Two expressions represent the same constant rate R. The first is R = 1/6m, where m is a positive constant....

1. TRANSLATE the problem information

• Given information:
  • First expression: \(\mathrm{R = \frac{1}{6m}}\)
  • Second expression: \(\mathrm{R = \frac{8n}{y}}\)
  • Both expressions represent the same rate R
• What this tells us: Since both expressions equal R, they must equal each other

2. INFER the solution approach

• Since both expressions equal the same rate R, we can set them equal to each other
• This gives us: \(\mathrm{\frac{1}{6m} = \frac{8n}{y}}\)
• Cross-multiplication will be the most efficient way to eliminate the fractions

3. SIMPLIFY by cross-multiplying

• Cross-multiply: \(\mathrm{y \times 1 = 6m \times 8n}\)
• This gives us: \(\mathrm{y = 48mn}\)

4. SIMPLIFY to solve for n

• Divide both sides by 48m: \(\mathrm{n = \frac{y}{48m}}\)
• This matches answer choice (A)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "two expressions represent the same rate" means they can be set equal to each other. Instead, they might try to manipulate each expression separately or get confused about what the problem is asking.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{1}{6m} = \frac{8n}{y}}\) but make arithmetic errors during cross-multiplication. They might forget to multiply by the 8, getting \(\mathrm{y = 6mn}\) instead of \(\mathrm{y = 48mn}\), leading them to \(\mathrm{n = \frac{y}{6m}}\).

This may lead them to select Choice B \(\mathrm{(\frac{y}{6m})}\).

The Bottom Line:

This problem tests whether students can recognize that equal expressions can be set equal to each other, then execute cross-multiplication accurately. The key insight is translating the verbal relationship into a mathematical equation.