Question

leah wants to buy a CD for $500 that earns 4% APR and is compounded quarterly for 5 years. she will be taxed on 20% of the interest that she earns. what is the total amount of interest leah will earn, after taxes?

Answer 1

To calculate the total amount of interest Leah will earn, we can follow these steps:

1. **Calculate the future value (FV) of the investment using the compound interest formula:**
\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \( P = 500 \) (the principal amount),
- \( r = 0.04 \) (the annual interest rate),
- \( n = 4 \) (the number of times interest is compounded per year),
- \( t = 5 \) (the number of years).

2. **Plug the values into the formula:**
\[
FV = 500 \left(1 + \frac{0.04}{4}\right)^{4 \times 5}
\]
\[
FV = 500 \left(1 + 0.01\right)^{20}
\]
\[
FV = 500 \left(1.01\right)^{20}
\]

3. **Calculate \( (1.01)^{20} \):**
\[
(1.01)^{20} \approx 1.22019
\]
Therefore,
\[
FV \approx 500 \times 1.22019 \approx 610.095
\]

4. **Calculate the total interest earned:**
The total interest earned is the future value minus the principal:
\[
\text{Interest} = FV - P = 610.095 - 500 \approx 110.095
\]

5. **Calculate the tax on the interest:**
Leah will be taxed on 20% of the interest she earns:
\[
\text{Tax} = 0.20 \times 110.095 \approx 22.019
\]

6. **Calculate the net interest after tax:**
\[
\text{Net Interest} = \text{Interest} - \text{Tax} = 110.095 - 22.019 \approx 88.076
\]

Thus, **the total amount of interest Leah will earn after taxes is approximately $88.08.**