The price of an item was increased from $40 to $45. The price was increased by x percent. What is...

Part 1: Brief Solution

Concepts tested: Percentage calculations, basic algebraic manipulation
Primary process skills: Translate, Infer

Essential steps:

• Translate the statement "13 is p% of 25" into the equation: \(13 = \frac{\mathrm{p}}{100} \times 25\)
• Simplify to get: \(13 = \frac{25\mathrm{p}}{100}\), which becomes \(13 = \frac{\mathrm{p}}{4}\)
• Solve for p: \(\mathrm{p} = 13 \times 4 = 52\)
• Final answer: p = 52

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Equation Setup Confusion - Phase: Devising Approach → Choice varies (often around 192)

   • Process skill failure: Translate
   • Students reverse the relationship and write \(25 = \frac{\mathrm{p}}{100} \times 13\), leading to \(\mathrm{p} \approx 192.3\)

2. Percentage Conversion Error - Phase: Executing Approach → Choice 0.52

   • Process skill failure: Infer
   • Students correctly calculate \(\frac{13}{25} = 0.52\) but forget that percentages require multiplying by 100

3. Arithmetic Mistakes - Phase: Executing Approach → Various wrong values

   • Computational error: Basic multiplication/division
   • Students make calculation errors when solving \(13 \times 4\) or when manipulating fractions

Part 3: Detailed Solution

Understanding the Problem Structure

When we encounter "13 is p% of 25," we're dealing with a classic percentage relationship problem. Think of this like asking: "If I have 25 apples total, and 13 of them are red, what percentage of my apples are red?"

Process Skill: TRANSLATE - The key insight is converting this English statement into mathematical language. The phrase "A is p% of B" always translates to the equation: \(\mathrm{A} = \frac{\mathrm{p}}{100} \times \mathrm{B}\).

In our case:

• A = 13 (the part)
• B = 25 (the whole)
• p = unknown (the percentage we're solving for)

So we get: \(13 = \frac{\mathrm{p}}{100} \times 25\)

Step-by-Step Solution

Starting with our equation:
\(13 = \frac{\mathrm{p}}{100} \times 25\)

Process Skill: INFER - We recognize that \(\frac{\mathrm{p}}{100} \times 25\) can be simplified by rewriting it as \(\frac{25\mathrm{p}}{100}\), which helps us see the algebraic structure more clearly.

\(13 = \frac{25\mathrm{p}}{100}\)

To eliminate the fraction, multiply both sides by 100:
\(13 \times 100 = 25\mathrm{p}\)
\(1300 = 25\mathrm{p}\)

Now divide both sides by 25 to isolate p:
\(\mathrm{p} = \frac{1300}{25}\)
\(\mathrm{p} = 52\)

Alternative Mental Math Approach:
Once we have \(13 = \frac{25\mathrm{p}}{100}\), we can simplify by recognizing that \(\frac{25}{100} = \frac{1}{4}\):
\(13 = \frac{\mathrm{p}}{4}\)

Therefore: \(\mathrm{p} = 13 \times 4 = 52\)

Verification Check:
Let's verify our answer makes sense: 52% of 25 should equal 13.
\(0.52 \times 25 = 13\) ✓

This confirms p = 52 is correct.

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

• Relationship Reversal (Process Skill: Translate) - Students often confuse which number is the "part" and which is the "whole." They might think "25 is p% of 13" and set up the equation as \(25 = \frac{\mathrm{p}}{100} \times 13\). This leads to \(\mathrm{p} = \frac{2500}{13} \approx 192.3\), which seems unreasonably large but doesn't trigger their sense-checking.

• Percentage Formula Confusion (Process Skill: Infer) - Some students struggle with the concept that "p%" means "p/100" and might write equations like \(13 = \mathrm{p} \times 25\), forgetting the percentage conversion entirely.

Errors while executing the approach:

• Incomplete Percentage Conversion (Process Skill: Infer) - Students correctly identify that \(\frac{13}{25} = 0.52\), but forget that percentages are expressed as numbers out of 100, not as decimals out of 1. They stop at 0.52 instead of multiplying by 100 to get 52.

• Fraction Manipulation Errors (Computational error) - When working with \(13 = \frac{25\mathrm{p}}{100}\), students might incorrectly cross-multiply or make arithmetic mistakes. For example, they might calculate \(13 \times 100 = 130\) instead of \(1300\).

• Mental Math Shortcuts Gone Wrong (Computational error) - Students who recognize that \(\frac{25}{100} = \frac{1}{4}\) might then incorrectly calculate \(13 \times 4\), getting 42 or 48 instead of 52.

Errors while selecting the answer:

• Unit Confusion - Students might arrive at the correct decimal 0.52 but select this as their final answer, not recognizing that the question asks for p where "p%" means the answer should be 52, not 0.52.

• Reasonableness Check Failure - Students who get answers like 192.3% don't pause to consider whether it makes sense for 13 to be nearly twice as much as 25 (which would be 200% of 25).