7 years ago, lillian won some money in the lottery and put it in a bank account that earns 8% interest…

Explanation:

Step1: Identify compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount (initial deposit), $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Here, $A = 800$, $r=0.08$, $n = 4$ (compounded quarterly), and $t = 7$.

Step2: Solve for the principal $P$

We can rewrite the formula as $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$. Substitute the values: [ \begin{align*} P&=\frac{800}{(1+\frac{0.08}{4})^{4\times7}}\ &=\frac{800}{(1 + 0.02)^{28}}\ &=\frac{800}{1.02^{28}} \end{align*} ] Calculate $1.02^{28}\approx1.741024$. Then $P=\frac{800}{1.741024}\approx459.49$.

Step3: Calculate the interest earned

The interest earned $I=A - P$. Substitute $A = 800$ and $P\approx459.49$. So $I=800 - 459.49 = 340.51$.

Answer:

$340.51$