Verified Answer :
- Identify the variables: \(A = 2000\), \(r = 0.03\), \(n = 12\), \(t = 9\).
- Use the compound interest formula rearranged for \(P\): \(P = \frac{A}{(1 + \frac{r}{n})^{nt}}\).
- Calculate the periodic rate and total compounding periods: \(\frac{0.03}{12} = 0.0025\) and \(12 \times 9 = 108\).
- Divide the future value by \((1.0025)^{108}\) to find the required principal: \(\boxed{\$1,527.27}\).
Explanation
- Analyzing the Problem and Variables
The goal is to determine the initial investment (principal) needed to reach a future value of \(\$2,000\) after 9 years. The annual interest rate is \(3\%\), which is \(0.03\) in decimal form, and interest is compounded monthly. Monthly compounding means the interest is calculated 12 times per year. Thus, we have the principal \(P\), future value \(A = \$2,000\), annual rate \(r = 0.03\), compounding periods per year \(n = 12\), and time \(t = 9\) years. The formula for compound interest is \(A = P(1 + \frac{r}{n})^{nt}\).
- Setting up the Calculation
To find the initial investment \(P\), we rearrange the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\) to isolate \(P\):
\[P = \frac{A}{(1 + \frac{r}{n})^{nt}}\]
Substituting the known values:
\[P = \frac{2000}{(1 + \frac{0.03}{12})^{12 \times 9}}\]
First, we calculate the periodic interest rate: \(\frac{0.03}{12} = 0.0025\).
Next, we determine the total number of compounding periods: \(12 \times 9 = 108\).
Now the equation is:
\[P = \frac{2000}{(1.0025)^{108}}\]
- Performing the Calculation
Now, we calculate the value of \((1.0025)^{108}\) and divide the future value of \(\$2,000\) by this factor:
\[(1.0025)^{108} \approx 1.309528\]
\[P = \frac{2000}{1.309528} \approx 1527.2735\]
Rounding to the nearest cent, the initial investment required is \(\$1,527.27\).
- Final Conclusion
The initial investment needed to reach a future value of \(\$2,000\) in 9 years with a \(3\%\) interest rate compounded monthly is \(\boxed{\$1,527.27}\).
Examples
Understanding how money grows with interest is crucial for personal finance, such as planning for future expenses like buying a home or saving for retirement. If you are setting aside money in a savings account or CD, knowing how much to invest today allows you to achieve your future financial targets reliably. By using compound interest calculations, you can make informed decisions about your savings strategy, ensuring your money works effectively for you over time.