the principal p is borrowed and the loans future value a at time t is given. determine the loans simple…

Explanation:

Step1: Recall simple - interest formula

The formula for simple interest is $A = P(1+rt)$, where $A$ is the future value, $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years. First, convert the time $t$ from months to years. Since $t = 9$ months, $t=\frac{9}{12}=\frac{3}{4}= 0.75$ years.

Step2: Rearrange the formula to solve for $r$

Starting with $A = P(1 + rt)$, we can expand it to $A=P + Prt$. Then, $A - P=Prt$. So, $r=\frac{A - P}{Pt}$.

Step3: Substitute the given values

We know that $P = 2300$, $A = 2786$, and $t = 0.75$. Substitute these values into the formula for $r$: $r=\frac{2786 - 2300}{2300\times0.75}$. First, calculate the numerator: $2786−2300 = 486$. Then, calculate the denominator: $2300\times0.75=1725$. So, $r=\frac{486}{1725}\approx0.282$. To express it as a percentage, multiply by 100: $r = 28.2%$.

Answer:

$28.2%$