Question:If x/y = 4 and \(\frac{30(\mathrm{x} - \mathrm{y})}{\mathrm{n} \mathrm{y}} = 5\), what is the value of n?Enter your answer as...

1. TRANSLATE the first equation into usable form

• Given: \(\frac{\mathrm{x}}{\mathrm{y}} = 4\)
• This means: \(\mathrm{x} = 4\mathrm{y}\)
• This relationship will let us substitute for x in the second equation

2. INFER the solution strategy

• We have one equation with a ratio and another with both variables
• Strategy: Use substitution to eliminate one variable
• Substitute \(\mathrm{x} = 4\mathrm{y}\) into the second equation

3. SIMPLIFY through substitution and algebraic manipulation

• Start with: \(\frac{30(\mathrm{x} - \mathrm{y})}{\mathrm{n} \mathrm{y}} = 5\)
• Substitute \(\mathrm{x} = 4\mathrm{y}\): \(\frac{30(4\mathrm{y} - \mathrm{y})}{\mathrm{n} \mathrm{y}} = 5\)
• Combine like terms: \(4\mathrm{y} - \mathrm{y} = 3\mathrm{y}\)
• Now we have: \(\frac{30 \cdot 3\mathrm{y}}{\mathrm{n} \mathrm{y}} = 5\)

4. SIMPLIFY further by canceling and solving

• Multiply out: \(\frac{90\mathrm{y}}{\mathrm{ny}} = 5\)
• Cancel the y terms: \(\frac{90}{\mathrm{n}} = 5\)
• Solve for n: \(\mathrm{n} = \frac{90}{5} = 18\)

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly interpret \(\frac{\mathrm{x}}{\mathrm{y}} = 4\)

Instead of recognizing this means \(\mathrm{x} = 4\mathrm{y}\), they might think it means \(\mathrm{x} = \frac{4}{\mathrm{y}}\) or get confused about which variable to solve for. This fundamental misunderstanding derails the entire substitution process, leading to algebraic equations that don't simplify properly and producing incorrect numerical answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic manipulation errors during the multi-step process

Common mistakes include not properly combining \(4\mathrm{y} - \mathrm{y}\) to get \(3\mathrm{y}\), incorrectly canceling the y terms, or making arithmetic errors when calculating \(\frac{90}{5}\). These execution errors lead to wrong final answers even when the overall strategy is correct.

The Bottom Line:

This problem tests whether students can convert ratio relationships into algebraic equations and then execute a clean substitution strategy. The key insight is recognizing that ratios give us direct variable relationships we can use for substitution.