Which expression is equivalent to \(20\mathrm{w} - (4\mathrm{w} + 3\mathrm{w})\)?

1. INFER the approach

• Key insight: Follow order of operations - work inside parentheses first
• Strategy: Combine like terms inside parentheses, then handle the subtraction

2. SIMPLIFY inside the parentheses

• Inside the parentheses: \(\mathrm{4w + 3w}\)
• Combine like terms: \(\mathrm{4w + 3w = 7w}\)
• Expression becomes: \(\mathrm{20w - (7w)}\)

3. SIMPLIFY the final expression

• Remove parentheses: \(\mathrm{20w - 7w}\)
• Combine like terms: \(\mathrm{20w - 7w = 13w}\)

Answer: B. \(\mathrm{13w}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with negative signs: Students incorrectly distribute the negative sign, treating the expression as \(\mathrm{20w - 4w + 3w}\) instead of \(\mathrm{20w - (4w + 3w)}\).

When they calculate \(\mathrm{20w - 4w + 3w}\), they get \(\mathrm{16w + 3w = 19w}\).

This leads them to select Choice C (\(\mathrm{19w}\)).

Second Most Common Error:

Poor INFER reasoning about order of operations: Students might try to combine all terms without recognizing the parentheses create a grouping that must be handled first.

They might incorrectly think they can rearrange to get something like \(\mathrm{20w + 4w - 3w}\) or make arithmetic errors in combining terms, leading to confusion and random answer selection.

The Bottom Line:

This problem tests whether students can correctly handle the interaction between parentheses and negative signs - a critical skill for more advanced algebra where sign errors become costly.