Use the ordinary interest method, 360 days, to solve the following word problem. Round to the nearest cent when necessary. Richie Powers is the owner of American Eagle Boats, a manufacturer of custom pleasure boats. Because of the economic recession and slow boat sales recently, American Eagle has begun accepting promissory notes from its dealers to help finance large orders. This morning American Eagle accepted a 90 day, \( 8.6 \% \) promissory note for \( \$ 800,000 \) from one of its sales dealers. You are a manager for a bank, and Richie is one of your clients. Your bank's discount A rate is currently \( 13 \% \). Richie's goal is to discount the note as soon as possible, but not until the proceeds are at least equal to the face value of the note, \( \$ 800,000 \). As his banker, Richie has asked you to "run the numbers" at ten day intervals starting with day 20 and advise him as to when he can discount the note and still receive his \( \$ 800,000 \). Find the bank discount (in \( \$ \) ) for days 20,30 , and 40 , and indicate for each discount whether it is too high or whether it is low enough for Richie to discount the note and still receive his \( \$ 800,000 \).

To solve this problem, we will follow these steps: 1. **Calculate the interest earned on the promissory note for the specified days (20, 30, and 40 days).** 2. **Calculate the bank discount for each of these days.** 3. **Determine the proceeds after the discount and check if they meet or exceed the face value of the note (\$800,000).** ### Step 1: Calculate the Interest Earned The formula for calculating ordinary interest is: \[ I = P \times r \times t \] Where: - \( I \) = Interest - \( P \) = Principal (face value of the note) - \( r \) = Interest rate (as a decimal) - \( t \) = Time in years (using a 360-day year) Given: - \( P = 800,000 \) - \( r = 8.6\% = 0.086 \) We will calculate the interest for 90 days first, and then for the remaining days (20, 30, and 40 days). ### Step 2: Calculate the Bank Discount The bank discount is calculated using the formula: \[ D = F \times d \times t \] Where: - \( D \) = Bank discount - \( F \) = Face value of the note - \( d \) = Discount rate (as a decimal) - \( t \) = Time in years Given: - \( d = 13\% = 0.13 \) ### Step 3: Calculate Proceeds The proceeds after discounting the note can be calculated as: \[ P_{\text{proceeds}} = F - D \] Now, let's perform the calculations for days 20, 30, and 40. 1. **Calculate the interest for 90 days:** \[ I_{90} = 800,000 \times 0.086 \times \frac{90}{360} \] 2. **Calculate the bank discount for days 20, 30, and 40:** - For day 20: \[ D_{20} = 800,000 \times 0.13 \times \frac{70}{360} \] - For day 30: \[ D_{30} = 800,000 \times 0.13 \times \frac{60}{360} \] - For day 40: \[ D_{40} = 800,000 \times 0.13 \times \frac{50}{360} \] 3. **Calculate proceeds for each day:** - For day 20: \[ P_{\text{proceeds, 20}} = 800,000 - D_{20} \] - For day 30: \[ P_{\text{proceeds, 30}} = 800,000 - D_{30} \] - For day 40: \[ P_{\text{proceeds, 40}} = 800,000 - D_{40} \] Now, let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(800000\times 0.13\left(\frac{60}{360}\right)\) - step1: Reduce the fraction: \(800000\times 0.13\times \frac{1}{6}\) - step2: Multiply the terms: \(104000\times \frac{1}{6}\) - step3: Multiply the numbers: \(\frac{52000}{3}\) Calculate or simplify the expression \( 800000 * 0.13 * (50/360) \). Calculate the value by following steps: - step0: Calculate: \(800000\times 0.13\left(\frac{50}{360}\right)\) - step1: Reduce the fraction: \(800000\times 0.13\times \frac{5}{36}\) - step2: Multiply the terms: \(104000\times \frac{5}{36}\) - step3: Multiply the numbers: \(\frac{130000}{9}\) Calculate or simplify the expression \( 800000 * 0.13 * (70/360) \). Calculate the value by following steps: - step0: Calculate: \(800000\times 0.13\left(\frac{70}{360}\right)\) - step1: Reduce the fraction: \(800000\times 0.13\times \frac{7}{36}\) - step2: Multiply the terms: \(104000\times \frac{7}{36}\) - step3: Multiply the numbers: \(\frac{182000}{9}\) Calculate or simplify the expression \( 800000 * 0.086 * (90/360) \). Calculate the value by following steps: - step0: Calculate: \(800000\times 0.086\left(\frac{90}{360}\right)\) - step1: Reduce the fraction: \(800000\times 0.086\times \frac{1}{4}\) - step2: Multiply the terms: \(68800\times \frac{1}{4}\) - step3: Reduce the numbers: \(17200\times 1\) - step4: Simplify: \(17200\) Calculate or simplify the expression \( 800000 - 20222.2222 \). Calculate the value by following steps: - step0: Calculate: \(800000-20222.2222\) - step1: Subtract the numbers: \(779777.7778\) Calculate or simplify the expression \( 800000 - 14444.4444 \). Calculate the value by following steps: - step0: Calculate: \(800000-14444.4444\) - step1: Subtract the numbers: \(785555.5556\) Calculate or simplify the expression \( 800000 - 17333.3333 \). Calculate the value by following steps: - step0: Calculate: \(800000-17333.3333\) - step1: Subtract the numbers: \(782666.6667\) Let's summarize the calculations step by step: ### Step 1: Calculate the Interest Earned for 90 Days The interest earned on the promissory note for 90 days is: \[ I_{90} = 800,000 \times 0.086 \times \frac{90}{360} = 17,200 \] ### Step 2: Calculate the Bank Discount for Days 20, 30, and 40 1. **For Day 20:** \[ D_{20} = 800,000 \times 0.13 \times \frac{70}{360} \approx 20,222.22 \] 2. **For Day 30:** \[ D_{30} = 800,000 \times 0.13 \times \frac{60}{360} \approx 17,333.33 \] 3. **For Day 40:** \[ D_{40} = 800,000 \times 0.13 \times \frac{50}{360} \approx 14,444.44 \] ### Step 3: Calculate Proceeds After Discount 1. **For Day 20:** \[ P_{\text{proceeds, 20}} = 800,000 - 20,222.22 \approx 779,777.78 \] 2. **For Day 30:** \[ P_{\text{proceeds, 30}} = 800,000 - 17,333.33 \approx 782,666.67 \] 3. **For Day 40:** \[ P_{\text{proceeds, 40}} = 800,000 - 14,444.44 \approx 785,555.56 \] ### Summary of Results - **Day 20:** - Bank Discount: \$20,222.22 - Proceeds: \$779,777.78 (Too low) - **Day 30:** - Bank Discount: \$17,333.33 - Proceeds: \$782,666.67 (Too low) - **Day 40:** - Bank Discount: \$14,444.44 - Proceeds: \$785,555.56 (Too low) ### Conclusion Richie cannot discount the note at days 20, 30, or 40, as the proceeds are below the face value of \$800,000. He will need to wait longer to achieve the desired proceeds.