New - Borrow $$ \$ 15,000 $$ at $$ 5 \% $$ interest rate 16. Pick a monthly payment $$ \$ 150 $$ (must be more than $$ \$ 50 $$ ) 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 minimum each month?

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Certainly! Let's break down your car loan scenario to determine how long it will take to pay off the loan with a monthly payment of \$150 and how much you’ll end up paying in total.

### **Loan Details**
- **Principal Amount (P):** \$15,000
- **Annual Interest Rate:** 5%
- **Monthly Interest Rate (r):** 5% / 12 = **0.4167%** (or 0.004167 in decimal)
- **Monthly Payment (M):** \$150

### **1. Calculating the Time to Pay Off the Loan**

To determine how long it will take to pay off the loan, we can use the **loan amortization formula**:

$$
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
$$

Where:
- $$ M $$ = Monthly payment
- $$ P $$ = Principal loan amount
- $$ r $$ = Monthly interest rate
- $$ n $$ = Number of monthly payments

**Rearranging the formula to solve for $$ n $$:**

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

**Plugging in the numbers:**

$$
n = \frac{\ln\left(\frac{150}{150 - 15,000 \times 0.004167}\right)}{\ln(1 + 0.004167)}
$$

$$
n = \frac{\ln\left(\frac{150}{150 - 62.50}\right)}{\ln(1.004167)}
$$

$$
n = \frac{\ln\left(\frac{150}{87.50}\right)}{\ln(1.004167)}
$$

$$
n = \frac{\ln(1.7142857)}{\ln(1.004167)} \approx \frac{0.53907}{0.004158} \approx 129.6
$$

**Result:**
- **Number of Months:** Approximately **130 months**
- **Converted to Years:** About **10 years and 10 months**

### **2. Calculating the Total Amount Paid**

To find out how much you'll pay in total over the life of the loan:

$$
\text{Total Payments} = \text{Monthly Payment} \times \text{Number of Months} = 150 \times 130 = \$19,500
$$

**Breakdown:**
- **Total Amount Paid:** \$19,500
- **Total Interest Paid:** \$19,500 (total payments) - \$15,000 (principal) = \$4,500

### **Summary**

- **Time to Pay Off the Loan:** **130 months** (approximately **10 years and 10 months**)
- **Total Amount Paid:** **\$19,500**

### **Additional Considerations**

- **Consistency:** This calculation assumes that you make **consistent monthly payments** of \$150 without any missed or extra payments.
- **Interest Rate Stability:** It also assumes that the **interest rate remains constant** at 5% throughout the loan period.
- **Fees and Taxes:** Any additional fees, insurance, or taxes are **not included** in this calculation and could affect the total amount paid and the time to repay the loan.

If you have the flexibility to increase your monthly payments above \$150, you can **significantly reduce** both the **loan term** and the **total interest paid**. It's always a good strategy to pay more than the minimum when possible to save on interest and become debt-free sooner.

If you have any more questions or need further clarification, feel free to ask!
It will take approximately 10 years and 10 months to pay off the car with monthly payments of \$150. Over this period, you'll end up paying a total of \$19,500, which includes \$4,500 in interest.

New - Borrow \( \$ 15,000 \) at \( 5 \% \) interest rate 16. Pick a monthly payment \( \$ 150 \) (must be more than \( \$ 50 \) ) 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 minimum each month?

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