New - Borrow $$ \$ 15,000 $$ at $$ 5 \% $$ interest rate 16 Pick a monthly payment 450 (must be marethan $$ \$ 50 $$ ) How long does it take to pay off the car? How much do you end up paying for the car if you just pay the minmum each month?

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To evaluate your car loan scenario, let's break down the details and answer each of your questions systematically.

### **Loan Details:**
- **Principal Amount (P):** \$15,000
- **Annual Interest Rate (APR):** 5%
- **Monthly Interest Rate (r):** 5% / 12 = **0.4167%** or **0.0041667** (in decimal)
- **Chosen Monthly Payment (A):** \$450 (must be more than \$50)

---

### **1. Time to Pay Off the Car with \$450 Monthly Payments**

To determine how long it will take to pay off the loan with monthly payments of \$450, we'll use the standard loan amortization formula:

$$
N = \frac{-\ln\left(1 - \frac{r \times P}{A}\right)}{\ln(1 + r)}
$$

Where:
- $$ N $$ = Number of monthly payments
- $$ P $$ = Principal loan amount (\$15,000)
- $$ r $$ = Monthly interest rate (0.0041667)
- $$ A $$ = Monthly payment (\$450)

**Calculations:**
1. **Calculate $$ \frac{r \times P}{A} $$:**
   $$
   \frac{0.0041667 \times 15,000}{450} = \frac{62.5}{450} \approx 0.1389
   $$
2. **Compute $$ 1 - 0.1389 = 0.8611 $$**
3. **Take the natural logarithm:**
   $$
   \ln(0.8611) \approx -0.1508
   $$
   $$
   \ln(1 + 0.0041667) \approx 0.004158
   $$
4. **Determine $$ N $$:**
   $$
   N = \frac{-(-0.1508)}{0.004158} \approx 36.28 \text{ months}
   $$

**Result:**
- **Time to Pay Off:** Approximately **36 months** (3 years)
- **Total Amount Paid:** $$ 36 \times \$450 = \$16,200 $$
- **Total Interest Paid:** $$ \$16,200 - \$15,000 = \$1,200 $$

---

### **2. Total Cost if Paying the Minimum Monthly Payment of \$50**

**Important Consideration:**
- **Minimum Payment (A):** \$50

First, we need to assess whether a \$50 monthly payment is sufficient to cover the interest accruing each month.

**Monthly Interest Accrued:**
$$
\text{Interest} = P \times r = 15,000 \times 0.0041667 = \$62.50
$$

**Analysis:**
- **Monthly Payment (\$50) < Monthly Interest (\$62.50):**
  - **Implication:** Each month, the unpaid interest (\$62.50 - \$50 = \$12.50) gets added to the principal.
  - **Result:** The loan balance **increases** over time, leading to **negative amortization**.

**Outcome:**
- **Loan Never Repaid:** Paying only \$50 monthly is insufficient to reduce the principal. Instead, the debt grows by \$12.50 each month.
- **Long-Term Implications:** Without increasing the monthly payment above \$62.50, the loan will never be paid off and will continue to accumulate interest indefinitely.

---

### **Summary**

- **With a \$450 Monthly Payment:**
  - **Loan Paid Off In:** 36 months (3 years)
  - **Total Paid:** \$16,200
  - **Total Interest:** \$1,200

- **With a \$50 Monthly Payment:**
  - **Loan Status:** **Never repaid**
  - **Loan Balance Increases:** By \$12.50 each month
  - **Total Cost:** Continues to grow indefinitely due to accruing interest

---

### **Recommendation**

To effectively manage and repay your car loan:

- **Ensure Monthly Payments Exceed:** At least the monthly interest (\$62.50) to prevent the loan balance from increasing.
- **Higher Payments Reduce Total Interest:** Paying more than the minimum required accelerates loan repayment and reduces the total interest paid.
- **Consider Financial Planning:** Align your monthly payments with your budget to maintain financial health and avoid increasing debt.

If you have any further questions or need assistance with different payment scenarios, feel free to ask!
To pay off the car loan: - **With \$450 monthly payments:** It will take approximately 3 years to pay off the \$15,000 loan with a 5% interest rate. You'll end up paying a total of \$16,200, which includes \$1,200 in interest. - **With \$50 monthly payments:** Paying only \$50 each month is not enough to cover the monthly interest of \$62.50. This means the loan balance will continue to grow, and you'll never fully repay the car.

New - Borrow \( \$ 15,000 \) at \( 5 \% \) interest rate 16 Pick a monthly payment 450 (must be marethan \( \$ 50 \) ) How long does it take to pay off the car? How much do you end up paying for the car if you just pay the minmum each month?

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