Question

Archie invests $27,000
into a savings account with an interest rate of 2.25%
compounded monthly.
What is the balance of Archies savings account after 8
years? Round your answer to the nearest hundredth.

Answer 1

To calculate the balance of Archie’s savings account after 8 years with monthly compounding interest, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial amount of money).
- \(r\) = the annual interest rate (decimal).
- \(n\) = the number of times that interest is compounded per year.
- \(t\) = the time the money is invested for, in years.

For this problem:
- \(P = 27,000\)
- \(r = 2.25\% = \frac{2.25}{100} = 0.0225\)
- \(n = 12\) (since the interest is compounded monthly)
- \(t = 8\)

Now, plug these values into the formula:

\[
A = 27000 \left(1 + \frac{0.0225}{12}\right)^{12 \times 8}
\]

Calculating \(\frac{0.0225}{12}\):
\[
\frac{0.0225}{12} = 0.001875
\]

Now substituting back into the formula:

\[
A = 27000 \left(1 + 0.001875\right)^{96}
\]
\[
A = 27000 \left(1.001875\right)^{96}
\]

Now we calculate \( (1.001875)^{96} \):

\[
(1.001875)^{96} \approx 1.189736
\]

Now, multiply by the principal:

\[
A \approx 27000 \times 1.189736 \approx 32121.82
\]

Rounding to the nearest hundredth:

The balance of Archie's savings account after 8 years is approximately **$32,121.82**.