To solve for the rate of interest [tex]\( r \)[/tex] in terms of the account balance [tex]\( P \)[/tex], the initial amount [tex]\( I \)[/tex], and time [tex]\( t \)[/tex] in years, we start from the given formula for simple interest:
[tex]\[
P = I (1 + rt)
\][/tex]
We need to isolate [tex]\( r \)[/tex] in this equation. Here are the detailed steps:
1. Start with the given formula:
[tex]\[
P = I (1 + rt)
\][/tex]
2. Divide both sides by [tex]\( I \)[/tex] to isolate the term containing [tex]\( r \)[/tex]:
[tex]\[
\frac{P}{I} = 1 + rt
\][/tex]
3. Subtract 1 from both sides to further isolate the term containing [tex]\( r \)[/tex]:
[tex]\[
\frac{P}{I} - 1 = rt
\][/tex]
4. Recognize that [tex]\((\frac{P}{I} - 1)\)[/tex] is a common denominator and can be simplified:
[tex]\[
\frac{P - I}{I} = rt
\][/tex]
5. Divide both sides by [tex]\( t \)[/tex] to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\frac{P - I}{I}}{t}
\][/tex]
6. Simplify the fraction by multiplying the numerator and the denominator by [tex]\( I \)[/tex]:
[tex]\[
r = \frac{P - I}{I t}
\][/tex]
Thus, the correct formula to solve for the rate [tex]\( r \)[/tex] in terms of the account balance [tex]\( P \)[/tex], the initial amount [tex]\( I \)[/tex], and time [tex]\( t \)[/tex] is:
[tex]\[
r = \frac{P - I}{I t}
\][/tex]
The corresponding choice is:
A) [tex]\( r = \frac{P - I}{I t} \)[/tex]
