Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25...

1. Translate the problem requirements: We need to express r (percent of revenue from Newspaper A) in terms of p (percent of newspapers sold that were Newspaper A). Newspaper A costs \(\$1.00\) and Newspaper B costs \(\$1.25\).
2. Set up variables for total quantities: Use a concrete total number of newspapers to make calculations simple and avoid working with complex fractions throughout.
3. Express revenues in terms of the given percentages: Calculate total revenue from both newspapers and determine what fraction comes from Newspaper A.
4. Convert the revenue fraction to percentage form: Transform the fraction into the required percentage expression r in terms of p.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday terms. We have a store that sells two different newspapers:
• Newspaper A costs \(\$1.00\) each
• Newspaper B costs \(\$1.25\) each

The problem tells us that:
• p percent of all newspapers sold were Newspaper A
• r percent of total revenue came from Newspaper A

We need to find a formula that tells us r when we know p.

Think about it this way: if more expensive newspapers (B) are sold, then even if Newspaper A makes up the same percentage of total newspapers sold, it will make up a smaller percentage of the revenue because each copy brings in less money.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships

2. Set up variables for total quantities

Instead of working with confusing percentages and fractions right away, let's use a concrete example that makes the math simple.

Let's say the store sold exactly 100 newspapers total. This makes percentage calculations very straightforward:
• If p percent of newspapers sold were Newspaper A, then the store sold p copies of Newspaper A
• This means the store sold \((100 - p)\) copies of Newspaper B

For example, if \(p = 40\), then:
• 40 copies of Newspaper A were sold
• 60 copies of Newspaper B were sold

Using 100 as our total eliminates fractions and makes the relationship between percentages and actual quantities obvious.

3. Express revenues in terms of the given percentages

Now let's calculate how much revenue came from each type of newspaper:

Revenue from Newspaper A = \(p \text{ copies} \times \$1.00 = \$p\)
Revenue from Newspaper B = \((100 - p) \text{ copies} \times \$1.25 = \$1.25(100 - p) = \$(125 - 1.25p)\)

Total Revenue = Revenue from A + Revenue from B
Total Revenue = \(\$p + \$(125 - 1.25p) = \$(125 - 0.25p)\)

Now we can find what fraction of total revenue comes from Newspaper A:
Fraction from A = \(\frac{\text{Revenue from A}}{\text{Total Revenue}} = \frac{p}{125 - 0.25p}\)

Let's simplify this fraction by factoring out 0.25 from the denominator:
\(125 - 0.25p = 0.25(500 - p)\)

So our fraction becomes: \(\frac{p}{0.25(500 - p)} = \frac{p}{0.25} \div (500 - p) = \frac{4p}{500 - p}\)

4. Convert the revenue fraction to percentage form

We found that the fraction of revenue from Newspaper A is \(\frac{4p}{500 - p}\).

To convert this fraction to a percentage, we multiply by 100:
\(r = \frac{4p}{500 - p} \times 100 = \frac{400p}{500 - p}\)

Let's verify this makes sense: if \(p = 100\) (all newspapers sold were type A), then:
\(r = \frac{400(100)}{500 - 100} = \frac{40,000}{400} = 100\)

This means 100% of revenue came from Newspaper A, which is exactly right!

If \(p = 0\) (no Newspaper A sold), then \(r = 0\), which also makes perfect sense.

4. Final Answer

Our formula is \(r = \frac{400p}{500 - p}\), which matches answer choice D exactly.

Answer: D

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Confusing what r and p represent
Students often mix up the definitions of r and p. They might think p represents the percentage of revenue from Newspaper A (instead of percentage of newspapers sold), or that r represents the percentage of newspapers sold (instead of percentage of revenue). This fundamental misunderstanding leads to setting up the wrong relationships from the start.

Faltering Point 2: Failing to establish the connection between quantities and revenue percentages
Many students struggle to see that they need to work with both the number of newspapers sold AND the different prices. They might try to directly relate p and r without considering that Newspaper B costs more (\(\$1.25\) vs \(\$1.00\)), which affects how quantity percentages translate to revenue percentages.

Faltering Point 3: Overcomplicating the setup with unnecessary variables
Instead of using the smart approach of assuming 100 total newspapers (which makes percentage calculations simple), students often introduce multiple variables like 'total newspapers = N' or 'total revenue = R', making the algebra much more complex and error-prone.

Errors while executing the approach

Faltering Point 1: Arithmetic errors when calculating total revenue
When computing Total Revenue = \(p + 1.25(100-p) = p + 125 - 1.25p = 125 - 0.25p\), students frequently make sign errors or incorrectly combine like terms, getting results like \(125 + 0.25p\) or \(125 - 1.25p\) instead of the correct \(125 - 0.25p\).

Faltering Point 2: Incorrectly simplifying the revenue fraction
When simplifying \(\frac{p}{125 - 0.25p}\), students often fail to properly factor out 0.25 from the denominator. They might write \(125 - 0.25p = 0.25(125 - p)\) instead of the correct \(0.25(500 - p)\), leading to wrong expressions like \(\frac{p}{125 - p}\) instead of \(\frac{4p}{500 - p}\).

Faltering Point 3: Forgetting to convert fraction to percentage
After finding that the fraction of revenue from Newspaper A is \(\frac{4p}{500 - p}\), students sometimes forget that r represents a percentage, so they need to multiply by 100. They might select an answer choice that gives \(\frac{4p}{500 - p}\) instead of \(\frac{400p}{500 - p}\).

Errors while selecting the answer

Faltering Point 1: Matching expressions with different denominators
Students who made errors in their calculation might end up with expressions like \(\frac{400p}{125 - p}\) or \(\frac{400p}{250 - p}\) and incorrectly match these to answer choices A or B, not realizing their algebraic mistakes led to the wrong denominator.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient smart numbers

Since we need to work with percentages and the problem involves newspapers sold at \(\$1.00\) and \(\$1.25\), let's choose a total number of newspapers that makes percentage calculations clean. Let's say the store sold 100 total newspapers (this makes percentage calculations straightforward since p% of 100 = p newspapers).

Step 2: Express quantities in terms of p

If p percent of newspapers sold were Newspaper A:
• Newspapers A sold = p newspapers
• Newspapers B sold = \((100 - p)\) newspapers

Step 3: Calculate revenues from each newspaper

• Revenue from Newspaper A = \(p \times \$1.00 = \$p\)
• Revenue from Newspaper B = \((100 - p) \times \$1.25 = \$1.25(100 - p) = \$(125 - 1.25p)\)

Step 4: Calculate total revenue

Total revenue = \(\$p + \$(125 - 1.25p) = \$(125 - 0.25p)\)

Step 5: Find what percent of revenue comes from Newspaper A

\(r = \frac{\text{Revenue from A}}{\text{Total revenue}} \times 100\)
\(r = \frac{p}{125 - 0.25p} \times 100\)
\(r = \frac{100p}{125 - 0.25p}\)

Step 6: Simplify to match answer format

To eliminate the decimal, multiply both numerator and denominator by 4:
\(r = \frac{100p \times 4}{(125 - 0.25p) \times 4}\)
\(r = \frac{400p}{500 - p}\)

This matches answer choice D.