Suppose that a certain car has the following average operating and ownership costs.
| | | |
| :-------- | :-------- | :---- |
| Operating | Ownership | Total |
| $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 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].
Answer (1)
Calculate the total annual expense: T o t a l A nn u a l E x p e n se = $0.96 × 30 , 000 = $28 , 800 .
Apply the future value of an annuity formula: 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 amount saved: A ≈ $170 , 653 . The final answer is 170653 .
Explanation
Understanding the Problem We are given the average operating and ownership costs per mile for a car. We need to find the total annual expense if we drive 30,000 miles per year, and then calculate the amount saved after 5 years if the total annual expense is deposited into an IRA with an interest rate of 8.5% compounded yearly.
Calculating Total Annual Expense First, we need to calculate the total annual expense for the car. The total cost per mile is $0.96, and we drive 30,000 miles per year. So, the total annual expense is: T o t a l A nn u a l E x p e n se = T o t a l C os t p er M i l e × M i l es Dr i v e n p er Y e a r 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 the amount saved at the end of 5 years. We are given the formula for the future value of an annuity: A = ( n r ) P [ ( 1 + n r ) n t − 1 ] Where:
A is the amount saved at the end of 5 years.
P is the total annual expense ($28,800).
r is the interest rate (8.5% or 0.085).
n is the number of times the interest is compounded per year (1, since it's compounded yearly).
t is the number of years (5).
Calculating the Amount Saved Now, we substitute the 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 14505.5017 A = 170652.9612 Rounding to the nearest dollar, we get $170,653.
Final Answer Therefore, the total annual expense for the car is $28,800, and the amount saved at the end of 5 years is approximately $170,653.
Examples
Understanding the future value of investments is crucial in personal finance. For instance, if you plan to save for your child's education, knowing how much your annual deposits will grow over time with a certain interest rate helps you determine the feasibility of reaching your savings goal. This calculation is also essential when planning for retirement, where consistent annual contributions to a retirement account can significantly accumulate over the years due to the power of compound interest. By using the future value of an annuity formula, you can make informed decisions about your savings and investments.
