\documentclass{exam}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx,wasysym,gensymb}
\usepackage[hmargin=1in,tmargin=.75in,bmargin=1in]{geometry}
\usepackage[dvipsnames,table]{xcolor}
\usepackage{leftidx}
\usepackage{pgfplots}
\pagestyle{empty}

\usepackage{amsthm}
\makeatletter
\newtheoremstyle{indented}
  {.15cm}% space before
  {.15cm}% space after
  {\addtolength{\@totalleftmargin}{3.5em}
   \addtolength{\linewidth}{-3.5em}
   \parshape 1 3.5em \linewidth}% body font
  {}% indent
  {\bfseries}% header font
  {.}% punctuation
  {.5em}% after theorem header
  {}% header specification (empty for default)
\makeatother
\theoremstyle{indented}
\newtheorem{exmp}{Example}
\newtheorem{prac}{Practice}

\usepackage[%
  colorlinks=true,%
  urlcolor=blue%
]{hyperref}

\usepackage{tikz}
\usepackage{tikz-qtree}
\usetikzlibrary{calc,shapes,arrows.meta}

\usepackage{grffile}

\usepackage{enumitem}

\usepackage{stackengine}

\usepackage{pdfpages}

\usepackage{blkarray, bigstrut}

\usepackage{caption}

\usepackage{pifont}

\newcommand\funderline[1]{\stackengine{0pt}{\underline{\qquad\qquad\vphantom{#1}}}{#1}{O}{c}{F}{F}{L}}

\newcommand\tline[2]{\underset{\text{#1}}{\funderline{$#2$}}}

\newcommand\myperm[2]{\leftidx{_#1}{P}{_#2}}

\newcommand\mycomb[2]{\leftidx{_#1}{C}{_#2}}

\newcommand*\NewPage{\newpage\null}

\pgfmathdeclarefunction{gauss}{3}{\pgfmathparse{1/(#3*sqrt(2*pi))*exp(-((#1-#2)^2)/(2*#3^2))}}

\newcommand\CircN[1]{%
  \tikz[baseline=(char.base)]\node[circle,draw,inner sep=2pt] (char) {#1};}

\newcommand\CircP[1]{%
  \tikz[baseline=(char.base)]\node[circle,draw,inner sep=4.75pt] (char) {#1};}
  
\newcommand\BLUE[1]{\textcolor{blue}{\textbf{#1}}}
\newcommand\WHITE[1]{\textcolor{white}{\textbf{#1}}}
\newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp}
\newcolumntype{C}[1]{>{\PreserveBackslash\centering}p{#1}}
\newcolumntype{R}[1]{>{\PreserveBackslash\raggedleft}p{#1}}
\newcolumntype{L}[1]{>{\PreserveBackslash\raggedright}p{#1}}
\newcommand{\cmark}{\ding{51}}
\newcommand{\xmark}{\ding{55}}



\begin{document}
\noindent Name: \rule{7cm}{.2pt} \vspace{.3cm} \hfill Date:  \rule{3.5cm}{.2pt}
\vspace{.3cm}

\noindent\hrulefill
\begin{center}
{\bf \Large{MA 162}} \\\vspace{.1cm} Week 13 Recitation Worksheet (Thursday)
\end{center}
\vspace{-.3cm}
\hrulefill

\begin{center}
\fbox{\textbf{You must show all work to receive full credit.}}
\end{center}

\noindent It costs money to borrow money.  The cost one pays to borrow money is called \emph{interest}.  The money being borrowed or loaned is called the \emph{principal} or \emph{present value}.  When interest is only paid on the original amount borrowed, it is called \BLUE{simple interest}.  The interest is charged for the amount of time the money is borrowed.  If an amount $P$ is borrowed for a time $t$ at an interest rate of $r$ per time period, then the interest $I$ that is charged is
\begin{equation*}
I=Prt.
\end{equation*}
The total amount $A$ of the transaction is called the \emph{accumulated value} or the \emph{future value}, and is the sum of the principal and interest:
\begin{equation*}
A=P+I=P+Prt=P(1+rt).
\end{equation*}\vspace{0.5cm}

\begin{enumerate}[ref=\#\arabic*]
%1
\item (HW17 \#1) What is the interest if \$600 is borrowed for 6 months at 8\% annual simple interest?\vfill


%2
\item (HW17 \#2) Find the amount due if \$400 is borrowed for 4 months at 7\% annual simple interest.\vfill


%3
\item (HW17 \#4) Find the length of the loan in months if \$700 is borrowed with an annual simple interest rate of 8\% and with \$774.67 repaid at the end of the loan.\vfill


\newpage
%4
\item (HW17 \#5) Find the annual simple interest rate of a loan where \$1000 is borrowed and \$1060 is repaid at the end of 13 months.\vfill
\end{enumerate}


\noindent Interest can also work in your favor!
\begin{enumerate}
\setcounter{enumi}{4}
%5
\item (HW17 \#3) Charlie wants to buy a \$200 stereo set in 9 weeks. How much should he invest now at 16\% annual simple interest to have the money in 9 weeks?\vfill


%6
\item (HW17 \#6) An investment pays simple interest, and doubles in 9 years. What is the annual simple interest rate?\vfill
\end{enumerate}


\newpage
\noindent Banks sometimes deduct the simple interest from the loan amount at the time the loan is made. When this happens, we say the loan has been \BLUE{discounted}. The interest that is deducted is called the \emph{discount} and the actual amount given to the borrower is called the \emph{proceeds}. The amount the borrower is obligated to repay is called the \emph{maturity value}.  If an amount $M$ is borrowed for a time $t$ at a discount rate of $r$ per time period, then the discount $D$ is
\begin{equation*}
D=Mrt.
\end{equation*}
The proceeds $P$ is given by
\begin{equation*}
P=M-D=M-Mrt=M(1-rt).
\end{equation*}\vspace{0.5cm}

\begin{enumerate}
\setcounter{enumi}{6}
%7
\item (HW17 \#7) What are the proceeds for a discounted loan of \$600 repaid in 9 months at 12.25\% annual simple interest?\vfill


%8
\item (HW17 \#8) Consider a discounted loan of \$800, where the proceeds equal \$704. The loan is repaid at the end of 16 months. Find the annual simple discount rate.\vfill\vfill
\end{enumerate}


\newpage
\noindent With simple interest, you only earn interest on the principal (amount borrowed or invested).  Often times, the interest earned on an investment is reinvested, allowing one to earn interest on interest.  This is called \BLUE{compound interest}.  If an amount $P$ is invested at an annual interest rate $r$, compounded $n$ times per year, then after $t$ years the accumulated value $A$ is
\begin{equation*}
A=P\left(1+\dfrac{r}{n}\right)^{nt}.
\end{equation*}\vspace{0.5cm}

\begin{enumerate}[ref=\#\arabic*]
\setcounter{enumi}{8}
%9
\item (HW18 \#2) If you invest \$3{,}953.47 in an account earning an annual interest rate of 2.576\% compounded weekly, how much will be in your account after 2 years?\vfill


%10
\item[10$^*$.] (HW18 \#4) A mutual fund pays 10\% compounded monthly. How much should I invest now so that 2 years from now I will have \$4500 in the account?\vfill


\setcounter{enumi}{10}
%11
\item (HW18 \#8) If you make a deposit into a bank account, at what interest rate, compounded monthly, should you invest if you would like to double your investment in 40 months?\vfill
\end{enumerate}


\newpage
\noindent When interest is compounded ``infinitely many times'' per year, it is said to be \emph{compounded continuously}.  In this case, the accumulated value $A$ of an amount $P$ invested at an annual interest rate $r$ for $t$ years is
\begin{equation*}
A=Pe^{rt}.
\end{equation*}\vspace{0.5cm}

\begin{enumerate}
\setcounter{enumi}{11}
%12
\item[12$^*$.] (HW18 \#5) John wants to buy a new sports car, and he estimates that he'll need to make a \$4,525.00 down payment towards his purchase. If he has 32 months to save up for the new car, how much should he deposit into his account if the account earns 5.976\% compounded continuously so that he may reach his goal?\vfill


\setcounter{enumi}{12}
%13
\item (HW18 \#6) Suppose you invest \$16,900.00 into an account earning an interest rate of 2.257\% \\compounded continuously for 1 year and thereafter earning an interest rate of 4.164\% compounded weekly. How much money is in the account after 12 years?\vfill\vfill
\end{enumerate}


\newpage
\noindent If given the option of two interest rates compounded over different periods of time, how do you know which is better?
\begin{enumerate}
\setcounter{enumi}{13}
%14
\item[14$^*$.] (HW18 \#9/10) Find the \emph{effective rate of interest} for 
\begin{parts}
\part 5\% compounded semiannually.\vspace{5cm}
\part 4\% compounded continuously.
\end{parts}
\end{enumerate}
\end{document}