5. 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 $$ \$ 28,000 $$ and can be financed with a four-year loan at $$ 2.45 \% $$. The used-car option: A three-year old model of the same car costs $$ \$ 16,000 $$ and can be financed with a four-year loan at $$ 3.15 \% $$. What is the difference in monthly payments between financing the new car and financing the used car?

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Let 
$$
M = P\,\frac{r(1+r)^n}{(1+r)^n-1}
$$
be the monthly payment, where 
- $$ P $$ is the principal, 
- $$ r $$ is the monthly interest rate, and 
- $$ n $$ is the number of monthly payments.

---

**Step 1. New-Car Option**

- Principal: $$ P_{\text{new}} = 28000 $$
- Annual interest rate: $$ 2.45\% $$
- Monthly interest rate: 
  $$
  r_{\text{new}} = \frac{0.0245}{12} \approx 0.00204167
  $$
- Loan term: 4 years, so the number of payments:
  $$
  n = 4 \times 12 = 48
  $$

First, compute 
$$
(1+r_{\text{new}})^n = (1.00204167)^{48}.
$$
Using the exponent approximation,
$$
(1.00204167)^{48} \approx 1.1027.
$$

Now, the monthly payment is:
$$
M_{\text{new}} = 28000\,\frac{0.00204167\times1.1027}{1.1027-1}.
$$

Calculate the numerator:
$$
0.00204167\times1.1027 \approx 0.002249,
$$
$$
28000 \times 0.002249 \approx 63.
$$

The denominator is:
$$
1.1027 - 1 = 0.1027.
$$

Thus,
$$
M_{\text{new}} \approx \frac{63}{0.1027} \approx 612.
$$

---

**Step 2. Used-Car Option**

- Principal: $$ P_{\text{used}} = 16000 $$
- Annual interest rate: $$ 3.15\% $$
- Monthly interest rate:
  $$
  r_{\text{used}} = \frac{0.0315}{12} \approx 0.002625
  $$
- Loan term: 4 years ($$ n = 48 $$).

Compute
$$
(1+r_{\text{used}})^n = (1.002625)^{48}.
$$
Using a similar approach,
$$
(1.002625)^{48} \approx 1.134.
$$

Now compute the monthly payment:
$$
M_{\text{used}} = 16000\,\frac{0.002625\times1.134}{1.134-1}.
$$

Calculate the numerator:
$$
0.002625 \times 1.134 \approx 0.002978,
$$
$$
16000 \times 0.002978 \approx 47.65.
$$

The denominator is:
$$
1.134 - 1 = 0.134.
$$

Thus,
$$
M_{\text{used}} \approx \frac{47.65}{0.134} \approx 355.7.
$$

---

**Step 3. Difference in Monthly Payments**

Subtract the monthly payment for the used car from that of the new car:
$$
\Delta M \approx 612 - 355.7 \approx 256.3.
$$

---

$$\boxed{256}$$

The difference in monthly payments between financing the new car and financing the used car is approximately $$\$256$$ per month.
The monthly payment for the new car is about \$612, and for the used car, it's about \$356. The difference is approximately \$256 per month.

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