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


Sagot :

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].