# You go to the bank and deposit $2,500 into your savings. Your bank has annual interest rate of 8%, compounded monthly. How long would it take the investment to reach$5,000?

It would take 8 years and nine months for the investment to surpass $5,000. #### Explanation: The general formula for compound interest is $F V = P V {\left(1 + \frac{i}{n}\right)}^{n t}$Where $t$is the number of years the investment is left to accumulate interest. This is what we are trying to solve for. $n$is the number of compounding periods per year. In this case, since the interest is compounded monthly, $n = 12$. $F V$is the future value of the investment after $n t$compounding periods. In this case FV=$5,000.
$P V$ is the present value of the investment which is the amount of money originally deposited before accumulation of any interest. In this case PV=$2,500. $i$is the annual interest rate the bank offers to depositors. In this case $i = 0.08$. Before we start plugging numbers into our equation, let's solve the equation for $t$. Divide both sides by $P V$. $\frac{F V}{P V} = {\left(1 + \frac{i}{n}\right)}^{n t}$Take the natural log of both sides. Why the NATURAL log? Because it's the natural thing to do. Sorry, a little math humor there. In reality it really doesn't matter what base you use as long as you apply the same base to both sides of the equation. Try it with ${\log}_{\sqrt{17}}$and you'll still get the right answer. $\ln \left(\frac{F V}{P V}\right) = \ln {\left(1 + \frac{i}{n}\right)}^{n t} = n t \ln \left(1 + \frac{i}{n}\right)$Divide both sides by $n \ln \left(1 + \frac{i}{n}\right)$. $t = \frac{\ln \left(\frac{F V}{P V}\right)}{n \ln \left(1 + \frac{i}{n}\right)}$NOW we start plugging in numbers! $t = \frac{\ln \left(\frac{5000}{2500}\right)}{12 \ln \left(1 + \frac{0.08}{12}\right)} \approx 8.693$years 8.693 years is 8 years and $0.693 \cdot 12 \approx 8.3\$ months. Thus, you would have to wait 8 years and 9 months since the interest is compounded monthly.