Suppose you take out a loan for 160 days in the amount of \( \$ 12,000 \) at \( 7 \% \) ordinary interest. After 40 days, you make a partial payment of \( \$ 1,500 \). After another 60 days, you make a second partial payment of \( \$ 2,000 \). What is the final amount (in \( \$ \) ) due on the loan? (Round your answer to the nearest cent.)

To solve this problem, we will calculate the interest accrued on the loan at different stages and then determine the final amount due after the payments. ### Step 1: Calculate the interest for the first 40 days The formula for ordinary interest is given by: \[ I = P \times r \times t \] where: - \( I \) is the interest, - \( P \) is the principal amount, - \( r \) is the interest rate (as a decimal), - \( t \) is the time in years. For the first 40 days: - \( P = 12000 \) - \( r = 0.07 \) - \( t = \frac{40}{360} \) (since ordinary interest uses a 360-day year) Calculating the interest: \[ I_1 = 12000 \times 0.07 \times \frac{40}{360} \] ### Step 2: Calculate the new balance after the first payment After 40 days, a payment of \( \$ 1,500 \) is made. The new balance will be: \[ \text{New Balance} = \text{Principal} + I_1 - 1500 \] ### Step 3: Calculate the interest for the next 60 days Now, we will calculate the interest on the new balance for the next 60 days. The time in years for 60 days is: \[ t = \frac{60}{360} \] Calculating the interest on the new balance: \[ I_2 = \text{New Balance} \times 0.07 \times \frac{60}{360} \] ### Step 4: Calculate the new balance after the second payment After 60 days, a second payment of \( \$ 2,000 \) is made. The new balance will be: \[ \text{New Balance} = \text{Previous Balance} + I_2 - 2000 \] ### Step 5: Calculate the interest for the remaining 60 days Finally, we will calculate the interest on the new balance for the remaining 60 days: \[ t = \frac{60}{360} \] Calculating the interest on the new balance: \[ I_3 = \text{New Balance} \times 0.07 \times \frac{60}{360} \] ### Step 6: Calculate the final amount due The final amount due will be: \[ \text{Final Amount} = \text{New Balance} + I_3 \] Now, let's perform the calculations step by step. 1. Calculate \( I_1 \). 2. Update the balance after the first payment. 3. Calculate \( I_2 \). 4. Update the balance after the second payment. 5. Calculate \( I_3 \). 6. Calculate the final amount due. Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(12000\times 0.07\left(\frac{60}{360}\right)\) - step1: Reduce the fraction: \(12000\times 0.07\times \frac{1}{6}\) - step2: Multiply the terms: \(840\times \frac{1}{6}\) - step3: Multiply the numbers: \(140\) Calculate or simplify the expression \( 12000 * 0.07 * (60/360) \). Calculate the value by following steps: - step0: Calculate: \(12000\times 0.07\left(\frac{60}{360}\right)\) - step1: Reduce the fraction: \(12000\times 0.07\times \frac{1}{6}\) - step2: Multiply the terms: \(840\times \frac{1}{6}\) - step3: Multiply the numbers: \(140\) Calculate or simplify the expression \( 12000 * 0.07 * (40/360) \). Calculate the value by following steps: - step0: Calculate: \(12000\times 0.07\left(\frac{40}{360}\right)\) - step1: Reduce the fraction: \(12000\times 0.07\times \frac{1}{9}\) - step2: Multiply the terms: \(840\times \frac{1}{9}\) - step3: Multiply the numbers: \(\frac{280}{3}\) Calculate or simplify the expression \( 12000 + (280/3) - 1500 \). Calculate the value by following steps: - step0: Calculate: \(12000+\frac{280}{3}-1500\) - step1: Subtract the numbers: \(10500+\frac{280}{3}\) - step2: Reduce fractions to a common denominator: \(\frac{10500\times 3}{3}+\frac{280}{3}\) - step3: Transform the expression: \(\frac{10500\times 3+280}{3}\) - step4: Multiply the numbers: \(\frac{31500+280}{3}\) - step5: Add the numbers: \(\frac{31780}{3}\) Calculate or simplify the expression \( (12000 + (280/3) - 1500) + 140 - 2000 \). Calculate the value by following steps: - step0: Calculate: \(\left(12000+\frac{280}{3}-1500\right)+140-2000\) - step1: Calculate: \(\frac{31780}{3}+140-2000\) - step2: Subtract the numbers: \(\frac{31780}{3}-1860\) - step3: Reduce fractions to a common denominator: \(\frac{31780}{3}-\frac{1860\times 3}{3}\) - step4: Transform the expression: \(\frac{31780-1860\times 3}{3}\) - step5: Multiply the numbers: \(\frac{31780-5580}{3}\) - step6: Subtract the numbers: \(\frac{26200}{3}\) Calculate or simplify the expression \( (12000 + (280/3) - 1500) * 0.07 * (60/360) \). Calculate the value by following steps: - step0: Calculate: \(\left(12000+\frac{280}{3}-1500\right)\times 0.07\left(\frac{60}{360}\right)\) - step1: Calculate: \(\frac{31780}{3}\times 0.07\left(\frac{60}{360}\right)\) - step2: Reduce the fraction: \(\frac{31780}{3}\times 0.07\times \frac{1}{6}\) - step3: Multiply the terms: \(\frac{11123}{15}\times \frac{1}{6}\) - step4: Multiply the fractions: \(\frac{11123}{15\times 6}\) - step5: Multiply: \(\frac{11123}{90}\) Calculate or simplify the expression \( (10593.3333) + (123.5888) \). Calculate the value by following steps: - step0: Calculate: \(10593.3333+123.5888\) - step1: Add the numbers: \(10716.9221\) Let's summarize the calculations step by step: ### Step 1: Calculate the interest for the first 40 days \[ I_1 = 12000 \times 0.07 \times \frac{40}{360} = \frac{280}{3} \approx 93.33 \] ### Step 2: Update the balance after the first payment \[ \text{New Balance} = 12000 + 93.33 - 1500 = 10593.33 \] ### Step 3: Calculate the interest for the next 60 days \[ I_2 = 10593.33 \times 0.07 \times \frac{60}{360} = 140 \] ### Step 4: Update the balance after the second payment \[ \text{New Balance} = 10593.33 + 140 - 2000 = 8733.33 \] ### Step 5: Calculate the interest for the remaining 60 days \[ I_3 = 8733.33 \times 0.07 \times \frac{60}{360} \approx 123.59 \] ### Step 6: Calculate the final amount due \[ \text{Final Amount} = 8733.33 + 123.59 \approx 8856.92 \] Thus, the final amount due on the loan after 160 days, rounded to the nearest cent, is: \[ \boxed{8856.92} \]