Question

Today you have exactly $600.38 in your savings account earning exactly 0.5% interest, compounded monthly. How many years will it take before that account reaches $1030.14? Use trial and error to find the number of years (t). Explain how you know your answer is correct.

Answer 1

ANSWER

$107.9\text{ years}$

EXPLANATION

To find the number of years before the account reaches $1030.14, we have to apply the formula for monthly compounded amount:

$A=P(1+\frac{r}{12})^{12t}$

where P = initial amount = $600.38

r = interest rate = 0.5% = 0.005

t = number of years

Therefore, solve for t:

$\begin{gathered} 1030.14=600.38(1+\frac{0.005}{12})^{12\cdot t} \\ \Rightarrow\frac{1030.14}{600.38}=(1.000417)^{12t} \\ \Rightarrow1.716=1.000417^{12t} \end{gathered}$

Convert the exponential equation above into a logarithmic equation as follows:

$\log _{1.000417}1.716=12t$

Therefore, we have that:

$\begin{gathered} \frac{\log _{10}1.716}{\log _{10}1.000417}=12t \\ \Rightarrow12t=1294.96 \\ t=\frac{1294.96}{12} \\ t\approx107.9\text{ years} \end{gathered}$

That is the number of years that it will take.