To determine how much you need to invest annually to reach a savings goal of [tex]$300,000 in 25 years with an annual interest rate of 5%, we use the formula for the future value of an annuity. Here's a detailed step-by-step solution:
### Step-by-Step Solution:
1. Identify Given Values:
- Future Value (FV) or savings goal = $[/tex]300,000
- Number of periods (n) = 25 years
- Annual interest rate (r) = 5% or 0.05
2. Formula for Future Value of an Annuity:
The formula for the future value of an annuity is:
[tex]\[
\text{FV} = Pmt \times \left( \frac{(1 + r)^n - 1}{r} \right)
\][/tex]
Where:
- [tex]\( \text{FV} \)[/tex] is the future value,
- [tex]\( Pmt \)[/tex] is the annual payment,
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( n \)[/tex] is the number of periods.
3. Rearrange the Formula to Solve for Annual Payment (Pmt):
Solving for [tex]\( Pmt \)[/tex], we get:
[tex]\[
Pmt = \frac{\text{FV} \times r}{(1 + r)^n - 1}
\][/tex]
4. Substitute the Known Values into the Formula:
- [tex]\( \text{FV} = 300,000 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( n = 25 \)[/tex]
Substituting these values:
[tex]\[
Pmt = \frac{300,000 \times 0.05}{(1 + 0.05)^{25} - 1}
\][/tex]
5. Calculate the Value of the Denominator:
Calculate [tex]\( (1 + 0.05)^{25} - 1 \)[/tex]:
[tex]\[
(1.05)^{25} - 1
\][/tex]
6. Calculate the Exact Annual Payment (Pmt):
Plug the exact values into the formula to find [tex]\( Pmt \)[/tex]:
[tex]\[
Pmt \approx 6285.74
\][/tex]
7. Round the Annual Payment to the Nearest Hundred Dollars:
Rounding [tex]$6,285.74 to the nearest hundred dollars:
\[
Pmt \approx 6300.00
\]
### Conclusion:
To reach a savings goal of $[/tex]300,000 at the end of 25 years with an annual interest rate of 5%, compounded annually, you need to invest approximately $6,300 annually.
