Next year, if Company Q produces x units of a certain product and sells y units of this product, its...

1. Translate the problem requirements: Company Q produces \(\mathrm{x}\) units, sells \(\mathrm{y}\) units where \(\mathrm{y = rx}\). Total profit is \(\mathrm{10y - 4x}\). We need the average profit per item produced to be at least $4.50, so we need to find the minimum value of \(\mathrm{r}\).
2. Express total profit in terms of production relationship: Substitute \(\mathrm{y = rx}\) into the profit formula to get everything in terms of \(\mathrm{x}\) and \(\mathrm{r}\).
3. Set up the profit per item constraint: Average profit per item produced equals total profit divided by items produced, which must be at least $4.50.
4. Solve the inequality for minimum r: Transform the constraint into an inequality and solve for \(\mathrm{r}\) to find the least possible value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening here in plain English. Company Q is in the business of making and selling products, but they don't always sell everything they make.

Here's what we know:
• They produce \(\mathrm{x}\) units
• They sell \(\mathrm{y}\) units (where \(\mathrm{y = rx}\), meaning they sell a fraction \(\mathrm{r}\) of what they produce)
• Their total profit formula is: \(\mathrm{10y - 4x}\) dollars

Think of it this way: they make $10 for every unit they sell (\(\mathrm{10y}\)), but it costs them $4 for every unit they produce (\(\mathrm{4x}\)). The difference is their profit.

The key requirement: their average profit per item produced must be at least $4.50. We need to find the minimum value of \(\mathrm{r}\) (the fraction of items they actually sell).

Process Skill: TRANSLATE - Converting the business scenario into mathematical relationships

2. Express total profit in terms of production relationship

Now let's substitute the relationship \(\mathrm{y = rx}\) into our profit formula.

We started with: Total Profit = \(\mathrm{10y - 4x}\)

Since \(\mathrm{y = rx}\), we can replace \(\mathrm{y}\):
Total Profit = \(\mathrm{10(rx) - 4x}\)
Total Profit = \(\mathrm{10rx - 4x}\)
Total Profit = \(\mathrm{x(10r - 4)}\)

This is much cleaner! Now everything is expressed in terms of \(\mathrm{x}\) (units produced) and \(\mathrm{r}\) (the fraction sold).

3. Set up the profit per item constraint

Here's the crucial step. We need the average profit per item produced to be at least $4.50.

Average profit per item produced = Total Profit ÷ Number of items produced

We know:
• Total Profit = \(\mathrm{x(10r - 4)}\)
• Number of items produced = \(\mathrm{x}\)

So: Average profit per item = \(\mathrm{x(10r - 4) ÷ x = 10r - 4}\)

The constraint becomes:
\(\mathrm{10r - 4 \geq 4.50}\)

Notice how the \(\mathrm{x}\) cancels out completely! This makes sense - the average profit per item depends only on the relationship between production and sales, not the absolute numbers.

4. Solve the inequality for minimum r

Now we solve the inequality to find the minimum value of \(\mathrm{r}\):

\(\mathrm{10r - 4 \geq 4.50}\)

Add 4 to both sides:
\(\mathrm{10r \geq 4.50 + 4}\)
\(\mathrm{10r \geq 8.50}\)

Divide both sides by 10:
\(\mathrm{r \geq 8.50 ÷ 10}\)
\(\mathrm{r \geq 0.85}\)

This means Company Q must sell at least 85% of what they produce to achieve the minimum average profit of $4.50 per item produced.

Process Skill: APPLY CONSTRAINTS - Using the minimum profit requirement to determine the boundary condition

4. Final Answer

The least possible value of \(\mathrm{r}\) is 0.85.

Let's verify: If \(\mathrm{r = 0.85}\), then average profit per item = \(\mathrm{10(0.85) - 4 = 8.5 - 4 = \$4.50}\) ✓

Looking at our answer choices, this corresponds to Answer E: 0.85

Common Faltering Points

Errors while devising the approach

Students often misread the constraint and think they need to find profit per item sold rather than profit per item produced. This leads them to divide total profit by \(\mathrm{y}\) (units sold) instead of \(\mathrm{x}\) (units produced), resulting in a completely different equation and wrong answer.

Students may incorrectly assume that \(\mathrm{10y}\) represents revenue and \(\mathrm{4x}\) represents total costs, when actually the formula \(\mathrm{10y - 4x}\) already gives the profit directly. This confusion can lead them to set up additional unnecessary calculations or misunderstand what the constraint is asking for.

Some students fail to recognize that they need to substitute \(\mathrm{y = rx}\) into the profit formula early in their approach. Instead, they try to work with both variables \(\mathrm{x}\) and \(\mathrm{y}\) separately, making the problem much more complex than necessary.

Errors while executing the approach

When substituting \(\mathrm{y = rx}\) into the profit formula \(\mathrm{10y - 4x}\), students might incorrectly write \(\mathrm{10rx - 4rx}\) instead of \(\mathrm{10rx - 4x}\). This error stems from not carefully tracking which terms contain which variables.

When calculating average profit per item produced as \(\mathrm{x(10r - 4)/x}\), some students either forget to cancel the \(\mathrm{x}\) terms or make errors in the cancellation process, leading to expressions that still contain \(\mathrm{x}\) when the final result should only depend on \(\mathrm{r}\).

Errors while selecting the answer

After solving \(\mathrm{10r - 4 \geq 4.50}\) and getting \(\mathrm{r \geq 0.85}\), students might incorrectly think they need the maximum value of \(\mathrm{r}\) from the choices rather than recognizing that 0.85 is the minimum value, leading them to select a smaller option like 0.75 or 0.80.

Students may incorrectly verify their answer by testing values. For instance, they might test \(\mathrm{r = 0.80}\) and find it gives an average profit of $4.00, but fail to recognize this is below the required $4.50 minimum, leading them to incorrectly select this as the answer.