nola invests a sum of money in a savings account with a fixed annual interest rate of 4.61% compounded…

Explanation:

Step1: Recall 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 (initial investment), $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. We are given that $A = 5385.53$, $r=0.0461$ (since $4.61%=0.0461$), $n = 4$ (compounded quarterly), and $t = 8$.

Step2: Rearrange the formula to solve for $P$

Starting with $A = P(1+\frac{r}{n})^{nt}$, we can solve for $P$ by dividing both sides of the equation by $(1+\frac{r}{n})^{nt}$. So, $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$.

Step3: Substitute the given values into the formula for $P$

First, calculate $(1+\frac{r}{n})^{nt}$: [ \begin{align*} 1+\frac{r}{n}&=1+\frac{0.0461}{4}=1 + 0.011525=1.011525\ nt&=4\times8 = 32 \end{align*} ] Then, $(1+\frac{r}{n})^{nt}=(1.011525)^{32}$. Using a calculator, $(1.011525)^{32}\approx1.44777$. Now, $P=\frac{5385.53}{1.44777}\approx3720$.

Answer:

$3720$