To determine how much money Allison should invest now to have [tex]$8,000 saved in 4 years at an annual interest rate of 2.75% compounded quarterly, we can use the compound interest formula, which calculates the present value (the amount that should be invested now). Here's a detailed step-by-step solution:
1. Identify the variables:
- Future Value (FV): The amount Allison wants in the future, which is $[/tex]8,000.
- Annual Interest Rate (r): The rate at which the investment will grow annually, which is 2.75%.
- Number of Years (t): The time period over which the investment will grow, which is 4 years.
- Number of Times Compounded per Year (n): Since the interest is compounded quarterly, it is 4 times per year.
2. Convert the annual interest rate to the quarterly interest rate:
- Quarterly Interest Rate (r/n): This is achieved by dividing the annual interest rate by the number of times interest is compounded per year.
[tex]\[
\text{Quarterly Interest Rate} = \frac{2.75\%}{4} = \frac{0.0275}{4} = 0.006875
\][/tex]
3. Calculate the total number of times the interest is compounded over the whole period:
- Total number of compounding periods (nt): This is the number of years multiplied by the number of compounding periods per year.
[tex]\[
\text{Total Compounding Periods} = 4 \text{ years} \times 4 \text{ periods per year} = 16 \text{ periods}
\][/tex]
4. Use the compound interest formula to find the present value (PV):
The compound interest formula is:
[tex]\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value,
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of times interest is compounded per year,
- [tex]\( t \)[/tex] is the number of years.
Plugging in the values:
[tex]\[
PV = \frac{8000}{(1 + 0.006875)^{16}}
\][/tex]
5. Calculate the term inside the parentheses and then apply the exponentiation:
[tex]\[
1 + 0.006875 = 1.006875
\][/tex]
[tex]\[
(1.006875)^{16} = \text{(compute using a calculator or mathematical software)}
\][/tex]
6. Divide the future value by the calculated term:
[tex]\[
PV = \frac{8000}{\text{Value computed in previous step}}
\][/tex]
7. The final present value:
Upon calculating the above, we find:
[tex]\[
PV \approx 7169.37
\][/tex]
Thus, Allison should invest approximately [tex]$7,169.37 now to have $[/tex]8,000 saved in 4 years with a 2.75% annual interest rate compounded quarterly.
