Question

If $3000 is deposited in an account that pays 5% interest, what is the difference in the amount after 4 years between the amount earned if the principal is compounded annually and the amount earned calculated using simple interest?

A. $30.72

B. $41.12

C. $46.52

D. $53.76

Answer 1

`\bf \qquad \textit{Simple Interest Earned Amount}\\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to& \$3000\\ r=rate\to 5\%\to (5)/(100)\to &0.05\\ t=years\to &4 \end{cases} \\\\\\ A=3000(1+0.05\cdot 4)\implies \boxed{A=3600}\\\\ -------------------------------\\\\`


`\bf \qquad \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}\to &\$3000\\ r=rate\to 5\%\to (5)/(100)\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\to &1\\ t=years\to &4 \end{cases}`


`\bf A=3000\left(1+(0.05)/(1)\right)^(1\cdot 4)\implies A=3000(1.05)^4\implies \boxed{A=3646.51875}\\\\ -------------------------------\\\\ \stackrel{\textit{compounded interest}}{3646.51875}~~-~~\stackrel{\textit{simple interest}}{3600}`