In Town X, 64 percent of the population are employed, and 48 percent of the population are employed males. What...

1. Translate the problem requirements: We need to clarify what "\(64\) percent employed" and "\(48\) percent employed males" mean in terms of the total population, then find what percentage of just the employed group are females.
2. Identify the population components: Break down the given percentages to understand what portion of the total population consists of employed males versus employed females.
3. Calculate employed females as percentage of total population: Since we know total employed and employed males, we can find employed females as a percentage of the total population.
4. Convert to percentage of employed population: Transform our employed females percentage from "percentage of total population" to "percentage of employed population" to answer the question.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

We're told that "\(64\) percent of the population are employed" - this means that out of everyone living in Town X, \(64\%\) have jobs.

We're also told that "\(48\) percent of the population are employed males" - this means that out of everyone living in Town X (not just the employed people), \(48\%\) are men who have jobs.

The question asks: "What percent of the employed people in Town X are females?" This is asking us to look only at the group of people who have jobs, and figure out what percentage of that employed group are women.

Process Skill: TRANSLATE - We need to be very careful about what each percentage refers to - total population vs. employed population

2. Identify the population components

Let's think about this with a concrete example. Imagine Town X has exactly \(100\) people (this makes the percentages easy to work with).

From our translation:

• Total employed people = \(64\% \text{ of } 100 = 64\) people

• Employed males = \(48\% \text{ of } 100 = 48\) people

Now we can see the components clearly:

• The employed group has \(64\) people total

• Of these \(64\) employed people, \(48\) are males

• This means the remaining employed people must be females

3. Calculate employed females as percentage of total population

Using our concrete example:

• Total employed people = \(64\)

• Employed males = \(48\)

• Therefore, employed females = \(64 - 48 = 16\)

So employed females make up \(16\) out of our \(100\) total people, which means employed females are \(16\%\) of the total population.

In general terms: Employed females = Total employed - Employed males = \(64\% - 48\% = 16\%\) of total population

4. Convert to percentage of employed population

Now comes the key step. We found that employed females are \(16\%\) of the total population, but the question asks what percent of the employed people are females.

Using our concrete example:

• Employed females = \(16\) people

• Total employed people = \(64\) people

• Percentage of employed who are female = \((16 ÷ 64) \times 100\% = 25\%\)

We can verify this makes sense: \(25\%\) of \(64\) employed people = \(0.25 \times 64 = 16\) employed females ✓

In general terms: \((16\% \text{ of total population}) ÷ (64\% \text{ of total population}) = \frac{16}{64} = \frac{1}{4} = 25\%\)

4. Final Answer

\(25\%\) of the employed people in Town X are females.

This matches answer choice (B) \(25\%\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the percentages refer to
Students often confuse whether percentages refer to the total population or just the employed population. When the problem states "\(48\) percent of the population are employed males," students might incorrectly think this means \(48\%\) of employed people are males, rather than \(48\%\) of the total population are employed males.

2. Misunderstanding what the question is asking
The question asks "What percent of the employed people are females?" Students may misread this and try to find what percent of the total population are employed females, rather than what percent of just the employed group are females.

3. Not recognizing the need for a two-step calculation
Students might not realize they need to first find the number/percentage of employed females, and then convert that to a percentage of the employed population rather than the total population.

Errors while executing the approach

1. Arithmetic errors in subtraction
When calculating employed females as \(64\% - 48\% = 16\%\), students might make simple arithmetic mistakes, especially if working under time pressure.

2. Division errors when converting to percentage of employed
When calculating \((16\% \text{ of total}) ÷ (64\% \text{ of total}) = \frac{16}{64}\), students might make errors in the division or incorrectly simplify the fraction \((\frac{16}{64} = \frac{1}{4} = 0.25 = 25\%))\).

Errors while selecting the answer

1. Selecting the intermediate result instead of the final answer
Students might correctly calculate that employed females are \(16\%\) of the total population, but then mistakenly select \(16\%\) as their final answer instead of continuing to find that this represents \(25\%\) of the employed population.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a logical smart number for total population

Since we're working with percentages, let's use \(100\) as the total population. This makes percentage calculations straightforward.

Total population = \(100\) people

Step 2: Calculate actual numbers from given percentages

• Total employed people = \(64\% \text{ of } 100 = 64\) people

• Employed males = \(48\% \text{ of } 100 = 48\) people

Step 3: Find employed females

• Employed females = Total employed - Employed males

• Employed females = \(64 - 48 = 16\) people

Step 4: Calculate percentage of employed people who are females

• Percentage = \((\text{Employed females} ÷ \text{Total employed}) \times 100\%\)

• Percentage = \((16 ÷ 64) \times 100\% = \frac{1}{4} \times 100\% = 25\%\)

Answer: (B) \(25\%\)

The smart number approach makes this problem very concrete - out of \(100\) people total, \(64\) are employed, \(48\) are employed males, so \(16\) must be employed females. Since \(16\) out of \(64\) employed people are females, that's \(25\%\).