- Translate the problem requirements: We need to identify what each percentage means - \(\mathrm{10\%}\) defective staplers, \(\mathrm{2\%}\) of nondefective staplers rejected by mistake, and we know \(\mathrm{90}\) nondefective staplers were rejected. We're looking for total staplers produced.
- Work backwards from the concrete number: Use the fact that \(\mathrm{90}\) nondefective staplers were rejected to find the total number of nondefective staplers.
- Apply the defective/nondefective relationship: Use the \(\mathrm{10\%}\) defective rate to determine how nondefective staplers relate to total production.
- Calculate total production: Combine the relationships to find the total number of staplers produced.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what the problem is telling us in everyday language:
What we know:
- \(\mathrm{10\%}\) of all staplers produced were defective
- This means \(\mathrm{90\%}\) of all staplers produced were nondefective (good staplers)
- Of the good staplers, \(\mathrm{2\%}\) were mistakenly rejected
- We know that exactly \(\mathrm{90}\) good staplers were rejected by mistake
What we need to find:
- The total number of staplers produced that day
Process Skill: TRANSLATE - Converting the percentage relationships into clear mathematical understanding
2. Work backwards from the concrete number
Let's start with what we know for certain: \(\mathrm{90}\) nondefective staplers were rejected.
If \(\mathrm{90}\) nondefective staplers represent \(\mathrm{2\%}\) of all nondefective staplers, then we can find the total number of nondefective staplers.
Think of it this way: If \(\mathrm{2\%}\) equals \(\mathrm{90}\) staplers, then \(\mathrm{1\%}\) equals \(\mathrm{45}\) staplers.
Therefore, \(\mathrm{100\%}\) (all nondefective staplers) equals \(\mathrm{45 \times 100 = 4{,}500}\) staplers.
In mathematical terms: If \(\mathrm{2\%}\) of nondefective staplers \(\mathrm{= 90}\)
Then total nondefective staplers \(\mathrm{= 90 \div 0.02 = 4{,}500}\)
3. Apply the defective/nondefective relationship
Now we know there were \(\mathrm{4{,}500}\) nondefective staplers produced.
The problem tells us that \(\mathrm{10\%}\) of all staplers were defective, which means \(\mathrm{90\%}\) of all staplers were nondefective.
So if \(\mathrm{4{,}500}\) staplers represent \(\mathrm{90\%}\) of total production:
- \(\mathrm{90\%}\) of total production \(\mathrm{= 4{,}500}\) staplers
- \(\mathrm{1\%}\) of total production \(\mathrm{= 4{,}500 \div 90 = 50}\) staplers
- \(\mathrm{100\%}\) of total production \(\mathrm{= 50 \times 100 = 5{,}000}\) staplers
In mathematical terms: Total staplers \(\mathrm{= 4{,}500 \div 0.90 = 5{,}000}\)
4. Calculate total production
Let's verify our answer makes sense:
- Total staplers produced: \(\mathrm{5{,}000}\)
- Defective staplers (\(\mathrm{10\%}\)): \(\mathrm{5{,}000 \times 0.10 = 500}\)
- Nondefective staplers (\(\mathrm{90\%}\)): \(\mathrm{5{,}000 \times 0.90 = 4{,}500}\) ✓
- Nondefective staplers rejected by mistake (\(\mathrm{2\%}\) of \(\mathrm{4{,}500}\)): \(\mathrm{4{,}500 \times 0.02 = 90}\) ✓
This matches our given information perfectly!
Final Answer
The total number of staplers produced that day was \(\mathrm{5{,}000}\).
Looking at our answer choices, this corresponds to Answer E: \(\mathrm{5{,}000}\).
Common Faltering Points
Errors while devising the approach
- Misinterpreting what "\(\mathrm{2\%}\) of nondefective staplers were rejected" means: Students might think this means \(\mathrm{2\%}\) of ALL staplers were rejected, rather than \(\mathrm{2\%}\) of only the nondefective staplers. This fundamental misunderstanding would lead them to set up the wrong equation from the start.
- Confusion about the relationship between defective and nondefective staplers: Students may struggle to recognize that if \(\mathrm{10\%}\) are defective, then \(\mathrm{90\%}\) must be nondefective. They might try to work with these as independent quantities rather than complementary percentages that add up to \(\mathrm{100\%}\).
- Not recognizing the need to work backwards: Students might try to assign a variable to the total number of staplers and work forward through multiple equations, making the problem unnecessarily complex instead of starting with the concrete number (\(\mathrm{90}\) rejected nondefective staplers) and working backwards.
Errors while executing the approach
- Percentage calculation errors: When calculating that \(\mathrm{90}\) staplers represent \(\mathrm{2\%}\) of nondefective staplers, students might incorrectly compute \(\mathrm{90 \times 0.02 = 1.8}\) instead of \(\mathrm{90 \div 0.02 = 4{,}500}\). This is a common error where students multiply by the percentage instead of dividing by it.
- Mixing up the \(\mathrm{10\%}\) and \(\mathrm{90\%}\) in the final calculation: After finding \(\mathrm{4{,}500}\) nondefective staplers, students might incorrectly think this represents \(\mathrm{10\%}\) of total production (since \(\mathrm{10\%}\) were defective) and calculate \(\mathrm{4{,}500 \div 0.10 = 45{,}000}\), rather than recognizing that \(\mathrm{4{,}500}\) represents \(\mathrm{90\%}\) of total production.
Errors while selecting the answer
- Selecting an intermediate result instead of the final answer: Students might arrive at \(\mathrm{4{,}500}\) (the number of nondefective staplers) during their calculation and mistakenly select answer choice C (\(\mathrm{4{,}500}\)) instead of continuing to find the total production of \(\mathrm{5{,}000}\) staplers.
