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 partial 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) Now, we can calculate the interest for the first 40 days. ### Step 2: Calculate the new balance after the first payment After 40 days, a payment of \( \$ 1,500 \) is made. We will subtract this payment from the total amount due (principal + interest). ### Step 3: Calculate the interest for the next 60 days After the first payment, we will calculate the interest on the remaining balance for the next 60 days. ### Step 4: Calculate the new balance after the second payment After 60 days, a second payment of \( \$ 2,000 \) is made. We will again subtract this payment from the total amount due. ### Step 5: Calculate the interest for the final 60 days Finally, we will calculate the interest on the remaining balance for the last 60 days. ### Step 6: Calculate the final amount due Now, let's perform these calculations step by step. #### Step 1: Interest for the first 40 days \[ I_1 = 12000 \times 0.07 \times \frac{40}{360} \] #### Step 2: New balance after the first payment \[ \text{New Balance}_1 = 12000 + I_1 - 1500 \] #### Step 3: Interest for the next 60 days \[ I_2 = \text{New Balance}_1 \times 0.07 \times \frac{60}{360} \] #### Step 4: New balance after the second payment \[ \text{New Balance}_2 = \text{New Balance}_1 + I_2 - 2000 \] #### Step 5: Interest for the final 60 days \[ I_3 = \text{New Balance}_2 \times 0.07 \times \frac{60}{360} \] #### Step 6: Final amount due \[ \text{Final Amount} = \text{New Balance}_2 + I_3 \] Now, let's calculate these values step by step. ### Calculating the values 1. Calculate \( I_1 \) 2. Calculate \( \text{New Balance}_1 \) 3. Calculate \( I_2 \) 4. Calculate \( \text{New Balance}_2 \) 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 \( 12000 * 0.07 * (40/360) \). 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 \( (12000 + (12000 * 0.07 * (40/360)) - 1500) * 0.07 * (60/360) \). Calculate the value by following steps: - step0: Calculate: \(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\) - step1: Reduce the fraction: \(\left(12000+\left(12000\times 0.07\times \frac{1}{9}\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\) - step2: Multiply the terms: \(\left(12000+\frac{280}{3}-1500\right)\times 0.07\left(\frac{60}{360}\right)\) - step3: Calculate: \(\frac{31780}{3}\times 0.07\left(\frac{60}{360}\right)\) - step4: Reduce the fraction: \(\frac{31780}{3}\times 0.07\times \frac{1}{6}\) - step5: Multiply the terms: \(\frac{11123}{15}\times \frac{1}{6}\) - step6: Multiply the fractions: \(\frac{11123}{15\times 6}\) - step7: Multiply: \(\frac{11123}{90}\) Calculate or simplify the expression \( ((12000 + (12000 * 0.07 * (40/360)) - 1500) + ((12000 + (12000 * 0.07 * (40/360)) - 1500) * 0.07 * (60/360)) - 2000) * 0.07 * (60/360) \). Calculate the value by following steps: - step0: Calculate: \(\left(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)+\left(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step1: Reduce the fraction: \(\left(\left(12000+\left(12000\times 0.07\times \frac{1}{9}\right)-1500\right)+\left(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step2: Multiply the terms: \(\left(\left(12000+\frac{280}{3}-1500\right)+\left(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step3: Calculate: \(\left(\frac{31780}{3}+\left(\left(12000+\left(12000\times 0.07\left(\frac{40}{360}\right)\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step4: Reduce the fraction: \(\left(\frac{31780}{3}+\left(\left(12000+\left(12000\times 0.07\times \frac{1}{9}\right)-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step5: Multiply the terms: \(\left(\frac{31780}{3}+\left(\left(12000+\frac{280}{3}-1500\right)\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step6: Calculate: \(\left(\frac{31780}{3}+\left(\frac{31780}{3}\times 0.07\left(\frac{60}{360}\right)\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step7: Reduce the fraction: \(\left(\frac{31780}{3}+\left(\frac{31780}{3}\times 0.07\times \frac{1}{6}\right)-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step8: Multiply the terms: \(\left(\frac{31780}{3}+\frac{11123}{90}-2000\right)\times 0.07\left(\frac{60}{360}\right)\) - step9: Calculate: \(\frac{784523}{90}\times 0.07\left(\frac{60}{360}\right)\) - step10: Reduce the fraction: \(\frac{784523}{90}\times 0.07\times \frac{1}{6}\) - step11: Multiply the terms: \(\frac{5491661}{9000}\times \frac{1}{6}\) - step12: Multiply the fractions: \(\frac{5491661}{9000\times 6}\) - step13: Multiply: \(\frac{5491661}{54000}\) 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 \( (140 + (11123/90) - 2000) \). Calculate the value by following steps: - step0: Calculate: \(140+\frac{11123}{90}-2000\) - step1: Subtract the numbers: \(-1860+\frac{11123}{90}\) - step2: Reduce fractions to a common denominator: \(-\frac{1860\times 90}{90}+\frac{11123}{90}\) - step3: Transform the expression: \(\frac{-1860\times 90+11123}{90}\) - step4: Multiply the numbers: \(\frac{-167400+11123}{90}\) - step5: Add the numbers: \(\frac{-156277}{90}\) - step6: Rewrite the fraction: \(-\frac{156277}{90}\) Calculate or simplify the expression \( (10593.3333 + (5491661/54000)) - 2000 \). Calculate the value by following steps: - step0: Calculate: \(\left(10593.3333+\frac{5491661}{54000}\right)-2000\) - step1: Add the numbers: \(\frac{360957287}{33750}-2000\) - step2: Reduce fractions to a common denominator: \(\frac{360957287}{33750}-\frac{2000\times 33750}{33750}\) - step3: Transform the expression: \(\frac{360957287-2000\times 33750}{33750}\) - step4: Multiply the numbers: \(\frac{360957287-67500000}{33750}\) - step5: Subtract the numbers: \(\frac{293457287}{33750}\) Calculate or simplify the expression \( (10593.3333 + (5491661/54000)) \). Calculate the value by following steps: - step0: Calculate: \(10593.3333+\frac{5491661}{54000}\) - step1: Convert the expressions: \(\frac{105933333}{10000}+\frac{5491661}{54000}\) - step2: Reduce fractions to a common denominator: \(\frac{105933333\times 27}{10000\times 27}+\frac{5491661\times 5}{54000\times 5}\) - step3: Multiply the numbers: \(\frac{105933333\times 27}{270000}+\frac{5491661\times 5}{54000\times 5}\) - step4: Multiply the numbers: \(\frac{105933333\times 27}{270000}+\frac{5491661\times 5}{270000}\) - step5: Transform the expression: \(\frac{105933333\times 27+5491661\times 5}{270000}\) - step6: Multiply the numbers: \(\frac{2860199991+5491661\times 5}{270000}\) - step7: Multiply the numbers: \(\frac{2860199991+27458305}{270000}\) - step8: Add the numbers: \(\frac{2887658296}{270000}\) - step9: Reduce the fraction: \(\frac{360957287}{33750}\) Calculate or simplify the expression \( (8695.0307 + (5491661/54000)) \). Calculate the value by following steps: - step0: Calculate: \(8695.0307+\frac{5491661}{54000}\) - step1: Convert the expressions: \(\frac{86950307}{10000}+\frac{5491661}{54000}\) - step2: Reduce fractions to a common denominator: \(\frac{86950307\times 27}{10000\times 27}+\frac{5491661\times 5}{54000\times 5}\) - step3: Multiply the numbers: \(\frac{86950307\times 27}{270000}+\frac{5491661\times 5}{54000\times 5}\) - step4: Multiply the numbers: \(\frac{86950307\times 27}{270000}+\frac{5491661\times 5}{270000}\) - step5: Transform the expression: \(\frac{86950307\times 27+5491661\times 5}{270000}\) - step6: Multiply the numbers: \(\frac{2347658289+5491661\times 5}{270000}\) - step7: Multiply the numbers: \(\frac{2347658289+27458305}{270000}\) - step8: Add the numbers: \(\frac{2375116594}{270000}\) - step9: Reduce the fraction: \(\frac{1187558297}{135000}\) Let's summarize the calculations step by step: ### Step 1: Interest for the first 40 days \[ I_1 = 12000 \times 0.07 \times \frac{40}{360} = \frac{280}{3} \approx 93.33 \] ### Step 2: New balance after the first payment \[ \text{New Balance}_1 = 12000 + 93.33 - 1500 = 10593.33 \] ### Step 3: Interest for the next 60 days \[ I_2 = 10593.33 \times 0.07 \times \frac{60}{360} = \frac{11123}{90} \approx 123.58 \] ### Step 4: New balance after the second payment \[ \text{New Balance}_2 = 10593.33 + 123.58 - 2000 = 8695.03 \] ### Step 5: Interest for the final 60 days \[ I_3 = 8695.03 \times 0.07 \times \frac{60}{360} = \frac{5491661}{54000} \approx 101.70 \] ### Step 6: Final amount due \[ \text{Final Amount} = 8695.03 + 101.70 \approx 8796.73 \] Thus, the final amount due on the loan after 160 days, rounded to the nearest cent, is: \[ \boxed{8796.73} \]