The number k is 12 more than one-third of the number m. Which equation represents m in terms of k?

1. TRANSLATE the verbal statement into mathematical notation

• Given statement: "The number k is 12 more than one-third of the number m"
• Break this down piece by piece:
  • "The number k is..." → \(\mathrm{k =}\)
  • "one-third of the number m" → \(\mathrm{\frac{1}{3}m}\) or \(\mathrm{\frac{m}{3}}\)
  • "12 more than [something]" → [something] + 12
• Putting it together: \(\mathrm{k = \frac{1}{3}m + 12}\)

2. SIMPLIFY to solve for m in terms of k

• Our goal is to isolate m on one side of the equation
• Start with: \(\mathrm{k = \frac{1}{3}m + 12}\)
• Subtract 12 from both sides: \(\mathrm{k - 12 = \frac{1}{3}m}\)
• Multiply both sides by 3 to eliminate the fraction: \(\mathrm{3(k - 12) = m}\)
• Distribute the 3: \(\mathrm{m = 3k - 36}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly translate to \(\mathrm{k = \frac{1}{3}m + 12}\) but make algebraic errors when solving for m. The most frequent mistake occurs with the distributive property: when they reach \(\mathrm{3(k - 12) = m}\), they incorrectly distribute as \(\mathrm{3k - 12}\) instead of \(\mathrm{3k - 36}\).

This leads them to select Choice C (\(\mathrm{m = 3k - 12}\)).

Second Most Common Error:

Conceptual confusion about equation solving: Students get to \(\mathrm{k - 12 = \frac{1}{3}m}\) but then become confused about how to isolate m. Instead of multiplying both sides by 3, they incorrectly think they should divide by 3, leading to \(\mathrm{m = \frac{k - 12}{3}}\).

This may lead them to select Choice A (\(\mathrm{m = \frac{k - 12}{3}}\)).

The Bottom Line:

This problem tests whether students can accurately translate verbal relationships into algebra AND maintain precision through multi-step algebraic manipulation. The key challenge is keeping track of negative signs and applying the distributive property correctly.