[tex]\bf ~~~~~~~~~~~~ \textit{Amortized Loan Value} \\\\ pymt=P\left[ \cfrac{\frac{r}{n}}{1-\left( 1+ \frac{r}{n}\right)^{-nt}} \right][/tex]
[tex]\bf ~~~~~~ \begin{cases} P= \begin{array}{llll} \textit{original amount deposited}\\ \end{array} & \begin{array}{llll} \end{array}\\ pymt=\textit{periodic payments}\dotfill &448.21\\ r=rate\to 9.9\%\to \frac{9.9}{100}\dotfill &0.099\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &10 \end{cases}[/tex]
[tex]\bf 448.21=P\left[ \cfrac{\frac{0.099}{12}}{1-\left( 1+ \frac{0.099}{12}\right)^{-12\cdot 10}} \right]\implies 448.21=P\left[ \cfrac{0.00825}{1-\left( 1.00825\right)^{-120}} \right] \\\\\\ 448.21\approx P \left[ \cfrac{0.00825}{0.6269111} \right]\implies 448.21\approx P(0.0131598) \\\\\\ \cfrac{448.21}{0.0131598}\approx P\implies 34059.02825 \approx P[/tex]
