Certainly! Let's break down the problem and solve it step-by-step.
We are given:
1. Future Value (FV): The amount we want to have in the future, which is [tex]$700.
2. Interest Rate (r): The annual interest rate, which is 7.5%. However, since the interest compounds quarterly, we need to convert this annual rate to a quarterly rate.
3. Number of Periods (n): The total time duration over which the money is invested, which is 3 years. Since the interest compounds quarterly, we need to convert this time duration into quarters.
To find the present value (PV) or the initial investment required to achieve the future value under these conditions, we use the formula for compound interest:
\[ PV = \frac{FV}{(1 + r/n)^{nt}} \]
Where:
- \( PV \) is the present value or the initial investment.
- \( FV \) is the future value (700).
- \( r \) is the annual interest rate (7.5% = 0.075).
- \( n \) is the number of times the interest is compounded per year (4 for quarterly).
- \( t \) is the number of years (3).
First, let's convert the annual interest rate to a quarterly rate:
\[ \text{Quarterly Interest Rate} = \frac{7.5\%}{4} = \frac{0.075}{4} = 0.01875 \]
Next, we need to calculate the total number of quarters in 3 years:
\[ \text{Number of Quarters} = 3 \text{ years} \times 4 \text{ quarters per year} = 12 \text{ quarters} \]
Now we can plug these values into the compound interest formula:
\[ PV = \frac{700}{(1 + 0.01875)^{12}} \]
Calculating the denominator:
\[ (1 + 0.01875)^{12} \approx 1.24928 \]
Finally, calculate the present value:
\[ PV = \frac{700}{1.24928} \approx 560.13 \]
Therefore, the amount that must be invested at 7.5% annual interest, compounding quarterly, to have $[/tex]700 at the end of 3 years is approximately $560.13.
