Future Value of an Investment
The formula to calculate the future value (FV) of an investment P for t years at a rate r is:
[tex]FV=P\mleft(1+\frac{r}{m}\mright)^{m\cdot t}[/tex]
Where m is the number of compounding periods per year.
Leyla needs FV = $7000 for a future project. She can invest P = $5000 now at an annual rate of r = 10.5% = 0.105 compunded monthly. This means m = 12.
It's required to find the time required for her to have enough money for her project.
Substituting:
[tex]\begin{gathered} 7000=5000(1+\frac{0.105}{12})^{12t} \\ \text{Calculating:} \\ 7000=5000(1.00875)^{12t} \end{gathered}[/tex]
Dividing by 5000:
[tex]\frac{7000}{5000}=(1.00875)^{12t}=1.4[/tex]
Taking natural logarithms:
[tex]\begin{gathered} \ln (1.00875)^{12t}=\ln 1.4 \\ \text{Operating:} \\ 12t\ln (1.00875)^{}=\ln 1.4 \\ \text{Solving for t:} \\ t=\frac{\ln 1.4}{12\ln (1.00875)^{}} \\ t=3.22 \end{gathered}[/tex]
It will take 3.22 years for Leila to have $7000
