- Translate the problem requirements: A preferred customer bought 6 CDs total, paying $15.95 for the first CD and $3.99 for each of the remaining 5 CDs. We need to find which expression represents the total amount paid.
- Calculate the total cost structure: Determine the cost for the first CD and the cost for the additional CDs separately.
- Express the total mathematically: Combine the costs into a single expression that matches the format of the answer choices.
- Match with answer choices using strategic approximation: Since the answer choices use rounded values like 4.00 and 15.90, identify which choice represents our calculated total cost.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- A customer bought 6 compact discs total
- The first CD cost $15.95
- Each additional CD after the first one cost $3.99
- Since there are 6 CDs total, there are 5 additional CDs (\(\mathrm{6 - 1 = 5}\))
Process Skill: TRANSLATE - Converting the word problem into clear mathematical components
2. Calculate the total cost structure
Now let's figure out the actual dollar amounts:
- Cost of first CD: $15.95
- Cost of additional CDs: \(\mathrm{5\,CDs \times \$3.99\,each = 5 \times \$3.99 = \$19.95}\)
- Total cost: \(\mathrm{\$15.95 + \$19.95 = \$35.90}\)
So the customer paid exactly $35.90 for all 6 compact discs.
3. Express the total mathematically
We can write the total cost as:
Total = (Cost of first CD) + (Number of additional CDs) × (Cost per additional CD)
\(\mathrm{Total = \$15.95 + 5 \times \$3.99}\)
This is the exact mathematical expression for what the customer paid.
4. Match with answer choices using strategic approximation
Now here's the key insight - the answer choices use approximations like 4.00 instead of 3.99, and 15.90 instead of 15.95.
Let's check each choice by calculating what it equals:
- \(\mathrm{5(4.00) + 15.90 = 20.00 + 15.90 = 35.90}\)
- \(\mathrm{5(4.00) + 15.95 = 20.00 + 15.95 = 35.95}\)
- \(\mathrm{5(4.00) + 16.00 = 20.00 + 16.00 = 36.00}\)
- \(\mathrm{5(4.00 - 0.01) + 15.90 = 5(3.99) + 15.90 = 19.95 + 15.90 = 35.85}\)
- \(\mathrm{5(4.00 - 0.05) + 15.95 = 5(3.95) + 15.95 = 19.75 + 15.95 = 35.70}\)
Our actual total was $35.90, which exactly matches choice (A).
Process Skill: INFER - Recognizing that the question asks for an equivalent expression, not necessarily the exact expression
Final Answer
The answer is (A) 5(4.00) + 15.90.
This choice represents the customer's total payment by rounding $3.99 to $4.00 for the additional CDs and $15.95 to $15.90 for the first CD. The small rounding adjustments cancel each other out, giving us the exact total of $35.90 that the customer actually paid.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the discount structure: Students may incorrectly think that ALL 6 CDs cost $3.99 each, missing that only the additional CDs (after the first one) get the discounted price. This leads to calculating \(\mathrm{6 \times \$3.99}\) instead of recognizing that there's 1 CD at $15.95 and 5 CDs at $3.99 each.
- Confusion about what constitutes "additional" CDs: Students might think there are 6 additional CDs instead of 5, not realizing that "additional" means "after the first one." This would lead them to calculate the cost as \(\mathrm{\$15.95 + 6 \times \$3.99}\).
- Missing the equivalence requirement: Students may focus only on finding the exact expression (\(\mathrm{\$15.95 + 5 \times \$3.99}\)) without recognizing that the question asks for an "equivalent" expression, meaning approximations that yield the same total are acceptable.
Errors while executing the approach
- Arithmetic errors in multiplication: When calculating \(\mathrm{5 \times \$3.99}\), students might incorrectly compute this as $19.50 or $20.95 instead of the correct $19.95, leading to wrong total amounts.
- Errors in evaluating answer choices: Students may make calculation mistakes when checking each option, such as computing \(\mathrm{5(4.00 - 0.01)}\) as \(\mathrm{5(3.90) = \$19.50}\) instead of the correct \(\mathrm{5(3.99) = \$19.95}\).
Errors while selecting the answer
- Choosing the exact mathematical expression over equivalent: Students who correctly identify that the exact expression is \(\mathrm{\$15.95 + 5 \times \$3.99}\) might look for choice (D) since it contains \(\mathrm{5(4.00 - 0.01) = 5(3.99)}\), without checking if other choices also give the correct total of $35.90.
- Selecting based on partial matching: Students might choose option (B) because it contains the exact first CD price of $15.95, without verifying that the total ($35.95) doesn't match their calculated total of $35.90.
