Question

Sara invests $1,200 each year in an IRA ( 12 years in an account that earned 5%
compounded annually.) At the end of 12 years, she stopped making payments to the account, but continued to invest her accumulated amount at 5% compounded annually for the next 11 years.
a.What was
the value of the Ira at the end of 12 years?
b.What was the value of the investment at the end of the next 11 years?
c.How much interest did she earn?

Please show work for how this is done.

Answer 1

A)

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

$\bf \begin{cases} A= \begin{array}{llll} \textit{original amount}\\ \textit{already compounded} \end{array} \begin{array}{llll} \end{array}\\ pymnt=\textit{periodic deposits}\to &1200\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{each year, once} \end{array}\to &1\\ t=years\to &12 \end{cases}$

B)

let's say after 12years, she ended up with a value of say "P"
so.. now she's just sitting on P, making no more deposits to it
just taking whatever the compound 5% interest will give, thus

$\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \qquad \begin{cases} A=\textit{compounded amount}\\ P=\textit{original amount deposited}\to &\boxed{P}\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{again, is only once} \end{array}\to &1\\ t=years\to &11 \end{cases}$


C)

from A) she made 1,200 every year, for 12 years that's 1200*12, that's how much she put out of pocket, if you got an amount P from A), then the interest is just the difference, or P - (1200*12)

from B), she started with an original amount of P, and ended up with a compounded amount of A after 11years, so the interest is just also the difference, or A - P

add those two folks together, and that's the total interest she got for the 23  years