Which expression is equivalent to \(5(4\mathrm{w}) - [9\mathrm{w} - 2\mathrm{w}]\)?10w12w13w14w

1. SIMPLIFY using order of operations

• First priority: Handle what's inside brackets/parentheses
• Second priority: Handle multiplication
• Final step: Perform subtraction and combine like terms

2. SIMPLIFY the multiplication term first

• \(\mathrm{5(4w) = 5 \times 4 \times w = 20w}\)

3. SIMPLIFY the expression inside the brackets

• \(\mathrm{[9w - 2w] = 7w}\)

4. SIMPLIFY by completing the subtraction

• Now we have: \(\mathrm{20w - 7w}\)
• Combine like terms: \(\mathrm{(20 - 7)w = 13w}\)

Answer: C) \(\mathrm{13w}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors when combining terms inside the brackets.

Some students incorrectly calculate \(\mathrm{9w - 2w = 10w}\) (perhaps thinking \(\mathrm{9 - 2 = 10}\) instead of 7), then compute \(\mathrm{20w - 10w = 10w}\). This leads them to select Choice A (\(\mathrm{10w}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in the final combination step.

Students correctly get to \(\mathrm{20w - 7w}\) but then miscalculate \(\mathrm{20 - 7 = 12}\) instead of 13, or mistakenly compute the bracket as \(\mathrm{9w - 2w = 6w}\), leading to \(\mathrm{20w - 6w = 14w}\). This may lead them to select Choice B (\(\mathrm{12w}\)) or Choice D (\(\mathrm{14w}\)).

The Bottom Line:

This problem tests careful execution of order of operations and accurate arithmetic with like terms. The algebraic concepts are straightforward, but computational accuracy in each step is crucial for success.