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?

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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.
To pay off a \$5,000 loan at a 5% annual interest rate with a monthly payment of at least \$50, it will take approximately 10 years and 9.5 months. If you pay \$50 each month, you'll end up paying a total of \$6,475, which includes \$1,475 in interest.

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?

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