Connie paid a sales tax of 8% on her purchase. If the sales tax had been only 5%, she would...

1. Translate the problem requirements: Connie paid 8% sales tax on her purchase. If she had paid only 5% tax instead, she would have saved $12. We need to find the total amount she actually paid (purchase price + 8% tax).
2. Identify the key relationship: The $12 difference represents exactly 3% of the original purchase price \(8\% - 5\% = 3\%\).
3. Calculate the original purchase price: Since 3% of the purchase price equals $12, we can find the full purchase price.
4. Determine the total amount paid: Add the 8% sales tax to the original purchase price to get Connie's total payment.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what happened to Connie step by step. She bought something and paid sales tax on it. The problem tells us two key facts:

• She actually paid 8% sales tax on her purchase
• If the sales tax had been only 5% instead, she would have paid $12 less in sales tax

We need to find the total amount she paid, which includes both the original purchase price and the 8% sales tax she actually paid.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Identify the key relationship

Here's the key insight that makes this problem simple: The $12 difference represents exactly the difference between 8% tax and 5% tax.

Think about it this way - if Connie had paid 5% tax instead of 8% tax on the same purchase, she would have saved exactly $12. This means:

8% of purchase price - 5% of purchase price = $12
\((8\% - 5\%)\) of purchase price = $12
3% of purchase price = $12

So the $12 represents exactly 3% of whatever Connie's original purchase price was (before any tax).

Process Skill: INFER - Recognizing that the $12 difference represents the gap between two different tax rates

3. Calculate the original purchase price

Now we can find the original purchase price using simple reasoning:

If 3% of the purchase price = $12, then we can find 100% of the purchase price.

Here's the straightforward way to think about it:

• 3% of purchase price = $12
• 1% of purchase price = \(\$12 \div 3 = \$4\)
• 100% of purchase price = \(\$4 \times 100 = \$400\)

So Connie's original purchase price (before tax) was $400.

4. Determine the total amount paid

Now we need to find how much Connie actually paid in total. Remember, she paid the original purchase price plus 8% sales tax:

Original purchase price: $400
Sales tax (8% of $400): \(0.08 \times \$400 = \$32\)
Total amount paid: \(\$400 + \$32 = \$432\)

Let's verify this makes sense: If she had paid only 5% tax instead:
5% tax would be: \(0.05 \times \$400 = \$20\)
Difference in tax: \(\$32 - \$20 = \$12\) ✓

This confirms our answer is correct.

5. Final Answer

Connie paid a total of $432 for her purchase including sales tax.

Looking at the answer choices, this matches choice E. $432.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the $12 represents

Students often confuse what the $12 difference represents. They might think it's the total tax amount Connie paid, or the difference between the total amounts paid (including purchase price), rather than understanding it's specifically the difference between two different tax amounts on the same purchase price.

2. Setting up incorrect equations due to unclear variable definition

Students may struggle to clearly define their variable. Some might let x = total amount paid (including tax), while others let x = purchase price before tax. This inconsistency leads to setting up wrong equations like "\(0.08x - 0.05x = 12\)" where x represents the total amount instead of the pre-tax amount.

3. Overlooking the constraint that both tax calculations use the same base amount

Students might not recognize that both the 8% and 5% tax rates apply to the same original purchase price. They could mistakenly think these are two different purchases or that the base amounts are different, leading to incorrect problem setup.

Errors while executing the approach

1. Arithmetic errors when calculating percentages

Students frequently make calculation mistakes when working with percentages, such as incorrectly computing 8% of $400 as $36 instead of $32, or making errors when converting between percentage and decimal forms (using 8 instead of 0.08).

2. Incorrect algebraic manipulation when solving for the purchase price

When students set up the equation \(0.03x = 12\), they might make algebraic errors such as multiplying both sides by 3 instead of dividing by 0.03, or incorrectly calculating \(12 \div 0.03\) as 40 instead of 400.

Errors while selecting the answer

1. Selecting the pre-tax amount instead of the total amount paid

The most common error in answer selection is choosing $400 (answer choice C), which represents the original purchase price before tax, instead of $432, which is the total amount including the 8% sales tax that Connie actually paid. Students solve correctly but forget that the question asks for the total amount paid including sales tax.