Of the 3,600 employees of Company X, 1/3 are clerical. If the clerical staff were to be reduced by 1/3,...

1. Translate the problem requirements: We start with 3,600 total employees where \(\frac{1}{3}\) are clerical. The clerical staff gets reduced by \(\frac{1}{3}\), and we need to find what percentage of the remaining total employees are clerical.
2. Calculate initial clerical staff and reduction: Find how many clerical employees we start with and how many remain after the \(\frac{1}{3}\) reduction.
3. Determine new totals after reduction: Calculate the new total number of employees after the clerical reduction.
4. Calculate percentage of remaining employees who are clerical: Use the reduced clerical count and new total to find the final percentage.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

• We start with a company that has 3,600 total employees
• One-third of these employees work in clerical positions
• The company decides to reduce the clerical staff by cutting one-third of the clerical workers
• We need to find what percentage of the remaining total workforce will be clerical workers

The key insight here is that when we reduce clerical staff, the total number of employees in the company also decreases. This means both our numerator (clerical employees) and denominator (total employees) will change.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Calculate initial clerical staff and reduction

Let's start with concrete numbers to make this easy to follow:

Initial clerical staff = \(\frac{1}{3}\) of 3,600 employees
= \(3{,}600 ÷ 3 = 1{,}200\) clerical employees

Now, the clerical staff gets reduced by \(\frac{1}{3}\):
Reduction = \(\frac{1}{3}\) of 1,200 = \(1{,}200 ÷ 3 = 400\) employees

Remaining clerical staff = \(1{,}200 - 400 = 800\) clerical employees

So after the reduction, we have 800 clerical employees left.

3. Determine new totals after reduction

Here's where we need to be careful - the total number of employees in the company has also changed:

Original total employees = 3,600
Employees who left = 400 (the clerical staff that was reduced)
New total employees = \(3{,}600 - 400 = 3{,}200\) employees

This is important because our percentage calculation will use this new total, not the original 3,600.

4. Calculate percentage of remaining employees who are clerical

Now we can find what percentage of the remaining workforce is clerical:

Percentage = \(\frac{\text{Remaining clerical employees}}{\text{New total employees}} × 100\%\)
= \(\frac{800}{3{,}200} × 100\%\)

Let's simplify this fraction:
\(\frac{800}{3{,}200} = \frac{8}{32} = \frac{1}{4} = 0.25\)

Therefore: \(0.25 × 100\% = 25\%\)

Process Skill: SIMPLIFY - Breaking down the fraction to its simplest form makes the calculation much easier

5. Final Answer

After the reduction, 25% of the remaining employees are clerical workers.

This matches answer choice (A) 25%.

To verify: We started with 1,200 clerical out of 3,600 total (33.3%). After reducing clerical by \(\frac{1}{3}\), we have 800 clerical out of 3,200 total, which equals exactly 25%.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what changes in the total employee count
Students often assume that only the clerical staff changes while the total employee count (3,600) remains the same. They fail to recognize that when clerical employees are let go, the total number of employees in the company also decreases. This leads them to use 3,600 as the denominator instead of the correct reduced total.

2. Confusion about the sequence of fraction calculations
Students may get confused about which "\(\frac{1}{3}\)" applies to what. The problem states "\(\frac{1}{3}\) are clerical" and then "clerical staff reduced by \(\frac{1}{3}\)". Some students might incorrectly think the reduction is \(\frac{1}{3}\) of the total employees (\(\frac{1}{3}\) of 3,600) rather than \(\frac{1}{3}\) of the clerical staff (\(\frac{1}{3}\) of 1,200).

Errors while executing the approach

1. Arithmetic errors in fraction calculations
Students may make computational mistakes when calculating \(\frac{1}{3}\) of 3,600 or \(\frac{1}{3}\) of 1,200, or when subtracting to find remaining employees. For example, incorrectly calculating \(3{,}600 ÷ 3\) or making errors in the subtraction steps (\(1{,}200 - 400\) or \(3{,}600 - 400\)).

2. Incorrect percentage calculation setup
Even if students correctly identify that both the numerator and denominator change, they might set up the final percentage calculation incorrectly. For instance, using \(\frac{800}{3{,}600}\) instead of \(\frac{800}{3{,}200}\), or forgetting to multiply by 100 to convert the decimal to a percentage.

Errors while selecting the answer

1. Using original total instead of reduced total
Students who calculated \(\frac{800}{3{,}600}\) would get approximately 22.2%, leading them to incorrectly select choice (B) instead of the correct answer (A) 25%. This stems from the fundamental error of not adjusting the total employee count after the reduction.