To find the interest rate charged given that [tex]$680 is worth $[/tex]698.70 after three months, we can utilize the simple interest formula:
[tex]\[ A = P(1 + rt) \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount.
- [tex]\( P \)[/tex] is the initial amount.
- [tex]\( r \)[/tex] is the interest rate.
- [tex]\( t \)[/tex] is the time period in years.
We are given:
- [tex]\( P = 680 \)[/tex]
- [tex]\( A = 698.70 \)[/tex]
- [tex]\( t = 3 \)[/tex] months, which is equivalent to [tex]\( \frac{3}{12} \)[/tex] years.
First, set up the equation with the known values:
[tex]\[ 698.70 = 680 \left(1 + r \cdot \frac{3}{12}\right) \][/tex]
Next, simplify the time period:
[tex]\[ 698.70 = 680 \left(1 + r \cdot 0.25\right) \][/tex]
Then, solve for [tex]\( r \)[/tex]:
[tex]\[ 698.70 = 680 (1 + 0.25r) \][/tex]
Divide both sides by 680:
[tex]\[ \frac{698.70}{680} = 1 + 0.25r \][/tex]
This simplifies to:
[tex]\[ 1.0275 = 1 + 0.25r \][/tex]
Subtract 1 from both sides to isolate the term with [tex]\( r \)[/tex]:
[tex]\[ 0.0275 = 0.25r \][/tex]
Divide both sides by 0.25 to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{0.0275}{0.25} = 0.11 \][/tex]
The interest rate [tex]\( r \)[/tex] is 0.11, which can be expressed as a percentage by multiplying by 100:
[tex]\[ r \times 100 = 0.11 \times 100 = 11 \% \][/tex]
Thus, the interest rate charged is:
[tex]\[ 11.0\% \][/tex]
Rounded to the nearest cent as required.
