Banks sometimes deduct the simple interest from the loan amount at the time the loan is made. When this happens, we say the loan has been discounted. The interest that is deducted is called the discount and the actual amount given to the borrower is called the proceeds. The amount the borrower is obligated to repay is called the maturity value. If an amount $M$ is borrowed for a time $t$ at a discount rate of $r$ per time period, then the discount $D$ is

$D=Mrt.$

The proceeds $P$ is given by

$P=M-D=M-Mrt=M(1-rt).$

$7*.($HW$17$ #$7)$ What are the proceeds for a discounted loan of $$600$ repaid in $9$ months at $12\mathrm{.}25\%$ annual simple interest?

$A=P(1+rt),6000\mathrm{.}1225\frac{3}{4}$

$P=600,55\mathrm{.}125$

$r=12\mathrm{.}25\%,P=m-D=600-55\mathrm{.}125=544\mathrm{.}87$

$t=\frac{9}{12r}=\frac{3}{4}rr,P$

$($HW$17$ #$8)$ Consider a discounted loan of $$800,$ where the proceeds equal $$704.$ The loan is repaid at the end of $16$ months. Find the annual simple discount rate.