Question

Sarah invested $800 in an account paying an interest rate of 3.5% compounded quarterly. Assuming no deposits or withdrawals are made, how much money, to the nearest cent, would be in the account after 8 years?

Answer 1

To calculate the future value of an investment with compound interest, we 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 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 for.

For this problem:
- \( P = 800 \)
- \( r = 0.035 \) (3.5% as a decimal)
- \( n = 4 \) (quarterly compounding)
- \( t = 8 \)

Now, plugging these values into the formula gives:

\[
A = 800 \left(1 + \frac{0.035}{4}\right)^{4 \times 8}
\]

First, calculate \( \frac{0.035}{4} \):

\[
\frac{0.035}{4} = 0.00875
\]

Now add 1 to this value:

\[
1 + 0.00875 = 1.00875
\]

Now, calculate \( 4 \times 8 \):

\[
4 \times 8 = 32
\]

Now raise \( 1.00875 \) to the power of 32:

\[
1.00875^{32} \approx 1.3002493
\]

Now, multiply this value by the principal amount:

\[
A \approx 800 \times 1.3002493 \approx 1040.19944
\]

Finally, rounding to the nearest cent, the amount in the account after 8 years would be:

\[
\boxed{1040.20}
\]