Question

Shelby invested $400 in account with a 2.5% interest rate compounded semi annually. She made no deposit of withdrawals on the account for five years. If interest is compounded semi annually which equation represents the balance in the account after five years.

Answer 1

To calculate the balance in an account with compound interest, you can use the 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 investment).
- \(r\) 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.

In Shelby's case:
- \(P = 400\)
- \(r = 0.025\) (2.5% expressed as a decimal)
- \(n = 2\) (since the interest is compounded semi-annually)
- \(t = 5\)

Substituting these values into the formula gives:

\[
A = 400 \left(1 + \frac{0.025}{2}\right)^{2 \times 5}
\]

Now simplifying it further:

1. Calculate \(\frac{r}{n}\):
\[
\frac{0.025}{2} = 0.0125
\]

2. Calculate \(nt\):
\[
2 \times 5 = 10
\]

3. Plug it back into the formula:
\[
A = 400 \left(1 + 0.0125\right)^{10}
\]
\[
= 400 \left(1.0125\right)^{10}
\]

Thus, the equation that represents the balance in the account after five years is:

\[
A = 400 \left(1.0125\right)^{10}
\]