College Algebra, Chapter 5, 5.4, Section 5.4, Problem 74

Suppose that a woman wants to invest $\$ 4000$ in an account that pays $9.75 \%$ per year, compounded semiannually. How long a time period should she choose to save an amount of $\$ 5000$?



Recall that the formula for interest compounded $n$ times per year is..

$\displaystyle A(t) = P \left( 1 + \frac{r}{n} \right)^{nt}$

So if the interest is compounded semiannually, then $n = 2$


$\begin{equation} \begin{aligned} 5000 =& 4000 \left( 1 + \frac{0.0975}{2} \right)^{(2) t} \\ \\ \frac{5000}{4000} =& \left( 1 + \frac{0.0975}{2} \right)^{2t} \\ \\ \ln \left( \frac{5}{4} \right) =& \ln \left( 1 + \frac{0.0975}{2} \right)^{2t} \\ \\ \ln \left( \frac{5}{4} \right) =& 2t \ln \left( 1 + \frac{0.0975}{2} \right) \\ \\ \frac{\displaystyle \ln \left( \frac{5}{4} \right) }{\displaystyle 2 \ln \left( 1 + \frac{0.0975}{2} \right)} =& t \\ \\ t =& 2.34 \text{ years} \\ \\ t =& 2 \text{ years} + 0.34 \text{ years} \left( \frac{12 \text{ months}}{1 \text{ year}} \right) \\ \\ t =& 2 \text{ years} + 4.08 \text{ months} \end{aligned} \end{equation}$


It shows that the woman will save an amount of $\$ 5000$ if the period is approximately $2$ years and $5$ months.