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Of the 800 employees of Company X, 70 percent have been with the company for at least ten years. If \(\mathrm{y}\) of these "long-term" members were to retire and no other employee changes were to occur, what value of \(\mathrm{y}\) would reduce the percent of "long-term" employees in the company to 60 percent ?
Let's break down what we know and what we need to find in everyday terms.
Currently, Company X has 800 employees total. We're told that 70% of these employees are "long-term" (meaning they've been with the company for at least 10 years).
So right now, the number of long-term employees = \(\mathrm{70\% \, of \, 800 = 0.70 \times 800 = 560}\) employees.
The question asks: if y long-term employees retire (and nobody else leaves or joins), how many would need to retire so that exactly 60% of the remaining employees are long-term?
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Let's think about what happens after y long-term employees retire:
In plain English: we need the fraction of remaining long-term employees to equal 60% of all remaining employees.
This gives us the relationship:
(Number of remaining long-term employees) ÷ (Total remaining employees) = 60%
Now we can express our requirement mathematically:
\(\mathrm{(560 - y) \div (800 - y) = 60\% = 0.60}\)
This can be written as the equation:
\(\mathrm{\frac{560 - y}{800 - y} = 0.60}\)
This equation captures exactly what we need: after y retirements, the remaining long-term employees make up exactly 60% of the workforce.
To solve \(\mathrm{\frac{560 - y}{800 - y} = 0.60}\), we'll cross-multiply:
\(\mathrm{560 - y = 0.60 \times (800 - y)}\)
\(\mathrm{560 - y = 0.60 \times 800 - 0.60 \times y}\)
\(\mathrm{560 - y = 480 - 0.6y}\)
Now let's collect the y terms on one side:
\(\mathrm{560 - 480 = y - 0.6y}\)
\(\mathrm{80 = 0.4y}\)
\(\mathrm{y = 80 \div 0.4}\)
\(\mathrm{y = 200}\)
Let's verify this makes sense: If 200 long-term employees retire:
The value of y that would reduce the percent of long-term employees to exactly 60% is 200.
Looking at our answer choices, this corresponds to choice A: 200.
Students often fail to recognize that when long-term employees retire, BOTH the numerator (long-term employees) AND the denominator (total employees) decrease by the same amount y. They might incorrectly think only the long-term employee count changes, leading to an equation like \(\mathrm{\frac{560-y}{800} = 0.60}\) instead of the correct \(\mathrm{\frac{560-y}{800-y} = 0.60}\).
Some students confuse the current percentage (70%) with the target percentage (60%) and accidentally set up their equation to maintain 70% instead of achieving 60%. This leads to solving for y = 0, which doesn't match any answer choice.
The problem specifically states that y "long-term" members retire, but students might incorrectly assume that ANY employees can retire. This would lead to a completely different equation structure where they don't subtract y from the long-term employee count.
When expanding \(\mathrm{0.60 \times (800 - y)}\), students commonly make the arithmetic mistake of getting \(\mathrm{0.60 \times 800 - 0.60 \times y = 480 - 0.60y}\), but then incorrectly write it as \(\mathrm{480 - 0.6y}\) or make sign errors when moving terms to opposite sides of the equation.
In the final step \(\mathrm{y = 80 \div 0.4}\), students often struggle with dividing by a decimal. They might incorrectly calculate this as \(\mathrm{80 \div 0.4 = 32}\) or 320 instead of the correct 200, especially if they don't convert 0.4 to 4/10 or multiply both numerator and denominator by 10.
During the solution process, students encounter several numbers: 560 (initial long-term employees), 480 (60% of 800), and 80 (the difference 560-480). Students might mistakenly select answer choice D (80) thinking this represents the final answer, when 80 is actually just an intermediate step before dividing by 0.4.