Answer :
Step 1. Gather all of the information.
We have the Principal P:
[tex]P=2300[/tex]The total amount A for the loan's future value:
[tex]A=2840[/tex]And the time t in moths:
[tex]t=9\text{months}[/tex]We are going to need the time in years, so we consider that 9 months are 9/12 of a year:
[tex]t=\frac{9}{12}\text{years}[/tex]Which can be simplified to 3/4 of a year:
[tex]t=\frac{3}{4}[/tex]Step 2. Remember the simple interest formula:
[tex]A=P(1+rt)[/tex]From this formula, we will need to find the simple interest rate r, so we will solve for r.
-The first step to solve for r is to divide both sides by P:
[tex]\frac{A}{P}=\frac{P}{P}(1+rt)[/tex]on the right-hand side P/P is 1 so we are left only with 1+rt:
[tex]\frac{A}{P}=1+rt[/tex]-the second step to solve for r is to subtract 1 to both sides:
[tex]\frac{A}{P}-1=1-1+rt[/tex]On the right-hand side, 1-1 cancels each other:
[tex]\frac{A}{P}-1=rt[/tex]-the last step to solve for r is to divide both sides by t:
[tex]\frac{\frac{A}{P}-1}{t}=r[/tex]This is the equation we will use to find the value of r.
Step 3. Substitute the known values into the formula to find r:
[tex]\begin{gathered} r=\frac{\frac{A}{P}-1}{t} \\ r=\frac{\frac{2840}{2300}-1}{\frac{3}{4}} \end{gathered}[/tex]We have substituted the values of A, P and t.
Simplifying the fractions:
[tex]r=\frac{1.2348-1}{0.75}[/tex]Solving the final operations:
[tex]\begin{gathered} r=\frac{0.2348}{0.75} \\ r=0.313 \end{gathered}[/tex]This is the simple interest rate represented in decimal form, to convert it to percentage, we need to multiply the result by 100:
[tex]\begin{gathered} r=0.313\times100 \\ r=31.3 \end{gathered}[/tex]The final result is 31.3%
Answer:
the simple interest rate to the nearest tenth is:
31.3%