Suppose that you are thinking about buying a car and have narrowed down your choices to two options. The new-car option: The new car costs $$ \$ 27,000 $$ and can be financed with a three-year loan at $$ 6.54 \% $$. The used-car option: A three-year old model of the same car costs $$ \$ 15,000 $$ and can be financed with a three-year loan at $$ 5.08 \% $$. What is the difference in monthly payments between financing the new car and financing the used car? Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$.

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We start from the formula

$$
\text{PMT} = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1+\frac{r}{n}\right)^{-nt}},
$$

where

- $$ P $$ is the purchase price,
- $$ r $$ is the annual interest rate (in decimal),
- $$ n $$ is the number of payments per year (here, $$ n=12 $$),
- $$ t $$ is the loan term in years (here, $$ t=3 $$).

**Step 1. Calculate the monthly payment for the new car**

For the new car, we have:
- $$ P_{\text{new}} = 27000 $$,
- $$ r_{\text{new}} = 0.0654 $$,
- $$ n = 12 $$,
- $$ t = 3 $$.

The monthly interest rate is

$$
\frac{r_{\text{new}}}{n} = \frac{0.0654}{12} \approx 0.00545.
$$

Substitute into the formula:

$$
\text{PMT}_{\text{new}} = \frac{27000 \times 0.00545}{1 - \left(1+0.00545\right)^{-36}}.
$$

**Step 2. Calculate the monthly payment for the used car**

For the used car, the values are:
- $$ P_{\text{used}} = 15000 $$,
- $$ r_{\text{used}} = 0.0508 $$,
- $$ n = 12 $$,
- $$ t = 3 $$.

The monthly interest rate is

$$
\frac{r_{\text{used}}}{n} = \frac{0.0508}{12} \approx 0.00423333.
$$

Thus, the monthly payment is

$$
\text{PMT}_{\text{used}} = \frac{15000 \times 0.00423333}{1 - \left(1+0.00423333\right)^{-36}}.
$$

**Step 3. Compute the values**

*For the new car:*

1. Compute the numerator:
   $$
   27000 \times 0.00545 \approx 147.15.
   $$

2. Compute the denominator:
   $$
   1 - \left(1.00545\right)^{-36}.
   $$
   
   First, calculate $$ \left(1.00545\right)^{36} $$. Using the fact that 
   $$
   \left(1.00545\right)^{36} = e^{36\ln(1.00545)},
   $$
   note that $$ \ln(1.00545) $$ is approximately $$ 0.005434 $$ so that
   $$
   36 \ln(1.00545) \approx 0.1956 \quad \Rightarrow \quad \left(1.00545\right)^{36} \approx e^{0.1956} \approx 1.2169.
   $$
   
   Then,
   $$
   \left(1.00545\right)^{-36} \approx \frac{1}{1.2169} \approx 0.8217,
   $$
   and the denominator becomes
   $$
   1 - 0.8217 \approx 0.1783.
   $$

3. Thus,
   $$
   \text{PMT}_{\text{new}} \approx \frac{147.15}{0.1783} \approx 825.3.
   $$

*For the used car:*

1. Compute the numerator:
   $$
   15000 \times 0.00423333 \approx 63.5.
   $$

2. Compute the denominator:
   $$
   1 - \left(1.00423333\right)^{-36}.
   $$
   
   Similarly, compute $$ \left(1.00423333\right)^{36} $$. Noting that 
   $$
   36 \ln(1.00423333) \approx 36 \times 0.004222 \approx 0.15199,
   $$
   hence,
   $$
   \left(1.00423333\right)^{36} \approx e^{0.15199} \approx 1.1641.
   $$
   
   Then,
   $$
   \left(1.00423333\right)^{-36} \approx \frac{1}{1.1641} \approx 0.8590,
   $$
   and the denominator becomes
   $$
   1 - 0.8590 \approx 0.1410.
   $$

3. So,
   $$
   \text{PMT}_{\text{used}} \approx \frac{63.5}{0.1410} \approx 450.35.
   $$

**Step 4. Find the difference in monthly payments**

Subtract the used car payment from the new car payment:

$$
\Delta \text{PMT} = \text{PMT}_{\text{new}} - \text{PMT}_{\text{used}} \approx 825.3 - 450.35 \approx 374.95.
$$

Rounded to the nearest dollar, the difference is approximately \$375 per month.
The monthly payment for the new car is approximately \$825, and for the used car, it's about \$450. The difference in monthly payments between financing the new car and the used car is approximately \$375.

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