In a corporation, 50% of the male employees and 40% of the female employees are at least 35 years old....

1. Translate the problem requirements: We need to find what fraction of all employees are female. We know that \(50\%\) of males are \(\geq 35\), \(40\%\) of females are \(\geq 35\), and overall \(42\%\) of all employees are \(\geq 35\).
2. Set up the weighted average relationship: The overall percentage (\(42\%\)) is a weighted average of the male percentage (\(50\%\)) and female percentage (\(40\%\)), where the weights are the proportions of males and females.
3. Apply the weighted average formula: Use the fact that (proportion of males × \(50\%\)) + (proportion of females × \(40\%\)) = \(42\%\) to create an equation.
4. Solve for the female proportion: Since males and females together make up \(100\%\) of employees, we can substitute and solve for the fraction of females.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what we need to find.

What we know:

• \(50\%\) of male employees are at least 35 years old
• \(40\%\) of female employees are at least 35 years old
• \(42\%\) of ALL employees are at least 35 years old

What we need to find:

• What fraction of all employees are female?

Think of this like mixing two groups together. We have males (where \(50\%\) are \(\geq 35\)) and females (where \(40\%\) are \(\geq 35\)), and when we combine them, we get \(42\%\) of everyone being \(\geq 35\).

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up the weighted average relationship

Imagine you're mixing two solutions with different concentrations. The final concentration depends on how much of each solution you use.

Here's the key insight: The overall percentage (\(42\%\)) sits between the male percentage (\(50\%\)) and female percentage (\(40\%\)). Since \(42\%\) is closer to \(40\%\) than to \(50\%\), this tells us there are probably more females than males.

Let's say:

• Fraction of employees who are male = \(\mathrm{M}\)
• Fraction of employees who are female = \(\mathrm{F}\)
• We know that \(\mathrm{M} + \mathrm{F} = 1\) (everyone is either male or female)

The weighted average relationship in plain English:
(Fraction of males) × (% of males \(\geq 35\)) + (Fraction of females) × (% of females \(\geq 35\)) = (% of all employees \(\geq 35\))

3. Apply the weighted average formula

Now let's put numbers into our relationship:

\(\mathrm{M} \times 50\% + \mathrm{F} \times 40\% = 42\%\)

Since \(\mathrm{M} + \mathrm{F} = 1\), we can write \(\mathrm{M} = 1 - \mathrm{F}\)

Substituting:
\((1 - \mathrm{F}) \times 50\% + \mathrm{F} \times 40\% = 42\%\)

Let's expand this:
\(50\% - 50\% \times \mathrm{F} + 40\% \times \mathrm{F} = 42\%\)
\(50\% - 10\% \times \mathrm{F} = 42\%\)

Solving for F:
\(-10\% \times \mathrm{F} = 42\% - 50\%\)
\(-10\% \times \mathrm{F} = -8\%\)
\(\mathrm{F} = -8\% \div (-10\%)\)
\(\mathrm{F} = \frac{8}{10} = \frac{4}{5}\)

Process Skill: MANIPULATE - Carefully working through the algebra while keeping fractions simple

4. Solve for the female proportion

Let's verify our answer makes sense:

If \(\mathrm{F} = \frac{4}{5}\), then \(\mathrm{M} = 1 - \frac{4}{5} = \frac{1}{5}\)

Checking our weighted average:

• Males: \(\frac{1}{5}\) of employees, \(50\%\) are \(\geq 35\)
• Females: \(\frac{4}{5}\) of employees, \(40\%\) are \(\geq 35\)
• Overall: \((\frac{1}{5} \times 50\%) + (\frac{4}{5} \times 40\%) = 10\% + 32\% = 42\%\) ✓

This confirms our answer is correct.

5. Final Answer

The fraction of employees who are female is \(\frac{4}{5}\).

Looking at our answer choices:

• A. \(\frac{3}{5} = 0.6\)
• B. \(\frac{2}{3} \approx 0.67\)
• C. \(\frac{3}{4} = 0.75\)
• D. \(\frac{4}{5} = 0.8\) ✓
• E. \(\frac{5}{6} \approx 0.83\)

The answer is D. \(\frac{4}{5}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the percentages represent

Students may confuse what "\(50\%\) of male employees are at least 35" means. They might think this means \(50\%\) of ALL employees (not just males) are males who are at least 35, rather than understanding it as a conditional percentage within the male group only. This fundamental misunderstanding leads to incorrect equation setup.

2. Not recognizing this as a weighted average problem

Many students fail to see that when two groups with different rates (\(50\%\) for males, \(40\%\) for females) combine to produce an overall rate (\(42\%\)), this creates a weighted average situation. Instead, they might try to use simple ratios or set up unnecessarily complex systems of equations, missing the elegant weighted average approach.

3. Incorrectly defining variables or relationships

Students often struggle with setting up the constraint that male fraction + female fraction = 1. They might define variables for actual numbers of people instead of fractions, or forget that the two groups must account for \(100\%\) of employees, leading to under-constrained or incorrectly constrained systems.

Errors while executing the approach

1. Sign errors when manipulating the weighted average equation

When expanding \((1-\mathrm{F}) \times 50\% + \mathrm{F} \times 40\% = 42\%\), students frequently make sign errors, especially when collecting like terms. They might incorrectly write \(50\% + 10\%\mathrm{F} = 42\%\) instead of \(50\% - 10\%\mathrm{F} = 42\%\), leading to \(\mathrm{F} = -\frac{4}{5}\), which should signal an error but often gets overlooked.

2. Arithmetic mistakes with percentage calculations

Students often struggle with the percentage arithmetic, particularly when converting between percentages and fractions. They might incorrectly calculate \(42\% - 50\% = -8\%\) or make errors when dividing \(-8\%\) by \(-10\%\), especially forgetting that dividing two negative numbers yields a positive result.

3. Algebraic manipulation errors

When solving \(-10\%\mathrm{F} = -8\%\) for F, students may incorrectly divide, getting \(\mathrm{F} = 8\%\) instead of \(\mathrm{F} = \frac{8}{10} = \frac{4}{5}\). They might also fail to properly substitute \(\mathrm{M} = 1-\mathrm{F}\), or make errors when expanding the distributed terms in the weighted average equation.

Errors while selecting the answer

1. Selecting the fraction for males instead of females

Since the calculation yields both male fraction (\(\frac{1}{5}\)) and female fraction (\(\frac{4}{5}\)), students might accidentally select the answer choice that corresponds to the male fraction. If they see \(\frac{1}{5} = 0.2\) and look for something close, they might incorrectly reason their way to one of the other fractions.

2. Not converting their decimal answer back to match answer choices

Students might correctly calculate \(\mathrm{F} = 0.8\) but then fail to recognize this equals \(\frac{4}{5}\) from the answer choices. They might look for \(0.8\) directly or convert incorrectly, especially if they're working quickly under time pressure and don't take the moment to verify their fraction conversion.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient total number of employees
Let's assume there are 100 total employees in the corporation. This makes percentage calculations straightforward since we can work directly with the given percentages.

Step 2: Determine employees aged 35 and older
\(42\%\) of all employees are at least 35 years old
Number of employees \(\geq 35 = 42\% \times 100 = 42\) employees

Step 3: Set up variables for male and female employees
Let \(\mathrm{M}\) = number of male employees
Let \(\mathrm{F}\) = number of female employees
We know: \(\mathrm{M} + \mathrm{F} = 100\)

Step 4: Express employees ≥35 in terms of M and F
Males \(\geq 35: 50\%\) of \(\mathrm{M} = 0.5\mathrm{M}\)
Females \(\geq 35: 40\%\) of \(\mathrm{F} = 0.4\mathrm{F}\)
Total employees \(\geq 35: 0.5\mathrm{M} + 0.4\mathrm{F} = 42\)

Step 5: Solve the system of equations
From Step 3: \(\mathrm{M} + \mathrm{F} = 100\), so \(\mathrm{M} = 100 - \mathrm{F}\)
Substituting into the equation from Step 4:
\(0.5(100 - \mathrm{F}) + 0.4\mathrm{F} = 42\)
\(50 - 0.5\mathrm{F} + 0.4\mathrm{F} = 42\)
\(50 - 0.1\mathrm{F} = 42\)
\(-0.1\mathrm{F} = -8\)
\(\mathrm{F} = 80\)

Step 6: Find the fraction of female employees
Fraction of females = \(\frac{\mathrm{F}}{\text{Total}} = \frac{80}{100} = \frac{4}{5}\)

Verification:
Males: \(\mathrm{M} = 100 - 80 = 20\)
Males \(\geq 35: 0.5 \times 20 = 10\)
Females \(\geq 35: 0.4 \times 80 = 32\)
Total \(\geq 35: 10 + 32 = 42\) ✓