Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
