Question

suppose you want to have $500,000 for retirement in 25 years. your account earns 10% interest. round your answers to the nearest cent.
a) how much would you need to deposit in the account each month?
b) how much interest will you earn?

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 monthly interest rate $r=\frac{0.1}{12}$. The number of years $t = 25$ years, and the number of months $n=25\times12 = 300$ months, and $FVA = 500000$.

Step2: Solve for $P$

From $FVA = P\times\frac{(1 + r)^{n}-1}{r}$, we can re - arrange it to solve for $P$: $P=\frac{FVA\times r}{(1 + r)^{n}-1}$.
Substitute $FVA = 500000$, $r=\frac{0.1}{12}$, and $n = 300$ into the formula:
\[

$$\begin{align*} P&=\frac{500000\times\frac{0.1}{12}}{(1+\frac{0.1}{12})^{300}-1}\\ \end{align*}$$

\]
First, calculate $(1+\frac{0.1}{12})^{300}$. Let $x=\frac{0.1}{12}\approx0.008333$. Then $(1 + x)^{300}\approx10.892554$.
\[

$$\begin{align*} P&=\frac{500000\times\frac{0.1}{12}}{10.892554 - 1}\\ &=\frac{\frac{500000\times0.1}{12}}{9.892554}\\ &=\frac{4166.67}{9.892554}\\ &\approx421.21 \end{align*}$$

\]

Step3: Calculate the total amount deposited

The total amount deposited over 300 months is $P\times n=421.21\times300 = 126363$.

Step4: Calculate the interest earned

The interest earned $I=FVA-(P\times n)$. Substitute $FVA = 500000$ and $P\times n=126363$. Then $I = 500000-126363=373637$.

Answer:

a) $\$421.21$
b) $\$373637$