Ramogi borrowed \( \$ 2900 \) from a bank for 20 months. The bank discounted the loan at \( 5 \% \). What is the loan's discount? What is the net amount of money Ramogi receives? What is the loan's actual rate of interest? (to the nearest hundredth of a percent)

To solve the problem, we will follow these steps: 1. **Calculate the loan's discount.** 2. **Determine the net amount of money Ramogi receives.** 3. **Calculate the loan's actual rate of interest.** ### Step 1: Calculate the Loan's Discount The formula for calculating the discount on a loan is given by: \[ D = P \times r \times t \] where: - \( D \) is the discount, - \( P \) is the principal amount (loan amount), - \( r \) is the rate of interest (as a decimal), - \( t \) is the time in years. Given: - \( P = 2900 \) - \( r = 5\% = 0.05 \) - \( t = \frac{20}{12} \) years (since 20 months is \(\frac{20}{12}\) years) Now, we can calculate the discount \( D \). ### Step 2: Determine the Net Amount of Money Ramogi Receives The net amount received can be calculated as: \[ N = P - D \] where: - \( N \) is the net amount received. ### Step 3: Calculate the Loan's Actual Rate of Interest The actual rate of interest can be calculated using the formula: \[ \text{Actual Rate} = \frac{D}{N} \times \frac{12}{t} \times 100 \] Now, let's perform the calculations step by step. First, we will calculate the discount \( D \). Let's calculate \( D \) and then proceed to find \( N \) and the actual rate of interest. Calculate the value by following steps: - step0: Calculate: \(2900\times 0.05\left(\frac{20}{12}\right)\) - step1: Reduce the fraction: \(2900\times 0.05\times \frac{5}{3}\) - step2: Multiply the terms: \(145\times \frac{5}{3}\) - step3: Multiply: \(\frac{145\times 5}{3}\) - step4: Multiply: \(\frac{725}{3}\) Calculate or simplify the expression \( 2900 - (2900 * 0.05 * (20/12)) \). Calculate the value by following steps: - step0: Calculate: \(2900-\left(2900\times 0.05\left(\frac{20}{12}\right)\right)\) - step1: Reduce the fraction: \(2900-\left(2900\times 0.05\times \frac{5}{3}\right)\) - step2: Multiply the terms: \(2900-\frac{725}{3}\) - step3: Reduce fractions to a common denominator: \(\frac{2900\times 3}{3}-\frac{725}{3}\) - step4: Transform the expression: \(\frac{2900\times 3-725}{3}\) - step5: Multiply the numbers: \(\frac{8700-725}{3}\) - step6: Subtract the numbers: \(\frac{7975}{3}\) Calculate or simplify the expression \( (2900 * 0.05 * (20/12)) / (2900 - (2900 * 0.05 * (20/12))) * (12 / (20/12)) * 100 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2900\times 0.05\left(\frac{20}{12}\right)\right)}{\left(2900-\left(2900\times 0.05\left(\frac{20}{12}\right)\right)\right)}\times \left(\frac{12}{\left(\frac{20}{12}\right)}\right)\times 100\) - step1: Remove the parentheses: \(\frac{2900\times 0.05\left(\frac{20}{12}\right)}{2900-\left(2900\times 0.05\left(\frac{20}{12}\right)\right)}\times \left(\frac{12}{\frac{20}{12}}\right)\times 100\) - step2: Reduce the fraction: \(\frac{2900\times 0.05\times \frac{5}{3}}{2900-\left(2900\times 0.05\left(\frac{20}{12}\right)\right)}\times \left(\frac{12}{\frac{20}{12}}\right)\times 100\) - step3: Reduce the fraction: \(\frac{2900\times 0.05\times \frac{5}{3}}{2900-\left(2900\times 0.05\times \frac{5}{3}\right)}\times \left(\frac{12}{\frac{20}{12}}\right)\times 100\) - step4: Multiply the terms: \(\frac{2900\times 0.05\times \frac{5}{3}}{2900-\frac{725}{3}}\times \left(\frac{12}{\frac{20}{12}}\right)\times 100\) - step5: Reduce the fraction: \(\frac{2900\times 0.05\times \frac{5}{3}}{2900-\frac{725}{3}}\times \left(\frac{12}{\frac{5}{3}}\right)\times 100\) - step6: Divide the terms: \(\frac{2900\times 0.05\times \frac{5}{3}}{2900-\frac{725}{3}}\times \frac{36}{5}\times 100\) - step7: Multiply the terms: \(\frac{\frac{725}{3}}{2900-\frac{725}{3}}\times \frac{36}{5}\times 100\) - step8: Subtract the numbers: \(\frac{\frac{725}{3}}{\frac{7975}{3}}\times \frac{36}{5}\times 100\) - step9: Divide the terms: \(\frac{1}{11}\times \frac{36}{5}\times 100\) - step10: Multiply the terms: \(\frac{36}{55}\times 100\) - step11: Reduce the numbers: \(\frac{36}{11}\times 20\) - step12: Multiply: \(\frac{36\times 20}{11}\) - step13: Multiply: \(\frac{720}{11}\) ### Step 1: Calculate the Loan's Discount The discount \( D \) is calculated as follows: \[ D = 2900 \times 0.05 \times \frac{20}{12} = 241.67 \text{ (approximately)} \] ### Step 2: Determine the Net Amount of Money Ramogi Receives Now, we can find the net amount \( N \): \[ N = 2900 - D = 2900 - 241.67 = 2658.33 \text{ (approximately)} \] ### Step 3: Calculate the Loan's Actual Rate of Interest The actual rate of interest is calculated as: \[ \text{Actual Rate} = \frac{D}{N} \times \frac{12}{\frac{20}{12}} \times 100 \] Substituting the values: \[ \text{Actual Rate} = \frac{241.67}{2658.33} \times \frac{12}{\frac{20}{12}} \times 100 \] Calculating this gives us: \[ \text{Actual Rate} \approx 65.45\% \] ### Summary of Results 1. **Loan's Discount:** \( D \approx 241.67 \) 2. **Net Amount Received:** \( N \approx 2658.33 \) 3. **Actual Rate of Interest:** \( \approx 65.45\% \) (to the nearest hundredth of a percent)