jose has determined he needs to have $500,000...
jose has determined he needs to have $500,000 for retirement in 20 years. his account earns 10% interest. do not round numbers while working on the problem. only round the final answer. since we are working with money, round to the cent. a) how much will he need to deposit in the account each month? b) how much total money will he put into the account? c) how much total interest will he earn?
Answer
# Explanation:
## Step1: Identify the relevant formula
The future - value of an ordinary annuity formula is $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, where $FVA$ is the future value of the annuity, $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods. The annual interest rate $i = 10\%=0.1$, the number of years $t = 20$ years, and since we are making monthly deposits, the number of periods $n=20\times12 = 240$ months, and the interest rate per period $r=\frac{0.1}{12}$. The future - value $FVA = 500000$.
## Step2: Solve for the monthly payment $P$
From $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, we can re - arrange it to $P=\frac{FVA\times r}{(1 + r)^{n}-1}$. Substitute $FVA = 500000$, $r=\frac{0.1}{12}$, and $n = 240$ into the formula.
\[
\begin{align*}
P&=\frac{500000\times\frac{0.1}{12}}{(1+\frac{0.1}{12})^{240}-1}\\
\end{align*}
\]
Let $x=\frac{0.1}{12}\approx0.008333$ and $m = 240$. Then $(1 + x)^{m}=(1+\frac{0.1}{12})^{240}\approx7.27847787$.
\[
\begin{align*}
P&=\frac{500000\times0.008333}{7.27847787 - 1}\\
&=\frac{4166.5}{6.27847787}\\
&\approx663.51
\end{align*}
\]
## Step3: Calculate the total amount deposited
The total amount deposited is the monthly deposit amount times the number of months. So, the total amount deposited $=P\times n=663.51\times240 = 159242.40$.
## Step4: Calculate the total interest earned
The total interest earned is the future value of the account minus the total amount deposited. So, the total interest earned $=500000 - 159242.40=340757.60$.
# Answer:
a) $\$663.51$
b) $\$159242.40$
c) $\$340757.60$