Of the employees at a certain company, 60 percent participate in the company retirement's plan; and of those employees, 20...

1. Translate the problem requirements: Break down the overlapping employee groups - retirement plan participants, health plan participants, and those in both plans. Identify what fraction we need to find.
2. Set up concrete numbers to avoid fraction confusion: Use a convenient total number of employees (like 100) to convert all percentages into actual employee counts.
3. Calculate the overlap between retirement and health plans: Find how many employees participate in both plans using the given percentage relationships.
4. Determine the target fraction: Calculate what fraction of health plan participants also participate in the retirement plan by comparing the overlap to the total health plan participants.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know about the employee groups in plain English:

We have three groups of employees to think about:

• Employees in the retirement plan
• Employees in the health plan
• Employees who are in BOTH plans (the overlap)

From the problem, we know:

• 60% of all employees participate in the retirement plan
• Of those retirement plan participants, 20% also participate in the health plan
• 48% of all employees participate in the health plan

What we need to find: Of the employees who participate in the health plan, what fraction also participate in the retirement plan?

Process Skill: TRANSLATE - Converting the overlapping percentages into clear group relationships

2. Set up concrete numbers to avoid fraction confusion

Let's use 100 employees as our total to make the percentages easy to work with. This turns all our percentages into actual people we can count:

With 100 total employees:

• Retirement plan participants: \(60\% \text{ of } 100 = 60\) employees
• Health plan participants: \(48\% \text{ of } 100 = 48\) employees

Now we can work with concrete numbers instead of getting confused by overlapping percentages.

3. Calculate the overlap between retirement and health plans

Here's the key insight: We know that 20% of retirement plan participants also participate in the health plan.

Since we have 60 employees in the retirement plan, and 20% of them are also in the health plan:

Employees in BOTH plans = \(20\% \text{ of } 60 = 0.20 \times 60 = 12\) employees

So 12 employees participate in both the retirement plan and the health plan.

4. Determine the target fraction

Now we can answer the question: "What fraction of health plan participants also participate in the retirement plan?"

We know:

• Total employees in health plan = 48
• Employees in both health plan AND retirement plan = 12

The fraction we want = (Employees in both plans) ÷ (Total employees in health plan)

Fraction = \(12 \div 48 = \frac{12}{48} = \frac{1}{4}\)

Final Answer

The fraction of health plan participants who also participate in the retirement plan is \(\frac{1}{4}\).

This matches answer choice A: \(\frac{1}{4}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what group the question is asking about: Students often confuse "what fraction of health plan participants also participate in retirement" with "what fraction of retirement plan participants also participate in health." This reverses the denominator and numerator, leading to calculating \(\frac{12}{60} = \frac{1}{5}\) instead of \(\frac{12}{48} = \frac{1}{4}\).

2. Confusion about overlapping percentages: Students may struggle to understand that "20% of retirement participants also participate in health" refers to a subset calculation (20% of the 60 retirement participants), not 20% of all employees. This can lead to incorrectly using 20 employees instead of 12 employees for the overlap.

3. Setting up the wrong relationship: Students might try to add percentages directly (60% + 48% - some overlap) instead of recognizing this as a conditional probability problem where they need to find what fraction of one specific group (health plan participants) belongs to another group (retirement plan).

Errors while executing the approach

1. Arithmetic errors in calculating the overlap: When finding 20% of 60 retirement participants, students might calculate incorrectly, getting 10 or 15 instead of 12. This leads to wrong fractions like \(\frac{10}{48}\) or \(\frac{15}{48}\).

2. Using wrong totals in fraction calculation: Even if students correctly find 12 employees in both plans, they might use the wrong denominator - such as 100 (total employees) or 60 (retirement participants) instead of 48 (health plan participants) when calculating the final fraction.

Errors while selecting the answer

1. Not simplifying the fraction completely: Students might correctly calculate \(\frac{12}{48}\) but fail to reduce it to \(\frac{1}{4}\), potentially selecting a wrong answer choice or being confused about which option matches their result.

2. Selecting the reciprocal: If students calculated the problem backwards (retirement participants who are in health plan), they would get \(\frac{12}{60} = \frac{1}{5}\). Since \(\frac{1}{5}\) isn't an option, they might incorrectly select \(\frac{1}{4}\) thinking it's close, or choose a completely different answer.

Alternate Solutions

Smart Numbers Approach

This problem involves overlapping groups and percentages, making it ideal for the smart numbers method. We can choose a convenient total number of employees to convert all percentages into concrete numbers.

Step 1: Choose a smart total number of employees
Let's use 100 employees as our total. This makes percentage calculations straightforward since each percent equals one employee.

Step 2: Calculate retirement plan participants
60% of 100 employees participate in the retirement plan
Retirement plan participants = \(0.60 \times 100 = 60\) employees

Step 3: Calculate employees in both plans
20% of retirement plan participants also participate in the health plan
Employees in both plans = \(0.20 \times 60 = 12\) employees

Step 4: Calculate total health plan participants
48% of all employees participate in the health plan
Health plan participants = \(0.48 \times 100 = 48\) employees

Step 5: Find the target fraction
We need the fraction of health plan participants who also participate in the retirement plan
This fraction = (Employees in both plans) ÷ (Total health plan participants)
= \(12 \div 48 = \frac{1}{4}\)

Verification: Our answer of \(\frac{1}{4}\) matches choice A, confirming our solution is correct.