Of the 800 employees of Company X, 70 percent have been with the company for at least ten years. If...

1. Translate the problem requirements: We need to find how many long-term employees (y) must retire so that the remaining long-term employees represent exactly 60% of the total remaining workforce. Currently 70% of 800 employees are long-term (560 people).
2. Set up the target scenario: After y retirements, we'll have (560 - y) long-term employees out of (800 - y) total employees, and this ratio must equal 60%.
3. Create the percentage equation: Express the requirement that (560 - y)/(800 - y) = 0.60 and solve for y.
4. Solve systematically: Cross-multiply and solve the resulting linear equation to find the exact number of retirees needed.

Execution of Strategic Approach

1. Translate the problem requirements

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

2. Set up the target scenario

Let's think about what happens after y long-term employees retire:

• We start with 560 long-term employees, so after y retire, we have \(\mathrm{(560 - y)}\) long-term employees remaining
• We start with 800 total employees, so after y retire, we have \(\mathrm{(800 - y)}\) total employees remaining
• We want the long-term employees to be exactly 60% of the total remaining employees

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%

3. Create the percentage equation

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.

4. Solve systematically

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:

• Remaining long-term employees: \(\mathrm{560 - 200 = 360}\)
• Total remaining employees: \(\mathrm{800 - 200 = 600}\)
• Percentage: \(\mathrm{\frac{360}{600} = 0.60 = 60\%}\) ✓

5. Final Answer

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.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what changes when employees retire

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 \(\mathrm{\frac{560-y}{800-y} = 0.60}\).

2. Setting up the equation with the wrong target percentage

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.

3. Misinterpreting "long-term" employee retirement constraint

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.

Errors while executing the approach

1. Cross-multiplication and distribution errors

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.

2. Decimal arithmetic mistakes

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.

Errors while selecting the answer

1. Selecting an intermediate calculation result

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.