To solve this we are going to use the compound interest formula [tex]A=P(1+ \frac{r}{n})^{nt} [/tex]
where:
[tex]P[/tex] is the investment
[tex]r[/tex] is the interest rate in decimal form
[tex]n[/tex] is the number of times the interest is compounded per year
[tex]t[/tex] is the time in years
[tex]A[/tex] is the amount after [tex]t[/tex] years
First, lets convert the interest rate to decimal dividing it by 100%:
[tex]r= \frac{7.03}{100} =0.0703[/tex]
Next, lets find [tex]n[/tex]. Since we know that the interest is compounded every 4 months (quarterly), it will be compounded [tex] \frac{12}{4} =3[/tex] times in a year, so [tex]n=4[/tex].
We also know that [tex]A=23000[/tex] and [tex]P=16000[/tex], so lets replace all the quantities into our compound interest formula:
[tex]25000=16000(1+ \frac{0.0703}{3})^{3t} [/tex]
[tex]25000=16000(1.0234)^{3t} [/tex]
Notice that the the number of years [tex]t[/tex] 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:
[tex] \frac{25000}{16000} =(1.0234)^{3t} [/tex]
[tex](1.0234)^{3t} = \frac{25}{16} [/tex]
[tex]ln(1.0234)^{3t} =ln( \frac{25}{16} )[/tex]
[tex]t= \frac{ln( \frac{25}{16}) }{3ln(1.0234)} [/tex]
[tex]t=6.43[/tex]
Now that we know [tex]t=6.43[/tex], the last thing to do is convert 0.43 years to months:
[tex](0.43 years)( \frac{12months}{1year} )=5.16months[/tex]
We can conclude that Jimmy's investment will take 6 years and 5 months to reach $25,000.
