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In a corporation, \(50\%\) of the male employees and \(40\%\) of the female employees are at least 35 years old. If \(42\%\) of all the employees are at least 35 years old, what fraction of the employees in the corporation are females?
Let's start by understanding what we know and what we need to find.
What we know:
What we need to find:
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
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:
The weighted average relationship in plain English:
(Fraction of males) × (% of males \(\geq 35\)) + (Fraction of females) × (% of females \(\geq 35\)) = (% of all employees \(\geq 35\))
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
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:
This confirms our answer is correct.
The fraction of employees who are female is \(\frac{4}{5}\).
Looking at our answer choices:
The answer is D. \(\frac{4}{5}\)
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.
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.
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.
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.
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.
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.
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.
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.
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 \(\geq 35: 10 + 32 = 42\) ✓