Question

Kristen invests $5,745 in a bank. The bank pays 6.5% interest compounded monthly. How long must she leave the money in the bank for it to double? Round the nearest tenth of a year.

Answer 1

formula is $A=P(1-\frac{r}{n})^{nt}$

where A=final amount

P=principal

r=interest rate in decimal

n=number of times per year it is compounded

t=time in years

we want to find where

A=2P

and P=5745

and r=6.5%=0.065

n=monthly=12

remember that $ln(x^a)=a(ln(x))$

also that $(a^b)^c=a^{bc}$

$2(5745)=5745(1+\frac{0.065}{12})^{12t}$, solving for t

divide both sides by 5745 to simplify things a bit

$2=(1+\frac{0.065}{12})^{12t}$ I'd rather not simplify this because it give us a decimals and those aren't exact, if we combine, we get 12.065/12 for inside parenthases

$2=(\frac{12.065}{12})^{12t}$

take ln of both sides

$ln(2)=ln((\frac{12.065}{12})^{12t})$

$ln(2)=(12t)ln(\frac{12.065}{12})$

divide both sides by $12ln(\frac{12.065}{12})$

$\frac{ln(2)}{12ln(\frac{12.065}{12})}=t$

using our calculator, t≈10.6927

so rounded, we get 10.7 years