At a post office last week, 25% of the employees were part-time employees and the rest were full-time employees. Each...

1. Translate the problem requirements: We need to find what fraction the part-time employees' total hours represents of ALL employees' total hours. Key relationships: \(25\%\) are part-time (\(75\%\) full-time), and each part-time employee works \(\frac{3}{5}\) as many hours as each full-time employee.
2. Set up concrete numbers to avoid fractions: Choose a total number of employees that makes calculations simple, ensuring we can work with whole numbers throughout.
3. Calculate total hours for each group: Determine total hours worked by part-time employees and total hours worked by full-time employees using the given ratios.
4. Find the fraction and verify: Express part-time hours as a fraction of total hours worked by all employees, then check against answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

• \(25\%\) of employees are part-time workers, which means \(75\%\) are full-time workers
• Each part-time employee works only \(\frac{3}{5}\) as many hours as each full-time employee
• We need to find: what fraction do the part-time workers' total hours represent out of ALL hours worked by everyone?

Think of it this way: if full-time workers are putting in their regular hours, part-time workers are doing about \(60\%\) of that amount each. But there are fewer part-time workers overall. So we need to figure out how all these hours add up.

Process Skill: TRANSLATE - Converting the percentage and fractional relationships into a clear mathematical setup

2. Set up concrete numbers to avoid fractions

Instead of working with messy percentages and fractions, let's pick smart numbers that make our calculations clean.

Let's say there are \(100\) total employees at the post office:

• Part-time employees: \(25\% \text{ of } 100 = 25\) people
• Full-time employees: \(75\% \text{ of } 100 = 75\) people

Now, let's say each full-time employee works \(10\) hours (we're picking \(10\) because it's easy to find \(\frac{3}{5}\) of it):

• Each full-time employee works: \(10\) hours
• Each part-time employee works: \(\frac{3}{5} \times 10 = 6\) hours

These concrete numbers will make our calculations much simpler and help us see the pattern clearly.

Process Skill: SIMPLIFY - Choosing convenient numbers to avoid complex fraction arithmetic

3. Calculate total hours for each group

Now we can easily calculate the total hours worked by each group:

Full-time employees' total hours:
\(75 \text{ employees} \times 10 \text{ hours each} = 750 \text{ total hours}\)

Part-time employees' total hours:
\(25 \text{ employees} \times 6 \text{ hours each} = 150 \text{ total hours}\)

All employees' total hours:
\(750 + 150 = 900 \text{ total hours}\)

Notice how our choice of simple numbers (\(100\) employees, \(10\) hours per full-time worker) made these calculations straightforward - no complex fractions to deal with!

4. Find the fraction and verify

Now we can answer the question: What fraction of total hours were worked by part-time employees?

Fraction = \(\text{Part-time hours} \div \text{Total hours} = 150 \div 900\)

Let's simplify this fraction:
\(\frac{150}{900} = \frac{15}{90} = \frac{1}{6}\)

To double-check: \(\frac{1}{6} \text{ of } 900 \text{ hours} = 150 \text{ hours}\) ✓

Looking at our answer choices, \(\frac{1}{6}\) matches choice D.

Mathematical summary: If \(\mathrm{P}\) = part-time employees, \(\mathrm{F}\) = full-time employees, and \(\mathrm{h}\) = hours per full-time employee, then:

• Part-time total hours = \(0.25\mathrm{P} \times \frac{3\mathrm{h}}{5} = 0.15\mathrm{P}\mathrm{h}\)
• Full-time total hours = \(0.75\mathrm{P} \times \mathrm{h} = 0.75\mathrm{P}\mathrm{h}\)
• Total hours = \(0.15\mathrm{P}\mathrm{h} + 0.75\mathrm{P}\mathrm{h} = 0.9\mathrm{P}\mathrm{h}\)
• Fraction = \(\frac{0.15\mathrm{P}\mathrm{h}}{0.9\mathrm{P}\mathrm{h}} = \frac{0.15}{0.9} = \frac{1}{6}\)

Final Answer

The total number of hours worked by part-time employees was \(\frac{1}{6}\) of the total hours worked by all employees.

Answer: D. \(\frac{1}{6}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the percentage relationships
Students often confuse which group represents \(25\%\) vs \(75\%\). They might incorrectly assume that \(75\%\) are part-time workers instead of full-time workers, completely reversing the setup and leading to an incorrect final answer.

2. Misunderstanding what "\(\frac{3}{5}\) as many hours" means
Some students interpret "\(\frac{3}{5}\) as many hours" as meaning part-time workers work \(\frac{3}{5}\) MORE hours than full-time workers, rather than \(\frac{3}{5}\) OF the hours that full-time workers put in. This fundamental misreading changes the entire mathematical relationship.

3. Confusing what fraction needs to be calculated
Students may set up to find the wrong ratio - such as part-time hours to full-time hours, or full-time employees to total employees - rather than the specific fraction asked: part-time hours to total hours worked by everyone.

Errors while executing the approach

1. Arithmetic errors in fraction calculations
When calculating \(\frac{3}{5}\) of the full-time hours, students often make basic multiplication errors, especially if they choose inconvenient numbers. For example, if they pick \(7\) hours for full-time work, calculating \(\frac{3}{5} \times 7 = \frac{21}{5}\) becomes messy and error-prone.

2. Incorrect total hours calculation
Students frequently add the employee counts instead of the total hours, or forget to multiply the number of employees by their respective hours worked. They might calculate \(25 + 75 = 100\) instead of \((25 \times 6) + (75 \times 10) = 900\).

3. Fraction simplification mistakes
When simplifying \(\frac{150}{900}\), students often make errors in finding common factors or reducing fractions. They might incorrectly simplify to \(\frac{1}{5}\) or \(\frac{3}{20}\) instead of properly reducing to \(\frac{1}{6}\).

Errors while selecting the answer

1. Selecting the reciprocal fraction
After correctly calculating that part-time hours are \(150\) and total hours are \(900\), students might flip the fraction and select \(\frac{900}{150} = \frac{6}{1}\), then look for \(6\) among the choices or incorrectly choose a related fraction like \(\frac{1}{5}\).

2. Choosing a fraction that represents a different relationship
Students who calculated correctly but got confused about what they were solving for might select \(\frac{1}{5}\) (thinking it represents the \(25\%\) part-time employees) or \(\frac{3}{20}\) (mixing up the \(\frac{3}{5}\) hours relationship with percentages).

Alternate Solutions

Smart Numbers Approach

This problem is well-suited for the smart numbers method because we can select concrete values that make the percentage and fraction calculations straightforward.

Step 1: Choose smart numbers for total employees

Since \(25\%\) of employees are part-time, let's choose a total number of employees that makes this percentage easy to work with. Let's use \(20\) total employees (since \(25\%\) of \(20 = 5\), giving us whole numbers).

• Part-time employees: \(25\%\) of \(20 = 5\) employees
• Full-time employees: \(75\%\) of \(20 = 15\) employees

Step 2: Choose smart numbers for hours worked

Each part-time employee works \(\frac{3}{5}\) as many hours as each full-time employee. To avoid messy fractions, let's say each full-time employee works \(5\) hours (chosen because \(\frac{3}{5} \times 5 = 3\), giving us a whole number).

• Hours per full-time employee: \(5\) hours
• Hours per part-time employee: \(\frac{3}{5} \times 5 = 3\) hours

Step 3: Calculate total hours for each group

• Total hours by part-time employees: \(5 \text{ employees} \times 3 \text{ hours} = 15 \text{ hours}\)
• Total hours by full-time employees: \(15 \text{ employees} \times 5 \text{ hours} = 75 \text{ hours}\)

Step 4: Find the required fraction

• Total hours by all employees: \(15 + 75 = 90 \text{ hours}\)
• Fraction of hours worked by part-time employees: \(\frac{15}{90} = \frac{1}{6}\)

Answer: D (\(\frac{1}{6}\))

The smart numbers approach works perfectly here because we can logically select values that eliminate fractions and percentages while maintaining all the given relationships in the problem.