uppose 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 comma 00027,000 

and can be financed with a 

fivefive​-year 

loan at 

6.716.71​%.

The​ used-car option: A​ three-year old model of the same car costs 

​$14 comma 00014,000 

and can be financed with a 

fourfour​-year 

loan at 

6.536.53​%.

What is the difference in monthly payments between financing the new car and financing the used​ car? Use 

PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFractionPMT=Prn1−1+rn−nt.

Question content area bottom

Part 1

The difference in monthly payments between financing the new car and financing the used car is 

​$enter your response here.

​(Round to the nearest cent as​ needed.)

To calculate the difference in monthly payments between financing the new car and the used car, we'll use the **loan payment formula (PMT)**:

\[
PMT = \frac{P \times \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\]

Where:
- $ P $ is the loan principal (the amount financed).
- $ r $ is the annual interest rate.
- $ n $ is the number of payments per year (typically 12 for monthly payments).
- $ t $ is the loan term in years.

### Step 1: Calculate the Monthly Payment for the New Car
For the new car:
- $ P = 27,000 $
- $ r = 6.71\% = 0.0671 $
- $ n = 12 $
- $ t = 5 $ years

Plugging these values into the formula:

\[
PMT_{\text{new}} = \frac{27,000 \times \left(\frac{0.0671}{12}\right)}{1 - \left(1 + \frac{0.0671}{12}\right)^{-12 \times 5}}
\]

Let's calculate it step by step:

1. **Calculate the monthly interest rate:**

\[
\frac{0.0671}{12} \approx 0.005592
\]

2. **Calculate $ 1 + $ monthly interest rate:**

\[
1 + 0.005592 = 1.005592
\]

3. **Raise to the power of $-12 \times 5$:**

\[
\left(1.005592\right)^{-60} \approx 0.71299
\]

4. **Subtract from 1:**

\[
1 - 0.71299 = 0.28701
\]

5. **Calculate the numerator:**

\[
27,000 \times 0.005592 \approx 150.984
\]

6. **Finally, divide the numerator by the denominator:**

\[
PMT_{\text{new}} = \frac{150.984}{0.28701} \approx 526.15
\]

So, the monthly payment for the new car is approximately **$526.15**.

### Step 2: Calculate the Monthly Payment for the Used Car
For the used car:
- $ P = 14,000 $
- $ r = 6.53\% = 0.0653 $
- $ n = 12 $
- $ t = 4 $ years

Plugging these values into the formula:

\[
PMT_{\text{used}} = \frac{14,000 \times \left(\frac{0.0653}{12}\right)}{1 - \left(1 + \frac{0.0653}{12}\right)^{-12 \times 4}}
\]

Let's calculate it step by step:

1. **Calculate the monthly interest rate:**

\[
\frac{0.0653}{12} \approx 0.005442
\]

2. **Calculate $ 1 + $ monthly interest rate:**

\[
1 + 0.005442 = 1.005442
\]

3. **Raise to the power of $-12 \times 4$:**

\[
\left(1.005442\right)^{-48} \approx 0.7666
\]

4. **Subtract from 1:**

\[
1 - 0.7666 = 0.2334
\]

5. **Calculate the numerator:**

\[
14,000 \times 0.005442 \approx 76.188
\]

6. **Finally, divide the numerator by the denominator:**

\[
PMT_{\text{used}} = \frac{76.188}{0.2334} \approx 326.30
\]

So, the monthly payment for the used car is approximately **$326.30**.

### Step 3: Find the Difference in Monthly Payments
The difference between the monthly payments for the new and used car is:

\[
\text{Difference} = 526.15 - 326.30 = 199.85
\]

### Final Answer:
The difference in monthly payments between financing the new car and financing the used car is **$199.85**.

Would you like more details or have any further questions? Here are some related questions:

1. How much total interest will be paid over the life of the loan for the new car?
2. What is the total amount paid (principal + interest) for the new car loan?
3. How much could be saved in interest by choosing the used car over the new car?
4. What would the monthly payment be if the loan term for the new car was 4 years instead of 5?
5. How does a change in the interest rate affect the monthly payment for both options?
6. What if the down payment was increased? How would that impact the monthly payments?
7. How would an increase in the loan term to 6 years affect the new car's monthly payment?
8. What would the monthly payment be if you financed only $10,000 of the used car?

**Tip:** When comparing loans, consider both the monthly payment and the total interest paid over the life of the loan to fully understand the cost.

Learn how to calculate monthly payments for new and used car financing options using the loan payment formula (PMT). Understand the difference in monthly payments between financing a new car priced at $27,000 with a 5-year loan at 6.71% and a used car priced at $14,000 with a 4-year loan at 6.53%.

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