, R is the interest rate, and T is the time in years. In this case, P = $12 , 500 , R = 9% = 0.09 , and T = 360 45 years. Therefore, the interest accrued during the first 45 days is:
Calculate I_1 I 1 = 12500 × 0.09 × 360 45 = $140.625 Rounding to the nearest cent, I 1 = $140.63 .
Calculate Balance Before First Payment Next, we calculate the balance after 45 days before the first payment. This is the original principal plus the accrued interest:
Calculate B_1 B 1 = P + I 1 = 12500 + 140.63 = $12 , 640.63
Calculate Principal After First Payment Now, Max makes a payment of $4 , 375 . Since this payment is greater than the interest accrued, we reduce the principal by the difference:
Calculate P_1 P 1 = B 1 − 4375 = 12640.63 − 4375 = $8 , 265.63
Calculate Interest for the Next 30 Days Next, we calculate the interest accrued during the next 30 days (from day 45 to day 75). Using the same simple interest formula, with P = $8 , 265.63 , R = 0.09 , and T = 360 30 years, the interest accrued is:
Calculate I_2 I 2 = 8265.63 × 0.09 × 360 30 = $61.99225 Rounding to the nearest cent, I 2 = $61.99 .
Calculate Balance Before Second Payment We calculate the balance after 75 days before the second payment. This is the principal after the first payment plus the accrued interest:
Calculate B_2 B 2 = P 1 + I 2 = 8265.63 + 61.99 = $8 , 327.62
Calculate Principal After Second Payment Now, Max makes a second payment of $3 , 750 . Since this payment is greater than the interest accrued, we reduce the principal by the difference:
Calculate P_2 P 2 = B 2 − 3750 = 8327.62 − 3750 = $4 , 577.62
Calculate Interest for the Remaining 45 Days Next, we calculate the interest accrued during the remaining 45 days (from day 75 to day 120). Using the same simple interest formula, with P = $4 , 577.62 , R = 0.09 , and T = 360 45 years, the interest accrued is:
Calculate I_3 I 3 = 4577.62 × 0.09 × 360 45 = $51.498225 Rounding to the nearest cent, I 3 = $51.50 .
Calculate Ending Balance Due Finally, we calculate the ending balance due. This is the principal after the second payment plus the accrued interest:
Calculate B_3 B 3 = P 2 + I 3 = 4577.62 + 51.50 = $4 , 629.12
Calculate Total Interest Paid The total interest paid is the sum of the two payments minus the original principal plus the ending balance:
Calculate Total Interest T o t a l I n t eres t = 4375 + 3750 − ( 12500 − 4629.12 ) = 8125 − 7870.88 = $254.12
Examples
Understanding the U.S. Rule is crucial in scenarios involving loans with partial payments. For instance, consider a small business owner who takes out a loan to purchase equipment. Unexpectedly, they receive a large order and decide to make partial payments on the loan ahead of schedule. By applying the U.S. Rule, the lender ensures that these early payments are first used to cover any accrued interest, thereby reducing the principal amount more effectively. This method benefits the borrower by potentially shortening the loan term and decreasing the total interest paid over the life of the loan. It's a fair way to handle loan repayments, ensuring that both the lender and borrower are treated equitably.
