Question

Jimmy invest $16,000 in an account that pays 7.03% compounded quarterly. How long (in years and months) will it take for his investment to reach $23,000?

Answer 1

To solve this we are going to use the compound interest formula $A=P(1+ \frac{r}{n})^{nt}$
where:
$P$ is the investment
$r$ is the interest rate in decimal form
$n$ is the number of times the interest is compounded per year
$t$ is the time in years
$A$ is the amount after $t$ years 

First, lets convert the interest rate to decimal dividing it by 100%:
$r= \frac{7.03}{100} =0.0703$
Next, lets find $n$. Since we know that the interest is compounded every 4 months (quarterly), it will be compounded $\frac{12}{4} =3$ times in a year, so $n=4$.
We also know that $A=23000$ and $P=16000$, so lets replace all the quantities into our compound interest formula:
$25000=16000(1+ \frac{0.0703}{3})^{3t}$
$25000=16000(1.0234)^{3t}$

Notice that the the number of years $t$ is in the exponent, so we have to use logarithms to bring it down. But first lets divide both sides by 16000 to isolate the  exponential expression:
$\frac{25000}{16000} =(1.0234)^{3t}$
$(1.0234)^{3t} = \frac{25}{16}$
$ln(1.0234)^{3t} =ln( \frac{25}{16} )$
$t= \frac{ln( \frac{25}{16}) }{3ln(1.0234)}$
$t=6.43$

Now that we know $t=6.43$, the last thing to do is convert 0.43 years to months:
$(0.43 years)( \frac{12months}{1year} )=5.16months$

We can conclude that Jimmy's investment will take 6 years and 5 months to reach $25,000.