The given equation relates the positive numbers m, n, and p. Which equation correctly gives n in terms of m...

1. TRANSLATE the problem goal

• Given equation: \(7\mathrm{m} = 5(\mathrm{n} + \mathrm{p})\)
• Goal: Find n in terms of m and p (isolate n on one side)

2. SIMPLIFY by removing the coefficient of the parentheses

• Divide both sides by 5 to eliminate the coefficient of (n + p):

\(\frac{7\mathrm{m}}{5} = \mathrm{n} + \mathrm{p}\)

3. SIMPLIFY by isolating n

• Subtract p from both sides:

\(\frac{7\mathrm{m}}{5} - \mathrm{p} = \mathrm{n}\)

• Therefore: \(\mathrm{n} = \frac{7\mathrm{m}}{5} - \mathrm{p}\)

Answer: B. \(\mathrm{n} = \frac{7\mathrm{m}}{5} - \mathrm{p}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic manipulation errors when working with the equation. Some may incorrectly distribute or combine terms, leading to expressions like \(5(7\mathrm{m}) + \mathrm{p}\) or other incorrect forms. This may lead them to select Choice C (\(\mathrm{n} = 5(7\mathrm{m}) + \mathrm{p}\)) or get confused about the proper order of operations.

Second Most Common Error:

Poor TRANSLATE reasoning: Students may misunderstand what "solve for n in terms of m and p" means and attempt to substitute values or rearrange incorrectly. They might think they need to eliminate variables rather than isolate n, leading to expressions like \(\frac{5\mathrm{p}}{7\mathrm{m}}\). This may lead them to select Choice A (\(\mathrm{n} = \frac{5\mathrm{p}}{7\mathrm{m}}\)).

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires methodically applying inverse operations in the correct order while maintaining the balance of the equation.