p = 20 + 16/nThe given equation relates the numbers p and n, where n neq 0 and p gt...

1. TRANSLATE the problem information

• Given: \(\mathrm{p = 20 + \frac{16}{n}}\)
• Goal: Express n in terms of p (solve for n)

2. SIMPLIFY to isolate the fraction containing n

• Subtract 20 from both sides:

\(\mathrm{p - 20 = \frac{16}{n}}\)

3. SIMPLIFY to solve for n

• Since we have \(\mathrm{\frac{16}{n} = p - 20}\), we need to "flip" this relationship
• If \(\mathrm{\frac{16}{n} = p - 20}\), then \(\mathrm{n = \frac{16}{p - 20}}\)
• This uses the principle that if \(\mathrm{\frac{a}{b} = c}\), then \(\mathrm{b = \frac{a}{c}}\)

Answer: D. \(\mathrm{n = \frac{16}{p - 20}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle with fraction manipulation, particularly when the variable appears in the denominator. They might incorrectly think that if \(\mathrm{p - 20 = \frac{16}{n}}\), then \(\mathrm{n = 16 - (p - 20)}\) or \(\mathrm{n = \frac{p - 20}{16}}\), confusing the process of isolating a variable in the denominator.

This may lead them to select Choice A (\(\mathrm{n = \frac{p - 20}{16}}\)), which represents the incorrect algebraic manipulation.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "express n in terms of p" means and attempt to manipulate the original equation without a clear strategy, leading to random algebraic operations that don't systematically isolate n.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

The key challenge is recognizing that when a variable appears in the denominator of a fraction, solving for that variable requires understanding reciprocal relationships, not just standard linear equation techniques.