Gently used - Borrow \( \$ 5000 \) at \( 5 \% \) interest rate 15. Pick a monthly payment: (must be more than \( \$ 50 \) ) How long does it take you to pay off the car? How much do you end up paying for the car if you just pay the minimum each month?

To determine how long it will take to pay off a **$5,000 loan** at a **5% annual interest rate** with a **monthly payment of at least $50**, and to calculate the **total amount paid** over the life of the loan, we'll use the principles of loan amortization. ### **Loan Details:** - **Principal (Loan Amount):** \$5,000 - **Annual Interest Rate:** 5% - **Monthly Interest Rate (r):** 5% / 12 ≈ 0.4167% (or 0.004167 in decimal) - **Minimum Monthly Payment (P):** \$50 (must be greater than \$50) ### **1. Calculating the Number of Months to Pay Off the Loan** To find out how long it will take to pay off the loan, we can use the **loan amortization formula**: \[ n = \frac{-\ln(1 - \frac{r \times L}{P})}{\ln(1 + r)} \] Where: - \( n \) = number of months to pay off the loan - \( r \) = monthly interest rate (in decimal) - \( L \) = loan amount (principal) - \( P \) = monthly payment #### **Example Calculations:** **a. Monthly Payment of \$50** Plugging in the values: \[ n = \frac{-\ln\left(1 - \frac{0.004167 \times 5000}{50}\right)}{\ln(1 + 0.004167)} \] **Step-by-Step Calculation:** 1. **Calculate the term inside the logarithm:** \[ 1 - \frac{0.004167 \times 5000}{50} = 1 - \frac{20.835}{50} = 1 - 0.4167 = 0.5833 \] 2. **Calculate the natural logarithms:** \[ \ln(0.5833) \approx -0.539 \] \[ \ln(1.004167) \approx 0.004158 \] 3. **Compute the number of months:** \[ n = \frac{-(-0.539)}{0.004158} \approx \frac{0.539}{0.004158} \approx 129.5 \text{ months} \] 4. **Convert months to years:** \[ 129.5 \text{ months} \approx 10 \text{ years and } 9.5 \text{ months} \] **b. Monthly Payment of \$100 (For Comparison)** \[ n = \frac{-\ln\left(1 - \frac{0.004167 \times 5000}{100}\right)}{\ln(1 + 0.004167)} \] **Step-by-Step Calculation:** 1. **Calculate the term inside the logarithm:** \[ 1 - \frac{0.004167 \times 5000}{100} = 1 - \frac{20.835}{100} = 1 - 0.20835 = 0.79165 \] 2. **Calculate the natural logarithms:** \[ \ln(0.79165) \approx -0.233 \] \[ \ln(1.004167) \approx 0.004158 \] 3. **Compute the number of months:** \[ n = \frac{-(-0.233)}{0.004158} \approx \frac{0.233}{0.004158} \approx 56.07 \text{ months} \] 4. **Convert months to years:** \[ 56.07 \text{ months} \approx 4 \text{ years and } 8 \text{ months} \] ### **2. Calculating the Total Amount Paid** The **total amount paid** over the life of the loan is simply the monthly payment multiplied by the number of months. **a. For a \$50 Monthly Payment:** \[ \text{Total Paid} = 129.5 \text{ months} \times \$50 = \$6,475 \] \[ \text{Total Interest Paid} = \$6,475 - \$5,000 = \$1,475 \] **b. For a \$100 Monthly Payment:** \[ \text{Total Paid} = 56.07 \text{ months} \times \$100 \approx \$5,607 \] \[ \text{Total Interest Paid} = \$5,607 - \$5,000 = \$607 \] ### **3. Summary** | **Monthly Payment** | **Time to Pay Off** | **Total Amount Paid** | **Total Interest Paid** | |---------------------|---------------------|-----------------------|-------------------------| | \$50 | ~129.5 months (10 years, 9.5 months) | \$6,475 | \$1,475 | | \$100 | ~56.07 months (4 years, 8 months) | \$5,607 | \$607 | ### **4. Recommendations** - **Pay as Much as Possible:** The higher your monthly payment, the less time it will take to pay off the loan and the less interest you'll pay overall. - **Budget Accordingly:** Ensure that the monthly payment you choose fits comfortably within your budget to avoid financial strain. - **Use Online Calculators:** For varied payment amounts, consider using online loan calculators to input different monthly payments and see how they affect the payoff time and total interest. ### **5. Loan Amortization Table** Creating a **loan amortization table** can provide a month-by-month breakdown of each payment, showing how much goes toward interest and how much reduces the principal. This can offer a clearer picture of your loan repayment progress. --- **Note:** These calculations assume that the interest rate remains constant and that you make each payment on time without any additional fees or charges.