5. Suppose that you are thinking about buying a car and have namowed down your choices to two options: The new-car option: The new car costs $$ \$ 25,000 $$ and can be financed with a four-ycar loan at $$ 2.45 \% $$. The used-car option: A threc-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?

Question

UpStudy AI · Accepted Answer

We begin by using the standard formula for the monthly payment on an amortized loan:

$$
m = P \cdot \frac{i(1+i)^n}{(1+i)^n - 1}
$$

where  
$$ P $$ is the principal,  
$$ i $$ is the monthly interest rate, and  
$$ n $$ is the number of months.

### New-Car Option

- Principal: $$ P_{\text{new}} = 25000 $$
- Annual interest rate: $$ 2.45\% $$, so the monthly rate is

$$
i_{\text{new}} = \frac{0.0245}{12} \approx 0.00204167
$$

- Loan term: 4 years, so

$$
n = 4 \times 12 = 48 \text{ months}
$$

The monthly payment is

$$
m_{\text{new}} = 25000 \cdot \frac{0.00204167 \, (1+0.00204167)^{48}}{(1+0.00204167)^{48} - 1}
$$

Calculating the factor:

$$
(1+0.00204167)^{48} \approx 1.1029
$$

So

$$
m_{\text{new}} \approx 25000 \cdot \frac{0.00204167 \times 1.1029}{1.1029 - 1} = 25000 \cdot \frac{0.002250}{0.1029} \approx 25000 \cdot 0.02185 \approx 546.25
$$

### Used-Car Option

- Principal: $$ P_{\text{used}} = 16000 $$
- Annual interest rate: $$ 3.15\% $$, so the monthly rate is

$$
i_{\text{used}} = \frac{0.0315}{12} \approx 0.002625
$$

- Same loan term: $$ n = 48 $$ months.

The monthly payment is

$$
m_{\text{used}} = 16000 \cdot \frac{0.002625 \, (1+0.002625)^{48}}{(1+0.002625)^{48} - 1}
$$

Calculating the factor:

$$
(1+0.002625)^{48} \approx 1.1342
$$

So

$$
m_{\text{used}} \approx 16000 \cdot \frac{0.002625 \times 1.1342}{1.1342 - 1} = 16000 \cdot \frac{0.002976}{0.1342} \approx 16000 \cdot 0.02218 \approx 354.88
$$

### Difference in Monthly Payments

The difference is

$$
\Delta m = m_{\text{new}} - m_{\text{used}} \approx 546.25 - 354.88 \approx 191.37
$$

Thus, the new car’s monthly payment is approximately $$\$191$$ higher than that of the used car.
The new car's monthly payment is about \$191 more than the used car's monthly payment.

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New-Car Option

Used-Car Option

Difference in Monthly Payments

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