The employees of a certain department of Company X currently constitute 22 percent of the employees of the company. If...

1. Translate the problem requirements: We need to find what percentage of Company X's total employees will be in this department after expansion. Currently the department has \(22\%\) of company employees. The expansion adds 2 new employees for every 1 current department employee.
2. Set up concrete numbers using the percentage relationship: Choose simple numbers that make the \(22\%\) relationship easy to work with, avoiding fractions in our base calculations.
3. Calculate the expansion impact: Determine how many new employees join both the department and the total company after the 2-for-1 addition.
4. Find the new percentage composition: Calculate what fraction the expanded department represents of the new total company size.

Execution of Strategic Approach

1. Translate the problem requirements

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

• Right now, a certain department makes up \(22\%\) of Company X's total employees
• The company wants to expand this department by adding 2 new employees for every 1 current employee in the department
• We need to find what percentage of the company's total employees will be in this department after the expansion

The key insight is that when we add employees to the department, we're also adding to the company's total employee count, so both the numerator and denominator of our percentage will change.

2. Set up concrete numbers using the percentage relationship

Instead of working with variables, let's use concrete numbers that make the \(22\%\) relationship easy to work with.

Since the department currently represents \(22\%\) of the company, let's say:

• Current department employees = 22
• Current total company employees = 100

This gives us exactly \(\frac{22}{100} = 22\%\), which matches our given information perfectly.

Using round numbers like this will keep our arithmetic simple and help us avoid fractions throughout our calculations.

3. Calculate the expansion impact

Now let's figure out what happens during the expansion:

The expansion rule is: "add 2 employees new to the company for every current employee of the department"

• Current department employees: 22
• New employees added to department: \(2 \times 22 = 44\)
• Department employees after expansion: \(22 + 44 = 66\)

Since these 44 new employees are "new to the company," they also increase the total company size:

• Current total company employees: 100
• New employees added to company: 44
• Total company employees after expansion: \(100 + 44 = 144\)

4. Find the new percentage composition

After the expansion:

• Department employees: 66
• Total company employees: 144
• Department percentage: \(\frac{66}{144}\)

Let's calculate this percentage:

\(\frac{66}{144} = 66 \div 144 = 0.458333...\)

Converting to percentage: \(0.458333... \times 100\% \approx 45.8\%\)

Looking at our answer choices, \(45.8\%\) is closest to \(46\%\).

Final Answer

The department would represent approximately \(46\%\) of Company X's employees after the expansion.

This matches answer choice C. \(46\%\).

We can verify this makes sense: the department roughly tripled in size (from 22 to 66 employees), while the company grew by less than \(50\%\) (from 100 to 144 employees), so it's reasonable that the department's percentage would increase significantly from \(22\%\) to \(46\%\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the expansion rule

Students often misread "add 2 employees new to the company for every current employee of the department" as adding 2 employees total to the department, rather than 2 employees for EACH current employee. This leads to calculating only \(22 + 2 = 24\) department employees instead of \(22 + (2 \times 22) = 66\) employees after expansion.

Faltering Point 2: Forgetting that new employees increase total company size

Students frequently focus only on how the department grows and forget that since the new employees are "new to the company," they also increase the denominator (total company employees). They mistakenly keep the total company size at 100 instead of updating it to 144, leading to an incorrect final percentage calculation.

Faltering Point 3: Setting up overly complex algebraic expressions

Instead of using simple concrete numbers (like 22 department employees out of 100 total), students often create complicated variable-based setups that make the arithmetic more error-prone and time-consuming, potentially leading to mistakes in the subsequent calculations.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in multiplication and addition

Students may incorrectly calculate \(2 \times 22 = 44\) (perhaps getting 42 or 46) or make errors when adding \(22 + 44 = 66\) or \(100 + 44 = 144\). These basic arithmetic mistakes compound and lead to wrong final percentages.

Faltering Point 2: Division and percentage conversion errors

When calculating \(\frac{66}{144}\), students might make computational errors or struggle with the decimal-to-percentage conversion. They might get the decimal correct (0.458) but then incorrectly convert it to a percentage (like \(4.58\%\) instead of \(45.8\%\)).

Errors while selecting the answer

Faltering Point 1: Choosing the exact calculated value instead of "closest to"

Students calculate \(45.8\%\) correctly but then look for exactly \(45.8\%\) in the answer choices. When they don't find it, they might panic or choose a less appropriate option instead of recognizing that \(46\%\) is the closest choice, as the question asks for the value that "would be closest to."

Alternate Solutions

Smart Numbers Approach

This problem is well-suited for the smart numbers approach because we can choose convenient values that make the \(22\%\) relationship easy to work with.

Step 1: Choose smart numbers based on the given percentage
Since the department currently has \(22\%\) of company employees, let's choose numbers that make this percentage easy to calculate:
• Let current department size = 22 employees
• Let current total company size = 100 employees
This gives us exactly \(\frac{22}{100} = 22\%\) ✓

Step 2: Calculate the expansion
The company adds 2 new employees for every current department employee:
• New employees added to department = \(2 \times 22 = 44\) employees
• These are new to the company, so total company also grows by 44

Step 3: Find new totals after expansion
• New department size = \(22 + 44 = 66\) employees
• New total company size = \(100 + 44 = 144\) employees

Step 4: Calculate the new percentage
New percentage = \(\left(\frac{66}{144}\right) \times 100\%\)
= \(0.458... \times 100\%\)
= \(45.8\%\)
≈ \(46\%\)

Why this smart numbers approach works:
• We chose 100 for total company size to make percentage calculations straightforward
• We chose 22 for department size to naturally satisfy the \(22\%\) condition
• These numbers eliminate fractions in our intermediate calculations
• The final answer doesn't depend on our specific number choices - any valid combination would yield the same result