The algebraic error in the student's work occurs in the step:
[tex]\[ A = P + rt \][/tex]
This step is incorrect because the formula \( A = P(1 + rt) \) should not be expanded or simplified incorrectly. The original formula \( A = P(1 + rt) \) means that the amount \( A \) is equal to the principal \( P \) multiplied by the sum of 1 and the product of the rate \( r \) and time \( t \).
Let's correct the solution step-by-step:
1. Start with the original formula:
[tex]\[ A = P(1 + rt) \][/tex]
2. Divide both sides of the equation by \( P \) to isolate the term \( (1 + rt) \):
[tex]\[ \frac{A}{P} = 1 + rt \][/tex]
3. Subtract 1 from both sides to solve for \( rt \):
[tex]\[ \frac{A}{P} - 1 = rt \][/tex]
4. Finally, divide both sides by \( t \) to solve for \( r \):
[tex]\[ r = \frac{\frac{A}{P} - 1}{t} \][/tex]
The correctly solved equation for \( r \) is:
[tex]\[ r = \frac{A / P - 1}{t} \][/tex]
This properly handles the algebraic manipulation and yields the correct result.
