Suppose you lent $\$ 5000$ to friend $1$ for $18$ months at an annual simple interest rate of $9 \%$. After $1$ year, you need money for an emergency and decide to sell the note to friend $2$.

(a) How much does friend $1$ owe when the loan is due?

(b) If your agreement with friend $2$ means that she earns simple interest at an annual rate of $12 \%$, how much did friend $2$ pay you for the note?

$\$ 1800$ is invested for $9$ months at an annual simple interest rate of $15 \%$.

(a) How much interest will be earned?

(b) What is the future value of the investment after $9$ months?

Economy models A simple economy has a shoe industry and a cattle industry. Each unit of shoe output requires inputs of $0.1$ unit of shoes and $0.2$ unit of cattle products. Each unit of cattle products output requires inputs of $0.1$ unit of shoes and $0.05$ unit of cattle products.

(a) Write the technology matrix for this simple economy.

(b) If surpluses of $850$ units of shoes and $275$ units of cattle products are desired, find the gross production of each industry.

A developer wants to buy a certain parcel of land. The developer believes she can afford payments of $\$ 44,000$ each half-year for the next $7$ years. How much can she borrow and hold to this budget at each of the following interest rates?

(a) $8.9 \%$ compounded semiannually

(b) $7.3 \%$ compounded semiannually

(a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

Parents agree to invest $\$ 500$ (at $10 \%$ compounded semiannually) for their son on the December $31$ or June $30$ following each semester that he makes the dean's list during his $4$ years in college. If he makes the dean's list in each of the $8$ semesters, how much money will his parents have to give him when he graduates?

(a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

An insurance settlement of $\$ 1.5$ million must replace Trixie Eden's income for the next $40$ years. What income will this settlement provide at the end of each month if it is invested in an annuity that earns $8.4 \%$ compounded monthly?

Chess legend Legend has it that when the king of Persia wanted to reward the inventor of chess with whatever he desired, the inventor asked for one grain of wheat on the first square of the chessboard, with the number of grains doubled on each square thereafter for the remaining $63$ squares. Find the number of grains on the $64$th square. (The sum of all the grains of wheat on all the squares would cover Alaska more than $3$ inches deep with wheat!)

Bill Casler bought a $\$ 2000$, $9$-month certificate of deposit ($\mathrm{CD}$) that would earn $8 \%$ annual simple interest. Three months before the $\mathrm{CD}$ was due to mature, Bill needed his $\mathrm{CD}$ money, so a friend agreed to lend him money and receive the value of the $\mathrm{CD}$ when it matured.

(a) What is the value of the $\mathrm{CD}$ when it matures?

(b) If their agreement allowed the friend to earn a $10 \%$ annual simple interest return on his loan to Bill, how much did Bill receive from his friend?

A woman makes $\$ 3000$ contributions at the end of each half-year to a retirement account for a period of $8$ years. For the next $20$ years, she makes no additional contributions and no withdrawals.

(a) If the account earns $7.5\%$ compounded semiannually, find the value of the account after the $28$ years.

(b) If this account is used to set up an annuity that pays her an amount at the beginning of each $6$-month period for the next $20$ years, how much will each payment be?

Find the annual percentage yield for an investment at

(a) $7.3 \%$ compounded monthly

(b) $6 \%$ compounded continuously.

A couple purchasing a home budget $\$ 1800$ per month for their loan payment. If they have $\$ 20,000$ available for a down payment and are considering a $25$-year loan, how much can they spend on the home at each of the following rates?

(a) $6.9 \%$ compounded monthly

(b) $7.5 \%$ compounded monthly

Seana wants to save $\$ 20,000$ toward the purchase of a new car. She can contribute $\$ 400$ at the beginning of each month into an account that earns $4.8 \%$ compounded monthly. How long until she reaches her goal?

A couple bought some stock for $\$ 30$ per share that pays an annual dividend of $\$ 0.90$ per share. After 1 year the price of the stock was $\$ 33$. Determine the simple interest rate on the growth of their investment.

A young couple with a $\$ 120,000$ inheritance wants to invest all this money. They plan to diversify their investments, choosing some of each of the following types of stock.

Their strategy is to have the total investment in growth stocks equal to the sum of the other investments, and their goal is a $13 \%$ growth for their investment. How many shares of each type of stock should they purchase?