Question

To pay for a home improvement project that totals $16,000, Genesis is choosing between taking out a simple interest bank loan at 8% for 3 years or paying with a credit card that compounds monthly at an annual rate of 15% for 7 years. Which plan would give Genesis the lowest monthly payment?

Answer 1

well, let's first check with the Bank

`~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$16000\\ r=rate\to 8\%\to (8)/(100)\dotfill &0.08\\ t=years\dotfill &3 \end{cases} \\\\\\ A=16000[1+(0.08)(3)] \implies A = 19840~\hfill \underset{monthly~payment}{\stackrel{19840~/ \stackrel{months}{36}}{\approx \text{\LARGE 551.11}}}`

now let's check with the Credit Card

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$16000\\ r=rate\to 15\%\to (15)/(100)\dotfill &0.15\\ n= \begin{array}{llll} \textit{times it compounds per year} \end{array}\dotfill &12\\ t=years\dotfill &7 \end{cases}`

`A=16000\left(1+(0.15)/(12)\right)^(12\cdot 7) \implies A \approx 45425.81~\hfill \underset{monthly~payment}{\stackrel{45425.81~/ \stackrel{months}{84}}{\approx\text{\LARGE 540.78}}}`

well, seems the Credit Card is the better deal monthly wise, though in the long run is a lot of dough.