Use the following formula. 
Upper A equals StartStartFraction Upper P left bracket left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt Baseline minus 1 right bracket OverOver left parenthesis StartFraction r Over n EndFraction right parenthesis EndEndFraction
Suppose that you drive 40 comma 000 miles per year and gas averages ​$4 per gallon. Complete parts a. and b. below. 
Question content area bottom
Part 1
a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SUV averaging 16 miles per​ gallon?
​$
select: 6,000
6,000
​(Round to the nearest dollar as​ needed.)
Part 2
b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8​% compounded​ monthly, how much will have saved at the end of five ​years?
​$
  
enter your response here
​(Round to the nearest dollar as​ needed.)

Let's go through each part of the question step by step.

### Part 1: Annual Fuel Savings

You are comparing the fuel expenses between a hybrid car that averages 40 miles per gallon (mpg) and an SUV that averages 16 mpg. 

1. **Miles driven per year**: 40,000 miles
2. **Fuel cost**: $4 per gallon
3. **Miles per gallon (mpg)** for:
   - **Hybrid**: 40 mpg
   - **SUV**: 16 mpg

To calculate the annual fuel cost for each vehicle, use the formula:

\[
\text{Fuel Cost} = \frac{\text{Miles driven per year}}{\text{Miles per gallon}} \times \text{Cost per gallon}
\]

For the **SUV**:
\[
\text{Fuel Cost (SUV)} = \frac{40,000}{16} \times 4 = 10,000 \, \text{dollars}
\]

For the **Hybrid**:
\[
\text{Fuel Cost (Hybrid)} = \frac{40,000}{40} \times 4 = 4,000 \, \text{dollars}
\]

Now, the annual savings by owning the hybrid instead of the SUV is:

\[
\text{Annual Savings} = 10,000 - 4,000 = 6,000 \, \text{dollars}
\]

Thus, the **annual savings** is $6,000.

### Part 2: Savings After 5 Years in an Annuity

Now that you know the annual savings is $6,000, we will calculate how much you'll save at the end of 5 years by depositing the monthly fuel savings into an annuity that earns 4.8% interest compounded monthly.

1. **Monthly savings**: Since the annual savings is $6,000, the monthly savings is:
   \[
   \frac{6,000}{12} = 500 \, \text{dollars}
   \]

2. **Interest rate**: 4.8% compounded monthly means the monthly interest rate is:
   \[
   \frac{4.8\%}{12} = 0.004 \, \text{(or 0.004 per month)}
   \]

3. **Number of periods**: Since the savings occur monthly for 5 years, the number of periods is:
   \[
   5 \times 12 = 60 \, \text{months}
   \]

We can use the annuity future value formula:

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

Where:
- $ A $ = future value of the annuity (what we want to find),
- $ P $ = monthly savings = $500,
- $ r $ = annual interest rate = 0.048,
- $ n $ = number of compounding periods per year = 12,
- $ t $ = number of years = 5.

Substituting the values into the formula:

\[
A = 500 \times \frac{\left(1 + \frac{0.048}{12}\right)^{12 \times 5} - 1}{\frac{0.048}{12}}
\]

Let's calculate that.

The future value of the annuity after 5 years, given monthly fuel savings of $500 and an interest rate of 4.8% compounded monthly, is approximately **$33,830** (rounded to the nearest dollar).

### Summary:
1. **Annual savings** by owning a hybrid car instead of an SUV: **$6,000**.
2. **Future value of your savings** after 5 years in an annuity: **$33,830**.

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

1. How would the savings change if the interest rate was 6% compounded monthly?
2. What if you saved $600 per month instead of $500, how would that affect the future value?
3. How does the number of years (e.g., 10 years) affect the annuity savings?
4. What if the gas price changes to $5 per gallon, how does that affect the fuel savings?
5. Can you explain the impact of compounding frequency on savings?

**Tip:** The higher the frequency of compounding (monthly, quarterly), the more you benefit from interest on your savings.

Use the following formula. Upper A equals StartStartFraction Upper P left bracket left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript nt Baseline minus 1 right bracket OverOver left parenthesis StartFraction r Over n EndFraction right parenthesis EndEndFraction Suppose that you drive 40,000 miles per year and gas averages $4 per gallon. Complete parts a. and b. below. Part 1 a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SUV averaging 16 miles per gallon? Part 2 b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays 4.8% compounded monthly, how much will have saved at the end of five years?

Calculate how much you can save in annual fuel expenses by driving a hybrid car compared to an SUV and how much you can accumulate by investing your monthly fuel savings in an annuity with 4.8% interest. This problem explores concepts of fuel efficiency, algebra, and compound interest. Suitable for Grades 9-12.

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