4. In 8 years, the investment Helen made at 8% compounded quarterly,will have grown to 25000 a) What was her initial investment?

## Step 1: Define Variables and Formula<br />### We are given the future value ($A$), interest rate ($r$), compounding periods per year ($n$), and time ($t$). We need to find the principal ($P$). The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$.<br /><br />## Step 2: Plug in Values<br />### $A = \$25000$, $r = 8\% = 0.08$, $n = 4$ (quarterly), and $t = 8$ years. Substituting these values into the formula gives us $25000 = P(1 + \frac{0.08}{4})^{4 \cdot 8}$.<br /><br />## Step 3: Simplify and Solve for P<br />### Simplify the equation: $25000 = P(1 + 0.02)^{32}$ which becomes $25000 = P(1.02)^{32}$. Calculate $(1.02)^{32} \approx 1.8674$. Now, the equation is $25000 = P \cdot 1.8674$. Divide both sides by $1.8674$ to solve for $P$: $P = \frac{25000}{1.8674} \approx 13386.66$.