The positive numbers P, N, and C satisfy the equation 2P + NC = 19N. Which equation correctly expresses C...

1. TRANSLATE the problem information

• Given equation: \(2\mathrm{P} + \mathrm{NC} = 19\mathrm{N}\)
• Goal: Express \(\mathrm{C}\) in terms of \(\mathrm{P}\) and \(\mathrm{N}\)
• All variables are positive (so \(\mathrm{N} ≠ 0\), making division by \(\mathrm{N}\) valid)

2. INFER the solving strategy

• To isolate \(\mathrm{C}\), I need to eliminate other terms from the side containing \(\mathrm{NC}\)
• Strategy: First remove the \(2\mathrm{P}\) term, then divide by \(\mathrm{N}\) to get \(\mathrm{C}\) alone

3. SIMPLIFY by eliminating the 2P term

• Subtract \(2\mathrm{P}\) from both sides:

\(2\mathrm{P} + \mathrm{NC} - 2\mathrm{P} = 19\mathrm{N} - 2\mathrm{P}\)

• This gives us: \(\mathrm{NC} = 19\mathrm{N} - 2\mathrm{P}\)

4. SIMPLIFY by dividing both sides by N

• Divide both sides by \(\mathrm{N}\):

\(\frac{\mathrm{NC}}{\mathrm{N}} = \frac{19\mathrm{N} - 2\mathrm{P}}{\mathrm{N}}\)

• This gives us: \(\mathrm{C} = \frac{19\mathrm{N} - 2\mathrm{P}}{\mathrm{N}}\)

5. SIMPLIFY by distributing the division

• Apply the distributive property:

\(\frac{19\mathrm{N} - 2\mathrm{P}}{\mathrm{N}} = \frac{19\mathrm{N}}{\mathrm{N}} - \frac{2\mathrm{P}}{\mathrm{N}}\)

• Simplify: \(\mathrm{C} = 19 - \frac{2\mathrm{P}}{\mathrm{N}}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with the distributive property of division, thinking that \(\frac{19\mathrm{N} - 2\mathrm{P}}{\mathrm{N}}\) should be written as \(\frac{19 - 2\mathrm{P}}{\mathrm{N}}\), incorrectly factoring out the \(\mathrm{N}\) from both terms in the numerator.

This leads them to select Choice A: \(\mathrm{C} = \frac{19 - 2\mathrm{P}}{\mathrm{N}}\)

Second Most Common Error:

Poor attention to coefficients during SIMPLIFY: Students correctly perform the algebraic steps but lose track of the coefficient 2 when subtracting \(2\mathrm{P}\), treating it as just \(\mathrm{P}\).

This leads them to select Choice C: \(\mathrm{C} = 19 - \frac{\mathrm{P}}{\mathrm{N}}\)

The Bottom Line:

The challenge lies in carefully executing multiple algebraic steps while maintaining proper order of operations and coefficient tracking. Success requires systematic application of algebraic properties rather than rushing to a final form.