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 :

Sure, let's carefully go through the student's steps and identify the algebraic error. We will then state the correct approach to solve for [tex]\( r \)[/tex].

### Given Steps by the Student:

1. [tex]\( A = P(1 + rt) \)[/tex]
2. [tex]\( A = P + rt \)[/tex] (student's interpretation)
3. [tex]\( A - P = rt \)[/tex]
4. [tex]\( r = \frac{A - P}{t} \)[/tex]

### Discussion of the Algebraic Error:

Step 2 is where the student made an algebraic error. The correct approach to distribute [tex]\( P \)[/tex] in the equation [tex]\( A = P(1 + rt) \)[/tex] should preserve the multiplication of [tex]\( P \)[/tex] with the entire expression in the parentheses.

### Correction Breakdown:

Let's correctly solve for [tex]\( r \)[/tex] step-by-step:

1. Start with the correct formula for the simple interest:
[tex]\[ A = P(1 + rt) \][/tex]

2. Divide both sides of the equation by [tex]\( P \)[/tex]:
[tex]\[ \frac{A}{P} = 1 + rt \][/tex]

3. Subtract 1 from both sides to isolate the term with [tex]\( r \)[/tex]:
[tex]\[ \frac{A}{P} - 1 = rt \][/tex]

4. Finally, divide both sides by [tex]\( t \)[/tex] to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{\left(\frac{A}{P} - 1\right)}{t} \][/tex]
or equivalently,
[tex]\[ r = \frac{A/P - 1}{t} \][/tex]
which simplifies to:
[tex]\[ r = \frac{A - P}{Pt} \][/tex]

### Correct Equation for [tex]\( r \)[/tex]:

The correct equation to solve for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{\left(\frac{A}{P} - 1\right)}{t} \][/tex]

### Summary of the Student's Error:

The student incorrectly simplified the expression [tex]\( P(1 + rt) \)[/tex] as [tex]\( P + rt \)[/tex], which neglects the proper distribution of [tex]\( P \)[/tex]. The correct interpretation involves dividing [tex]\( A \)[/tex] by [tex]\( P \)[/tex] first, before isolating [tex]\( r \)[/tex].

### Algebraic Error Explanation:

The precise algebraic error occurred in the student's Step 2:
[tex]\[ A = P + rt \][/tex]
This should have been:
[tex]\[ \frac{A}{P} = 1 + rt \][/tex]

### Final Corrected Step:

To correct the student's final step, the equation [tex]\( r = \frac{A - P}{t} \)[/tex] needs to be modified to include proper distribution and isolation of [tex]\( r \)[/tex] as follows:
[tex]\[ r = \frac{A/P - 1}{t} \][/tex]

This ensures the correct formula for solving [tex]\( r \)[/tex] from the initial simple interest equation.

So the correct statement for the algebraic error and the correct equation is:

Algebraic error: Error: A = P + rt is incorrect; the correct step after distributing [tex]\( P \)[/tex] should be:
[tex]\[ A = P(1 + rt) \][/tex]

Correct modification of the equation:
[tex]\[ r = \frac{(A / P) - 1}{t} \][/tex]

This approach accurately isolates [tex]\( r \)[/tex] in the context of the given simple interest formula.