Question

Alan placed $15,000 in a savings account in which the interest is compounded semiannually at a 1.3% annual interest rate. How much money will be in Alan's account after 5 years?

Answer 1

To calculate the amount of money in Alan's account after 5 years with semiannual compounding interest, we can use the compound interest formula:

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

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

Given:
- \(P = 15000\)
- \(r = 1.3\% = 0.013\)
- \(n = 2\) (since the interest is compounded semiannually)
- \(t = 5\)

Now, we can plug the values into the formula:

\[
A = 15000 \left(1 + \frac{0.013}{2}\right)^{2 \times 5}
\]

Calculating \(\frac{0.013}{2}\):

\[
\frac{0.013}{2} = 0.0065
\]

Now we can proceed to calculate the accumulated amount:

\[
A = 15000 \left(1 + 0.0065\right)^{10}
\]
\[
A = 15000 \left(1.0065\right)^{10}
\]

Now calculating \(1.0065^{10}\):

\[
1.0065^{10} \approx 1.067394
\]

Now substitute back into the equation for \(A\):

\[
A = 15000 \times 1.067394 \approx 16010.91
\]

Therefore, after 5 years, the amount of money in Alan's account will be approximately **$16,010.91**.