At a loading dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the...

1. Translate the problem requirements: We need to understand that night crew workers load \(\frac{3}{4}\) as many boxes per worker compared to day crew workers, and there are \(\frac{4}{5}\) as many night workers as day workers. We want to find what fraction of total boxes the day crew loaded.
2. Set up variables for comparison: Choose simple variables to represent the number of day workers and boxes per day worker to avoid complex fractions.
3. Calculate total boxes for each crew: Multiply number of workers by boxes per worker for both day and night crews using our established relationships.
4. Find the day crew's fraction of total: Express day crew boxes as a fraction of the sum of both crews' boxes and simplify to match answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

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

Think of this like comparing two work shifts at a warehouse. We have a day crew and a night crew, and we want to figure out what portion of all the work the day crew did.

Here's what the problem tells us:
• Each night worker loads fewer boxes than each day worker - specifically, each night worker loads \(\frac{3}{4}\) as many boxes as each day worker
• There are fewer night workers than day workers - specifically, there are \(\frac{4}{5}\) as many night workers as day workers

Our goal is to find: What fraction of all boxes loaded came from the day crew?

Process Skill: TRANSLATE - Converting the word relationships into mathematical understanding

2. Set up variables for comparison

To avoid working with messy fractions, let's choose smart numbers that make our calculations clean.

Let's say the day crew has 5 workers (this makes the \(\frac{4}{5}\) relationship easy to work with).
Let's say each day worker loads 4 boxes (this makes the \(\frac{3}{4}\) relationship easy to work with).

Now we can figure out the night crew details:
• Number of night workers = \(\frac{4}{5} \times 5\) day workers = 4 workers
• Boxes per night worker = \(\frac{3}{4} \times 4\) boxes per day worker = 3 boxes per worker

These numbers give us a concrete picture to work with, and the fractions work out nicely.

3. Calculate total boxes for each crew

Now let's calculate how many boxes each crew loaded in total:

Day crew total boxes:
Number of day workers × Boxes per day worker = \(5 \times 4 = 20\) boxes

Night crew total boxes:
Number of night workers × Boxes per night worker = \(4 \times 3 = 12\) boxes

Total boxes loaded by both crews = \(20 + 12 = 32\) boxes

4. Find the day crew's fraction of total

Now we can find what fraction of all boxes the day crew loaded:

Day crew's fraction = Day crew boxes ÷ Total boxes
Day crew's fraction = \(20 ÷ 32 = \frac{20}{32}\)

Let's simplify this fraction:
\(\frac{20}{32} = \frac{5}{8}\) (dividing both numerator and denominator by 4)

Let's verify this makes sense: \(\frac{5}{8} = 0.625\), which means the day crew loaded 62.5% of all boxes. Since day workers are more productive per person AND there are more of them, it makes sense they'd do the majority of the work.

Final Answer

The day crew loaded \(\frac{5}{8}\) of all the boxes.

Looking at our answer choices, this matches choice (E) \(\frac{5}{8}\).

We can double-check: The night crew would have loaded \(\frac{12}{32} = \frac{3}{8}\) of the boxes, and \(\frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1\), confirming our fractions account for all boxes loaded.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the fraction relationships
Students often misread "\(\frac{3}{4}\) as many" and "\(\frac{4}{5}\) as many" statements. They might think "\(\frac{3}{4}\) as many boxes" means the night crew loads MORE boxes (since \(\frac{3}{4}\) looks like a big fraction), when it actually means they load FEWER boxes than the day crew. Similarly, they might confuse which crew the fractions refer to - thinking the day crew has \(\frac{4}{5}\) as many workers as the night crew instead of the other way around.

Faltering Point 2: Setting up the wrong equation structure
Students may try to work directly with variables and fractions instead of using concrete numbers. This leads to complex algebraic expressions like setting up equations with multiple variables (D for day workers, N for night workers, etc.) which becomes unnecessarily complicated and error-prone for this type of problem.

Faltering Point 3: Confusing what the question is asking for
The question asks for "what fraction of ALL boxes did the DAY crew load" but students might set up to find what fraction the NIGHT crew loaded, or confuse this with finding ratios between the crews rather than each crew's portion of the total.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in basic calculations
When calculating totals, students often make simple multiplication or addition errors. For example, miscalculating \(5 \times 4 = 20\) or \(4 \times 3 = 12\), or incorrectly adding \(20 + 12 = 32\). These seemingly simple calculations are where many students lose points under time pressure.

Faltering Point 2: Incorrect fraction simplification
When simplifying \(\frac{20}{32}\), students might reduce incorrectly. Common errors include dividing by the wrong number (like dividing by 2 to get \(\frac{10}{16}\) instead of dividing by 4), or making errors in the division process itself, leading to fractions like \(\frac{4}{6}\) or other incorrect results.

Errors while selecting the answer

Faltering Point 1: Selecting the night crew's fraction instead
After calculating both \(\frac{20}{32}\) for the day crew and \(\frac{12}{32} = \frac{3}{8}\) for the night crew, students might accidentally select \(\frac{3}{8}\) if it were among the answer choices, or look for an answer choice that represents the night crew's portion instead of the day crew's portion.

Faltering Point 2: Choosing an unsimplified or incorrectly simplified version
Students might see that \(\frac{20}{32}\) equals \(\frac{5}{8}\) but then second-guess themselves and select a different answer choice that represents their intermediate calculations, such as selecting \(\frac{2}{5}\) or \(\frac{3}{5}\) if they made errors in their fraction reduction process.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers that work well with the given fractions

Since we're dealing with fractions \(\frac{3}{4}\) and \(\frac{4}{5}\), let's choose numbers that make these calculations clean:
- Number of day crew workers = 20
- Boxes loaded per day worker = 4

These choices eliminate fractions in our intermediate calculations.

Step 2: Calculate night crew values using the given relationships

Night crew workers = \(\left(\frac{4}{5}\right) \times 20 = 16\) workers

Boxes per night worker = \(\left(\frac{3}{4}\right) \times 4 = 3\) boxes

Step 3: Calculate total boxes for each crew

Day crew total boxes = \(20 \times 4 = 80\) boxes

Night crew total boxes = \(16 \times 3 = 48\) boxes

Step 4: Find the fraction loaded by day crew

Total boxes by both crews = \(80 + 48 = 128\) boxes

Fraction loaded by day crew = \(\frac{80}{128} = \frac{5}{8}\)

Answer: (E) \(\frac{5}{8}\)

The smart numbers approach works particularly well here because our chosen values (20 and 4) are specifically selected to work cleanly with the given fractions \(\frac{4}{5}\) and \(\frac{3}{4}\), eliminating messy fraction arithmetic while maintaining the exact relationships described in the problem.