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Shown below are the steps a student took to solve the simple interest formula [tex]A = P(1 + rt)[/tex] for [tex]r[/tex].

[tex]\[
\begin{array}{l}
A = P(1 + rt) \\
A = P + rt \\
A - P = rt \\
r = \frac{A - P}{t}
\end{array}
\][/tex]

Explain the algebraic error in the student's work. Then, state how the equation [tex]r = \frac{A - P}{t}[/tex] can be modified to produce the correct result.



Answer :

GinnyAnswer
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.

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