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In a factory that produces computer circuit boards, 4.5 percent of all boards produced are found to be defective and are repaired before being sold, but 10 percent of all defective boards are sold without being repaired. What percentage of boards produced in the factory are defective?
Let's break down what the problem is telling us in plain English:
The factory statement says "\(4.5\) percent of all boards produced are found to be defective and are repaired before being sold." This means that out of every \(100\) boards made, \(4.5\) are discovered to have problems and get fixed.
But then we're told "\(10\) percent of all defective boards are sold without being repaired." This is the key insight - there are MORE defective boards than just the \(4.5\%\) that get caught and repaired!
Think of it like this: imagine defective boards as fish in a lake, and the factory's quality control as a fishing net. The net catches most of the fish (\(90\%\) of defective boards), but some slip through (\(10\%\) of defective boards).
Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships
Here's where we need to understand what that \(4.5\%\) really represents:
The \(4.5\%\) is NOT the total percentage of defective boards produced. It's only the percentage of defective boards that were caught and repaired.
Since \(10\%\) of defective boards are sold without repair, that means \(90\%\) of defective boards ARE caught and repaired. So the \(4.5\%\) represents \(90\%\) of all defective boards produced.
It's like saying "We caught \(90\%\) of the criminals, and that was \(45\) people." We'd need to figure out how many total criminals there were.
Now let's organize our information clearly:
We can see that if \(90\%\) of defective boards equals \(4.5\%\) of total production, we need to find what \(100\%\) of defective boards would be.
Using our plain English logic:
If \(90\%\) of defective boards = \(4.5\%\) of total production
Then \(100\%\) of defective boards = ? % of total production
To find the total, we can set up the relationship:
\(90\%\) of defective boards corresponds to \(4.5\%\)
\(100\%\) of defective boards corresponds to X%
Since \(100\% ÷ 90\% = \frac{10}{9}\), we multiply: \(4.5\% \times \frac{10}{9} = 4.5\% \times 1.111... = 5.0\%\)
Alternatively, we can think: if \(4.5\%\) represents \(9\) parts out of \(10\) parts total, then each part is \(4.5\% ÷ 9 = 0.5\%\). So all \(10\) parts would be \(10 \times 0.5\% = 5.0\%\).
Technically: Total defective % = \(4.5\% ÷ 0.9 = 5.0\%\)
The total percentage of boards produced that are defective is \(5.0\%\).
To verify: If \(5.0\%\) of boards are defective, then:
The answer is B. \(5.0\%\)
1. Misinterpreting what the \(4.5\%\) represents
Many students assume that \(4.5\%\) represents ALL defective boards produced, when it actually represents only the defective boards that are FOUND and repaired. This leads them to incorrectly conclude that \(4.5\%\) is the final answer without considering that some defective boards slip through undetected.
2. Confusion about the \(10\%\) statistic
Students often struggle to understand that "\(10\%\) of all defective boards are sold without being repaired" means there's a hidden portion of defective boards beyond the visible \(4.5\%\). They may incorrectly think this \(10\%\) should be added to \(4.5\%\) to get \(14.5\%\), not realizing it's describing a breakdown within the total defective population.
3. Setting up the wrong equation relationship
Students may incorrectly set up the problem by thinking: Total defective = \(4.5\% + 10\%\), rather than recognizing that \(4.5\% = 90\%\) of total defective boards. This fundamental misunderstanding of the part-to-whole relationship leads to the wrong mathematical approach.
1. Arithmetic errors in percentage calculations
Even when students correctly identify that they need to divide \(4.5\%\) by \(0.9\), they may make computational mistakes such as: \(4.5 ÷ 0.9 = 4.05\) or other incorrect calculations, leading to wrong numerical results.
2. Using incorrect fraction conversions
Students might incorrectly convert \(90\%\) to \(0.09\) instead of \(0.9\), or confuse the relationship between \(90\%\) and \(10\%\), leading to calculations like \(4.5\% ÷ 0.09 = 50\%\), which gives an unrealistic answer.
1. Failing to verify the answer makes logical sense
Students may arrive at \(5.0\%\) but then second-guess themselves because they expect the answer to be exactly \(4.5\%\) (the given percentage). They might change their answer to \(4.5\%\) without checking whether their calculated result actually satisfies the given conditions.
2. Choosing \(14.5\%\) due to simple addition error
Some students who misunderstood the problem setup may incorrectly add \(4.5\% + 10\% = 14.5\%\) and select answer choice E, not realizing this approach completely ignores the part-to-whole relationship described in the problem.
Step 1: Choose a convenient total number of boards
Let's say the factory produces \(1000\) boards total. This makes percentage calculations straightforward since \(1\% = 10\) boards.
Step 2: Identify what we know in concrete numbers
• \(4.5\%\) of all boards are found defective and repaired = \(4.5\% \times 1000 = 45\) boards
• These \(45\) boards represent the defective boards that were caught and repaired
Step 3: Understand the defective board breakdown
The problem states that \(10\%\) of all defective boards are sold without repair. This means \(90\%\) of defective boards are caught and repaired.
• \(90\%\) of defective boards = \(45\) boards (the ones that were caught)
• So if \(90\%\) of defective boards = \(45\), then \(100\%\) of defective boards = \(45 ÷ 0.9 = 50\) boards
Step 4: Calculate the total defective percentage
Total defective boards = \(50\) out of \(1000\) total boards
Percentage of defective boards = \(\frac{50}{1000} \times 100\% = 5.0\%\)
Verification:
• Total defective boards: \(50\)
• Defective boards caught and repaired: \(90\% \times 50 = 45\) boards ✓
• Defective boards sold without repair: \(10\% \times 50 = 5\) boards
• \(45\) boards represent \(4.5\%\) of total production ✓