A factory that employs 1{,}000 assembly-line workers pays each of these workers $6 per hour for the first 35 hours...

1. Translate the problem requirements: We need to find the total weekly payroll for 1,000 workers where pay is \(\$6\)/hour for first 35 hours and \(\$7.50\)/hour (\(1.25 \times \$6\)) for overtime hours. Workers are distributed as: \(60\%\) worked 20 hours, \(30\%\) worked 40 hours, \(10\%\) worked 50 hours.
2. Calculate worker distribution by hours: Determine how many workers fall into each hour category to organize our payroll calculations.
3. Apply pay structure to each group: Calculate total pay for each group by applying regular and overtime rates based on the 35-hour threshold.
4. Sum all group payrolls: Add the payroll amounts from all three groups to get the total factory payroll.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

We have a factory with 1,000 workers total. Each worker gets paid using this rule:
• For the first 35 hours in a week: \(\$6\) per hour (regular pay)
• For any hours beyond 35: \(\$6 \times 1.25 = \$7.50\) per hour (overtime pay)

The workers didn't all work the same hours that week:
• \(60\%\) of workers worked 20 hours
• \(30\%\) of workers worked 40 hours
• The remaining workers worked 50 hours

We need to find the total amount the factory paid all these workers for that week.

Process Skill: TRANSLATE - Converting the problem's wage structure and worker distribution into clear mathematical understanding

2. Calculate worker distribution by hours

Let's figure out how many actual workers fall into each category:

Group 1 (20 hours): \(60\%\) of 1,000 workers
\(60\% = \frac{60}{100} = 0.6\)
\(0.6 \times 1000 = 600\) workers

Group 2 (40 hours): \(30\%\) of 1,000 workers
\(30\% = \frac{30}{100} = 0.3\)
\(0.3 \times 1000 = 300\) workers

Group 3 (50 hours): The rest of the workers
Since \(60\% + 30\% = 90\%\), the remaining percentage is \(100\% - 90\% = 10\%\)
\(10\%\) of 1,000 = \(0.1 \times 1000 = 100\) workers

Let's verify: \(600 + 300 + 100 = 1000\) workers ✓

3. Apply pay structure to each group

Now we'll calculate the total pay for each group by considering the 35-hour regular/overtime threshold:

Group 1 (600 workers, 20 hours each):
Since 20 hours < 35 hours, all their hours are at regular pay
Pay per worker = \(20 \text{ hours} \times \$6/\text{hour} = \$120\)
Total for Group 1 = \(600 \text{ workers} \times \$120 = \$72,000\)

Group 2 (300 workers, 40 hours each):
Since 40 hours > 35 hours, they get regular pay for first 35 hours and overtime for the extra 5 hours
Regular hours per worker = \(35 \text{ hours} \times \$6/\text{hour} = \$210\)
Overtime hours per worker = \((40 - 35) = 5 \text{ hours} \times \$7.50/\text{hour} = \$37.50\)
Total pay per worker = \(\$210 + \$37.50 = \$247.50\)
Total for Group 2 = \(300 \text{ workers} \times \$247.50 = \$74,250\)

Group 3 (100 workers, 50 hours each):
Since 50 hours > 35 hours, they get regular pay for first 35 hours and overtime for the extra 15 hours
Regular hours per worker = \(35 \text{ hours} \times \$6/\text{hour} = \$210\)
Overtime hours per worker = \((50 - 35) = 15 \text{ hours} \times \$7.50/\text{hour} = \$112.50\)
Total pay per worker = \(\$210 + \$112.50 = \$322.50\)
Total for Group 3 = \(100 \text{ workers} \times \$322.50 = \$32,250\)

4. Sum all group payrolls

Now we add up the payroll from all three groups to get the total factory payroll:

Group 1 total: \(\$72,000\)
Group 2 total: \(\$74,250\)
Group 3 total: \(\$32,250\)

Total payroll = \(\$72,000 + \$74,250 + \$32,250 = \$178,500\)

Looking at our answer choices, this matches choice E: \(\$178,500\)

Final Answer: E. \(\$178,500\)

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the overtime rate calculation: Students often misinterpret "1.25 times that rate" and incorrectly calculate the overtime rate as \(\$6 + \$1.25 = \$7.25\) instead of \(\$6 \times 1.25 = \$7.50\) per hour.

2. Confusion about the overtime threshold: Students may think that overtime kicks in after 40 hours (standard full-time) rather than carefully reading that this factory uses 35 hours as the threshold for overtime pay.

3. Misinterpreting "the rest worked 50 hours": Students might assume this refers to a specific percentage given elsewhere or make up their own percentage instead of calculating that "the rest" means \(100\% - 60\% - 30\% = 10\%\) of workers.

Errors while executing the approach

1. Arithmetic errors in percentage calculations: When converting \(60\%\) and \(30\%\) of 1,000 workers, students may make basic multiplication errors, such as calculating \(30\%\) of 1,000 as 30 workers instead of 300 workers.

2. Incorrect overtime hour calculations: For workers who worked 40 hours, students might calculate overtime as \(40 \times \$7.50 = \$300\) instead of recognizing that only \((40-35) = 5\) hours qualify for overtime rate, making it \(5 \times \$7.50 = \$37.50\).

3. Computational errors in final multiplication: When multiplying the per-worker totals by the number of workers in each group (e.g., \(300 \times \$247.50\)), students often make calculation mistakes, especially with decimal amounts.

Errors while selecting the answer

1. Adding wrong group totals: Students may correctly calculate individual group payrolls but make errors when adding the three final amounts (\(\$72,000 + \$74,250 + \$32,250\)), potentially getting close but incorrect totals that might match other answer choices.

2. Forgetting to include all worker groups: Students might calculate the payroll for only two of the three worker groups and select an answer choice that matches this incomplete total, missing that they need to account for all 1,000 workers.