Question

Elise invests $4500 in an account that compounds interest monthly and earns 6%. How long will it take for her money to double?

Short answer

The time required to double the principal would be $12$years

Step-by-step solution

Step 1. Given information

$$\begin{gathered} Principal=4500\$ \\ Rateofinterest=6\% \\ Amount=9000\$ \end{gathered}$$

Step 2. Writing the concept.

A principal P, invested at interest rate r for time period t, compounded n times a year will result into an amount given as:

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

Step 3. Calculation

Put the values of respective variables in the given formula:

$$\begin{gathered} P=4500 \\ R=6\% \\ A=9000 \\ Now, \\ A=P{\left(1+\frac{r}{n}\right)}^{nt} \\ 9000=4500{\left(1+\frac{0.06}{12}\right)}^{12t} \\ 2={\left(\frac{12.06}{12}\right)}^{12t} \\ {\left(\frac{12.06}{12}\right)}^{12t}=2 \\ Taking\log bothsides, \\ \log {\left(\frac{12.06}{12}\right)}^{12t}=\log 2 \\ 12t(\log 12.06-\log 12)=\log 2 \\ 12t(0.0021)=0.301 \\ 0.025t=0.301 \\ t\approx 12 \end{gathered}$$