Certainly! Let's carefully examine where the student's algebraic steps went wrong and then find the correct method to isolate [tex]\( r \)[/tex].
### Student's Solution:
1. [tex]\( A = P(1 + rt) \)[/tex]
2. [tex]\( A = P + rt \)[/tex]
3. [tex]\( A - P = rt \)[/tex]
4. [tex]\( r = \frac{A - P}{t} \)[/tex]
Algebraic Error Explanation:
The error occurs in Step 2, where the student incorrectly expanded the expression [tex]\( P(1 + rt) \)[/tex]:
[tex]\[ P(1 + rt) \neq P + rt \][/tex]
Instead, the correct expansion of [tex]\( P(1 + rt) \)[/tex] is:
[tex]\[ P(1 + rt) = P \cdot 1 + P \cdot rt = P + Prt \][/tex]
### Correct Solution:
To solve for [tex]\( r \)[/tex], let's start from the correct equation:
[tex]\[ A = P(1 + rt) \][/tex]
Follow these steps:
1. First, isolate the term with [tex]\( r \)[/tex] by dividing both sides by [tex]\( P \)[/tex]:
[tex]\[ \frac{A}{P} = 1 + rt \][/tex]
2. Next, subtract 1 from both sides to further isolate the term containing [tex]\( r \)[/tex]:
[tex]\[ \frac{A}{P} - 1 = rt \][/tex]
3. Finally, divide both sides by [tex]\( t \)[/tex] to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\frac{A}{P} - 1}{t} \][/tex]
So, the correct modified equation to solve for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{\frac{A}{P} - 1}{t} \][/tex]
Putting it into a simpler form:
[tex]\[ r = \frac{A / P - 1}{t} \][/tex]
This result correctly isolates [tex]\( r \)[/tex] from the original simple interest formula [tex]\( A = P(1 + rt) \)[/tex].
