Question

If you need $8000 to travel to Europe after you graduate in 4 year. How much would your monthly deposits need to be if your account earns interest at 5% compounded monthly?

Answer 1

$\bf \qquad \qquad \textit{Future Value of an ordinary annuity}\\ \left. \qquad \qquad \right.(\textit{payments at the end of the period}) \\\\ A=pymnt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right]$

$\bf \qquad \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\to & \begin{array}{llll} 8000 \end{array}\\ pymnt=\textit{periodic payments}\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years\to &4 \end{cases}$

$\bf 8000=pymnt\left[ \cfrac{\left( 1+\frac{0.05}{12} \right)^{12\cdot 4}-1}{\frac{0.05}{12}} \right] \\\\\\ \cfrac{8000}{\left[ \frac{\left( 1+\frac{0.05}{12} \right)^{12\cdot 4}-1}{\frac{0.05}{12}} \right]}=pymnt\implies \cfrac{8000}{\frac{\left( \frac{241}{240} \right)^{48}-1}{\frac{1}{240}}}=pymnt \\\\\\ \cfrac{8000}{53.0148852}\approx pymnt\implies 150.9010152318\approx pymnt$