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Sagot :
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.
### 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.
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