If x/y = 4 and 24x/ny = 4, what is the value of n?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{\frac{x}{y} = 4}\)
  • \(\mathrm{\frac{24x}{ny} = 4}\)
• Find: the value of n

2. INFER the strategic approach

• The key insight is recognizing that the second equation contains the relationship \(\mathrm{\frac{x}{y}}\) from the first equation
• I need to factor \(\mathrm{\frac{24x}{ny}}\) to separate out the \(\mathrm{\frac{x}{y}}\) term
• Strategy: Rewrite \(\mathrm{\frac{24x}{ny}}\) as \(\mathrm{\frac{24}{n} \times \frac{x}{y}}\), then substitute \(\mathrm{\frac{x}{y} = 4}\)

3. SIMPLIFY by factoring the complex fraction

• Start with: \(\mathrm{\frac{24x}{ny} = 4}\)
• Factor the fraction: \(\mathrm{\frac{24x}{ny} = \frac{24}{n} \times \frac{x}{y}}\)
• So the equation becomes: \(\mathrm{\frac{24}{n} \times \frac{x}{y} = 4}\)

4. INFER the substitution opportunity

• Since \(\mathrm{\frac{x}{y} = 4}\), I can substitute this value
• \(\mathrm{\frac{24}{n} \times 4 = 4}\)

5. SIMPLIFY to solve for n

• \(\mathrm{\frac{24}{n} \times 4 = 4}\)
• \(\mathrm{\frac{96}{n} = 4}\)
• Multiply both sides by n: \(\mathrm{96 = 4n}\)
• Divide both sides by 4: \(\mathrm{n = 24}\)

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that \(\mathrm{\frac{24x}{ny}}\) contains the given relationship \(\mathrm{\frac{x}{y} = 4}\). Instead, they might try to solve the system by elimination or attempt to find individual values for x and y, leading to unnecessary complexity and confusion. This causes them to get stuck and abandon systematic solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret the fraction \(\mathrm{\frac{24x}{ny}}\), perhaps reading it as \(\mathrm{\frac{24x}{n \times y}}\) but not recognizing they can factor out \(\mathrm{\frac{x}{y}}\). They might try to cross-multiply incorrectly or manipulate the fractions in ways that don't preserve the relationship structure. This leads to algebraic errors and incorrect answer selection.

The Bottom Line:

This problem tests whether students can recognize embedded relationships within complex algebraic expressions. The critical insight is seeing that one given equation contains a component \(\mathrm{(\frac{x}{y})}\) that equals a known value from another equation.