To find how much [tex]$550 invested at the end of each quarter would be worth in 9 years at an annual interest rate of 4%, we can use the future value formula for an ordinary annuity. Here's a detailed, step-by-step solution:
### Step-by-Step Solution:
1. Determine Quarterly Investment (Principal):
- The amount invested at the end of each quarter is $[/tex]550.
2. Identify Number of Quarters per Year:
- There are 4 quarters in a year.
3. Calculate the Total Number of Quarters Over 9 Years:
- Total number of quarters = 9 years * 4 quarters/year = 36 quarters.
4. Annual Interest Rate:
- The given annual interest rate is 4%.
5. Convert Annual Interest Rate to Quarterly Interest Rate:
- Quarterly interest rate = Annual interest rate / Number of quarters per year
- Quarterly interest rate = 0.04 / 4 = 0.01 (or 1%).
6. Use the Future Value of An Ordinary Annuity Formula:
- The future value of an ordinary annuity formula is:
[tex]\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount per period ([tex]$550).
- \( r \) is the interest rate per period (0.01).
- \( n \) is the total number of periods (36).
7. Substitute the Values into the Formula:
- \( P = 550 \)
- \( r = 0.01 \)
- \( n = 36 \)
\[
FV = 550 \times \frac{(1 + 0.01)^{36} - 1}{0.01}
\]
8. Calculate the Future Value:
- First, calculate \( (1 + 0.01)^{36} - 1 \):
- \( 1.01^{36} \approx 1.430768 \)
- \( 1.430768 - 1 = 0.430768 \)
- Then divide by 0.01:
- \( \frac{0.430768}{0.01} = 43.0768 \)
- Multiply by $[/tex]550:
- [tex]\( 550 \times 43.0768 = 23692.28 \)[/tex]
### Final Answer:
- The amount after 9 years, when [tex]$550 is invested at the end of each quarter at an annual interest rate of 4%, is $[/tex]23,692.28.
So, $23,692.28 would be the worth of the investments after 9 years.
