Suppose that a certain car has the following average operating and ownership costs.
| | | |
| :--------- | :--------- | :---- |
| | Ownership | Total |
| Operating | | |
| \$0.28 | \$0.68 | \$0.96 |
a. If you drive 30,000 miles per year, what is total annual expense for this car?
b. If the total annual expense for this car is deposited at the end of each year into an IRA paying [tex]$8.5 \%$[/tex] compounded yearly, how much will be saved at the end of five years? Use the formula [tex]$A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)}$[/tex].
a. If you drive 30,000 miles per year, the total annual expense for this car is \$\square.
Answer (1)
Calculate the total annual expense by multiplying the total average cost per mile by the annual miles driven: 0.96 × 30 , 000 = $28 , 800 .
Use the future value of an annuity formula to calculate the amount saved at the end of five years: A = ( n r ) P [ ( 1 + n r ) n t − 1 ] .
Substitute the given values into the formula: A = ( 1 0.085 ) 28800 [ ( 1 + 1 0.085 ) 1 × 5 − 1 ] .
Calculate the future value: A ≈ $170 , 652 . The total annual expense is $28,800 and the amount saved after 5 years is 170652 .
Explanation
Understanding the Problem Let's break down this problem step by step. We're given the average operating and ownership costs per mile for a car, and we need to calculate the total annual expense and the future value of an IRA investment based on that expense.
Calculating Total Annual Expense First, we need to calculate the total annual expense for the car. We know the total average cost per mile is $0.96, and you drive 30,000 miles per year. To find the total annual expense, we multiply these two values: T o t a l A nn u a l E x p e n se = T o t a l A v er a g e C os t p er M i l e × A nn u a l M i l es Dr i v e n T o t a l A nn u a l E x p e n se = $0.96 × 30 , 000 = $28 , 800
Understanding the Future Value Formula Next, we need to calculate how much will be saved at the end of five years if the total annual expense is deposited at the end of each year into an IRA paying 8.5% compounded yearly. We'll use the future value of an annuity formula: A = ( n r ) P [ ( 1 + n r ) n t − 1 ] Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
Calculating the Future Value In this case:
P = $28,800 (the total annual expense)
r = 0.085 (8.5% annual interest rate as a decimal)
n = 1 (compounded yearly)
t = 5 (number of years) Now, we substitute these values into the formula: A = ( 1 0.085 ) 28800 [ ( 1 + 1 0.085 ) 1 × 5 − 1 ] A = 0.085 28800 [ ( 1 + 0.085 ) 5 − 1 ] A = 0.085 28800 [ ( 1.085 ) 5 − 1 ] A = 0.085 28800 [ 1.50365669 − 1 ] A = 0.085 28800 [ 0.50365669 ] A = 0.085 14505.41787 A = 170651.97494 ≈ $170 , 652 Rounding to the nearest dollar, the amount saved at the end of five years will be $170,652.
Final Answer Therefore, if you drive 30,000 miles per year, the total annual expense for this car is $28,800. If this amount is deposited at the end of each year into an IRA paying 8.5% compounded yearly, you will have approximately $170,652 saved at the end of five years.
Examples
Understanding the costs associated with owning and operating a car is crucial for budgeting and financial planning. This type of calculation can help you determine whether it's more cost-effective to lease or buy a car, or to choose a more fuel-efficient vehicle. Additionally, knowing how to calculate the future value of an investment, like an IRA, can help you plan for retirement or other long-term financial goals. By understanding these concepts, you can make informed decisions about your finances and investments, ensuring a more secure financial future. For example, if you are considering purchasing an electric car, you can compare the operating costs (electricity vs. gasoline) and ownership costs (insurance, maintenance) with a traditional gasoline car to see which is more economical over the long term. This analysis can extend to other areas, such as home energy efficiency upgrades or investment choices.
