[tex]\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill&\$6000\\ P=\textit{original amount deposited}\dotfill &\$5000\\ r=rate\to 7\%\to \frac{7}{100}\dotfill &0.07\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years \end{cases}[/tex]
[tex]\bf 6000=5000\left(1+\frac{0.07}{12}\right)^{12\cdot t}\implies \cfrac{6000}{5000}\approx (1.0058)^{12t}\implies \cfrac{6}{5}\approx(1.0058)^{12t} \\\\\\ \log\left( \cfrac{6}{5} \right)\approx \log[(1.0058)^{12t}]\implies \log\left( \cfrac{6}{5} \right)\approx 12t\log(1.0058) \\\\\\ \cfrac{\log\left( \frac{6}{5} \right)}{12\log(1.0058)}\approx t\implies 2.63\approx t\impliedby \textit{about 2 years, 7 months and 16 days}[/tex]
It takes 2 years to have $6000 without depositing additional funds.
Given that,
Amount = $6000
Principal amount = $5000
rate = 7 % compounded monthly
Time = ?
In mathematics it deals with numbers of operations according to the statements.
It is defined as the fee you pay to borrow money or the fee you levy to lend money. Interest is considerable and frequently recalled as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.
[tex]6000/5000 = (1 + 0.07/12)^{12t}\\6/5 = (1.005)^{12t}\\[/tex]
[tex]1.20 = (1.005)^t\\1.20/ = (1.005)^{12t}[/tex]
taking ln both side
ln 1.13 = 12t ln(1.005)
t = ln 13/12 ln 1.005
t = 2.04 year
Thus, it takes 2 years to have $6000 without depositing additional funds.
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