Respuesta :
SOLUTION
We will use the formula
[tex]\begin{gathered} PMT=P(\frac{\frac{r}{n}}{1-(1+\frac{r}{n})^{-nt}}) \\ where\text{ PMT = per month deposit = \$850} \\ P=1,000,000 \\ r=interest\text{ rate = 4.8\% = }\frac{4.8}{100}=0.048 \\ n=number\text{ of compounding = 12} \\ t=time\text{ in years = ?} \end{gathered}[/tex]Applying we have
[tex]\begin{gathered} 850=1,000,000(\frac{\frac{0.048}{12}}{1-(1+\frac{0.048}{12})^{-12t}}) \\ 850=1,000,000(\frac{0.004}{1-(1+0.004)^{-12t}} \\ 850=\frac{4000}{1-(1.004)^{-12t}} \\ 850(1-(1.004)^{-12t})=4000 \\ (1-(1.004)^{-12t})=\frac{4000}{850} \end{gathered}[/tex]Continuing we have
[tex]\begin{gathered} 1-(1.004)^{-12t}=4.7058824 \\ (1.004)^{-12t}=1-4.7058824 \\ (1.004)^{-12t}=-3.7058824 \\ taking\text{ log} \\ log(1.004)^{-12t}=-log(3.7058824) \\ -12tlog(1.004)=-log(3.7058824) \\ t=\frac{-log(3.7058824)}{-12log(1.004)} \\ t=27.34457 \\ t\approx27\text{ years } \end{gathered}[/tex]So when you retire you will be
[tex]23+27=50\text{ years }[/tex]So you will be approximately 50 years at retirement
