To calculate how much Harry Taylor will have at the end of three years from his annuity payments, we first need to determine the future value of the ordinary annuity at the specified interest rate.
Step 1: Calculate the Future Value of the Annuity After 3 Years
The formula for the future value of an ordinary annuity is given by:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = Future Value of the annuity
- \( P \) = Payment per period
- \( r \) = Interest rate per period
- \( n \) = Total number of payments
In this case:
- \( P = 6,400 \)
- \( r = \frac{3.1}{100} = 0.031 \) (annual interest rate)
- \( n = 3 \) (since we are only looking at the first three years)
Now, plug in the values:
\[ FV = 6,400 \times \frac{(1 + 0.031)^3 - 1}{0.031} \]
Calculating \((1 + 0.031)^3\):
\[ (1 + 0.031)^3 = (1.031)^3 \approx 1.09686 \]
Now, substituting this back into the formula:
\[ FV = 6,400 \times \frac{1.09686 - 1}{0.031} \] \[ FV = 6,400 \times \frac{0.09686}{0.031} \approx 6,400 \times 3.12806 \approx 19,998.88 \]
Thus, the total amount Harry will have at the end of three years is approximately $19,998.88.
Step 2: Calculate the Total Contributions
Now, let's calculate how much he actually contributed to the annuity over these three years:
\[ Total , Contributions = P \times n = 6,400 \times 3 = 19,200 \]
Step 3: Calculate Total Interest Earned
Now, to find out how much interest he earns, we subtract the total contributions from the future value:
\[ \text{Interest Earned} = FV - \text{Total Contributions} \] \[ = 19,998.88 - 19,200 = 798.88 \]
Final Results:
- At the end of three years, Harry will have approximately $19,998.88.
- The total interest earned on the investment after three years will be approximately $798.88.