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 EndEndFractionA=P1+rnnt−1rn

Suppose that you drive 

40 comma 00040,000 

miles per year and gas averages 

​$44 

per gallon. Complete parts **a.** and **b.** below. 

Question content area bottom

Part 1

a.a. 

What will you save in annual fuel expenses by owning a hybrid car averaging 

4040 

miles per gallon rather than an SUV averaging 

1616 

miles per​ gallon?

​$60006000

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

Part 2

b.b. 

If you deposit your monthly fuel savings at the end of each month into an annuity that pays 

5.25.2​% 

compounded​ monthly, how much will have saved at the end of 

fourfour 

​years?

​$enter your response here

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

Let's break this down step by step for both **a** and **b**.

### **Part a: Calculating annual fuel savings**

1. You drive 40,000 miles per year.
2. Gas costs \$4 per gallon.
3. The hybrid car averages 40 miles per gallon, and the SUV averages 16 miles per gallon.

#### Hybrid Car Fuel Consumption:
\[
\text{Fuel used} = \frac{40,000 \text{ miles}}{40 \text{ miles per gallon}} = 1,000 \text{ gallons}
\]
\[
\text{Annual fuel cost} = 1,000 \text{ gallons} \times 4 = \$4,000
\]

#### SUV Fuel Consumption:
\[
\text{Fuel used} = \frac{40,000 \text{ miles}}{16 \text{ miles per gallon}} = 2,500 \text{ gallons}
\]
\[
\text{Annual fuel cost} = 2,500 \text{ gallons} \times 4 = \$10,000
\]

#### Fuel Savings:
\[
\text{Fuel savings} = 10,000 - 4,000 = \$6,000
\]

So, the annual fuel savings is **\$6,000**.

### **Part b: Using the annuity formula**

Now, you want to deposit your monthly fuel savings into an annuity, which pays 5.2% interest compounded monthly. 

#### Monthly Savings:
\[
\text{Monthly savings} = \frac{\$6,000}{12} = \$500
\]

We will use the annuity formula:
\[
A = P \left( \frac{ \left( 1 + \frac{r}{n} \right)^{nt} - 1 }{ \frac{r}{n} } \right)
\]

Where:
- $A$ = the future value of the annuity
- $P$ = monthly deposit = \$500
- $r$ = annual interest rate = 5.2% = 0.052
- $n$ = number of compounding periods per year = 12 (since it's compounded monthly)
- $t$ = time in years = 4

Now, we can plug in the values:

\[
A = 500 \left( \frac{ \left( 1 + \frac{0.052}{12} \right)^{12 \times 4} - 1 }{ \frac{0.052}{12} } \right)
\]

Let's calculate this.The total amount saved at the end of four years, with monthly fuel savings deposited into an annuity paying 5.2% compounded monthly, would be approximately **\$26,615**.

Would you like any further explanation or details on this?

### Related Questions:
1. How would the savings change if the interest rate were higher, say 6%?
2. What if the deposit period were 6 years instead of 4? How would that affect the total savings?
3. How much more could you save if your monthly savings were \$600 instead of \$500?
4. What would the impact be if the interest were compounded quarterly instead of monthly?
5. How does an annuity differ from a standard savings account?

### Tip:
When saving regularly, small increases in the interest rate or time period can significantly impact the total savings due to the power of compound interest!

Use the following formula: A = P * ((1 + (r / n)) ^ (nt) - 1) / (r / n). Suppose that you drive 40,000 miles per year and gas averages $4 per gallon. Complete parts a and b below. Part 1: 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: If you deposit your monthly fuel savings at the end of each month into an annuity that pays 5.2% compounded monthly, how much will you have saved at the end of four years?

Learn how to calculate annual fuel savings by switching from an SUV to a hybrid car and how to grow those savings using an annuity with monthly deposits. This problem covers basic arithmetic, fuel efficiency, and compound interest through the annuity formula, suitable for Grades 11-12.

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