Answer :
To solve the problem, we need to determine which scenario is correctly modeled by the equation [tex]\((x)(0.65) = \$36.48\)[/tex].
This equation means that [tex]\(65\%\)[/tex] of the original price [tex]\(x\)[/tex] results in a sale price of [tex]\(\$36.48\)[/tex]. To find out which scenario fits, let's analyze each step by step:
1. Understanding the Equation:
[tex]\((x)(0.65) = \$36.48\)[/tex]
- [tex]\(x\)[/tex] represents the original price of the boots.
- [tex]\(0.65\)[/tex] is the percentage (written as a decimal) of the original price after the discount.
- [tex]\(\$36.48\)[/tex] is the sale price after applying the discount.
2. Solving for [tex]\(x\)[/tex]:
- To find the original price, rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\$36.48}{0.65}
\][/tex]
- Performing the calculation gives us the original price of the boots:
[tex]\[
x = \$56.12
\][/tex]
3. Identifying the Correct Scenario:
- We know that the sale price, after a [tex]\(65\%\)[/tex] discount from the original price, is [tex]\(\$36.48\)[/tex].
- The original price we found is [tex]\(\$56.12\)[/tex].
Now, let's match this information with the given scenarios:
- Scenario 1: "A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The sale price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This isn't correct because [tex]\(\$56.12\)[/tex] is the original price, not the sale price.
- Scenario 2: "A pair of boots is on sale for [tex]\(35\%\)[/tex] of the original cost. The sale price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This isn't correct because the equation indicates a [tex]\(65\%\)[/tex] discount, not [tex]\(35\%\)[/tex].
- Scenario 3: "A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This matches our analysis perfectly, as it indicates the boots are sold at [tex]\(65\%\)[/tex] of the original price and the original price is indeed [tex]\(\$56.12\)[/tex].
- Scenario 4: "A pair of boots is on sale for [tex]\(35\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex]."
- Again, this isn't accurate because the discount mentioned is incorrect.
Based on the given scenarios and our calculations, the correct scenario is:
Scenario 3: A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex].
This equation means that [tex]\(65\%\)[/tex] of the original price [tex]\(x\)[/tex] results in a sale price of [tex]\(\$36.48\)[/tex]. To find out which scenario fits, let's analyze each step by step:
1. Understanding the Equation:
[tex]\((x)(0.65) = \$36.48\)[/tex]
- [tex]\(x\)[/tex] represents the original price of the boots.
- [tex]\(0.65\)[/tex] is the percentage (written as a decimal) of the original price after the discount.
- [tex]\(\$36.48\)[/tex] is the sale price after applying the discount.
2. Solving for [tex]\(x\)[/tex]:
- To find the original price, rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\$36.48}{0.65}
\][/tex]
- Performing the calculation gives us the original price of the boots:
[tex]\[
x = \$56.12
\][/tex]
3. Identifying the Correct Scenario:
- We know that the sale price, after a [tex]\(65\%\)[/tex] discount from the original price, is [tex]\(\$36.48\)[/tex].
- The original price we found is [tex]\(\$56.12\)[/tex].
Now, let's match this information with the given scenarios:
- Scenario 1: "A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The sale price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This isn't correct because [tex]\(\$56.12\)[/tex] is the original price, not the sale price.
- Scenario 2: "A pair of boots is on sale for [tex]\(35\%\)[/tex] of the original cost. The sale price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This isn't correct because the equation indicates a [tex]\(65\%\)[/tex] discount, not [tex]\(35\%\)[/tex].
- Scenario 3: "A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex]."
- This matches our analysis perfectly, as it indicates the boots are sold at [tex]\(65\%\)[/tex] of the original price and the original price is indeed [tex]\(\$56.12\)[/tex].
- Scenario 4: "A pair of boots is on sale for [tex]\(35\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex]."
- Again, this isn't accurate because the discount mentioned is incorrect.
Based on the given scenarios and our calculations, the correct scenario is:
Scenario 3: A pair of boots is on sale for [tex]\(65\%\)[/tex] of the original cost. The original price of the boots is [tex]\(x, \$56.12\)[/tex].
