Walton just got his first job after college! He is making a budget and can afford to put $$200$ a month away for retirement. If his goal is to retire in $40$ years with $$500,000,$ what annual interest rate, compounding monthly, does his retirement savings account need to have?

Use the formula fo ind solve for the

Situation $3$

L'Dona is buying a house! How much will her monthly mortgage payment be if she is borrowing $$150,000$ at $3\mathrm{.}2\%$ interest for $30$ years?

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Situation $4$

Charlene has retired with $$400,000$ in the bank, earning $2\%$ annual interest, compounded quarterly. If she withdraws $$10,000$ per quarter, how long until the account is empty?

Use the formula fo and solve for the

Simple One$-$timerest Intere

$I=P_{0}r$

$A=P_0+I=P_0+P_{0}r=P_0(1+r)$

I is the interest

$A$ is the end amount: principal plus interest

$P_0$ is the principal $($starting amount$)$

$r$ is the interest rate $($in decimal form. Example: $5\%=0\mathrm{.}05)$

Compound Interest

$P_N=P_0(1+\frac{r}{k}{)}^{Nk}$

$P_N$ is the balance in the account after $N$ years.

$P_0$ is the starting balance of the account $($also called initial deposit, or principal$)$

$r$ is the annual interest rate in decimal form

$k$ is the number of compounding periods in one year.

Annuity Formula

$P_N=\frac{d((1+\frac{r}{k}{)}^{Nk}-1)}{(\frac{r}{k})}$

$P_N$ is the balance in the account after $N$ years.

$d$ is the regular deposit $($the amount you deposit each year, each month, etc.$)$

$r$ is the annual interest rate in decimal form.

Annuity Formula

$P_N=\frac{d((1+\frac{r}{k}{)}^{Nk}-1)}{(\frac{r}{k})}$

$P_N$ is the balance in the account after $N$ years.

$d$ is the regular deposit $($the amount you deposit each year, each month, etc.$)$

$r$ is the annual interest rate in decimal form.

$k$ is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.

Payout Annuity Formula

$P_0=\frac{d(1-(1+\frac{r}{k}{)}^{-Nk})}{(\frac{r}{k})}$

$P_0$ is the balance in the account at the beginning $($starting amount, or principal$).d$ is the regular withdrawal $($the amount you take out each year, each month, etc.$)r$ is the annual interest rate $($in decimal form. Example: $5\%=0\mathrm{.}05)$

$k$ is the number of compounding periods in one year.

$N$ is the number of years we plan to take withdrawals