In a factory that produces computer circuit boards, 4.5 percent of all boards produced are found to be defective and...

1. Translate the problem requirements: The problem states that \(4.5\%\) of boards are found defective and repaired, but there's an additional \(10\%\) of defective boards that are sold without repair. We need to find the total percentage of defective boards produced.
2. Recognize the relationship between categories: Understand that the \(4.5\%\) represents only the defective boards that are caught and repaired, not all defective boards produced.
3. Set up the defective board breakdown: If \(90\%\) of defective boards are found and repaired (the \(4.5\%\)), then \(10\%\) of defective boards go undetected and are sold as-is.
4. Calculate total defective percentage: Use the fact that \(4.5\%\) represents \(90\%\) of all defective boards to find the total defective rate.

Execution of Strategic Approach

1. Translate the problem requirements

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

2. Recognize the relationship between categories

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.

3. Set up the defective board breakdown

Now let's organize our information clearly:

• \(90\%\) of defective boards = boards that are found and repaired = \(4.5\%\) of total production
• \(10\%\) of defective boards = boards that slip through and are sold defective
• \(100\%\) of defective boards = what we're trying to find

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.

4. Calculate total defective percentage

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 % = \(4.5\% ÷ 0.9 = 5.0\%\)

4. Final Answer

The total percentage of boards produced that are defective is \(5.0\%\).

To verify: If \(5.0\%\) of boards are defective, then:

• \(90\%\) of these defective boards (which is \(4.5\%\) of total production) get repaired ✓
• \(10\%\) of these defective boards (which is \(0.5\%\) of total production) are sold without repair ✓

The answer is B. \(5.0\%\)

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the 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.

Errors while selecting the 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.

Alternate Solutions

Smart Numbers Approach

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 ✓