The following pairs of investment plans are identical except for a small difference in interest rates. Compute the balance in the accounts after $10$ and $30$ years. Discuss the difference.

Chang invests $\$ 500$ in an account that earns $3.5 \%$ compounded annually. Kio invests $\$ 500$ in a different account that earns $3.75\%$ compounded annually.

Mario deposits $140 in an account that earns simple interest at a rate of 5% per year. On the same day, Andy deposits$150 in an account that earns simple interest at a rate of 3% per year. Mario and Andy plan to keep the accounts for at least 5 years, but make no additional deposits. For each account, write a rule for the the amount I (in dollars) of interest as a function of the time t (in years) that the account has been open.

Rosa invests $\$ 3000$ in an account with an $\mathrm{APR}$ of $4\%$ and annual compounding. Julian invests $\$ 2500$ in an account with an $\mathrm{APR}$ of $5 \%$ and annual compounding.

a. Compute the balance in each account after $5$ and $20$ years.

b. Determine, for each account and for $5$ and $20$ years, the percentage of the balance that is interest.

c. Comment on the effect of interest rates and patience.

What is continuous compounding? How does the APY for continuous compounding compare to the APY for, say, daily compounding? Explain the formula for continuous compounding.

Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.

A $\$ 3000$ deposit in an account with an $\mathrm{APR}$ of $7.5 \%$

Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.

A $\$ 500$ deposit in an account with an $\mathrm{APR}$ of $2.7 \%$

Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.

A $\$ 3000$ deposit in an account with an $\mathrm{APR}$ of $6 \%$

Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.

A $\$ 7000$ deposit in an account with an $\mathrm{APR}$ of $4.5 \%$

How much must you deposit today into the following accounts in order to have $\$ 25,000$ in $8$ years for a down payment on a house? Suppose that no additional deposits are made. An account with annual compounding and an APR of $6 \%$

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 4000$ is invested for $20$ years with an APR of $6 \%$ and monthly compounding.

Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.

A $\$ 2000$ deposit in an account with an $\mathrm{APR}$ of $3.1 \%$

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 15,000$ is invested for $15$ years with an APR of $4.2 \%$ and monthly compounding.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 25,000$ is invested for $5$ years with an APR of $3 \%$ and daily compounding.

Use the appropriate compound interest formula to compute the balance in account after the stated period of time. $\$ 5000$ is invested for $10$ years with an APR of $2 \%$ and quarterly compounding.