Which expression is equivalent to \(5(7\mathrm{r} - 4\mathrm{r})\)? 3r 8r 15r 31r...

1. TRANSLATE the problem information

• Given expression: \(\mathrm{5(7r - 4r)}\)
• Need to find: An equivalent expression

2. SIMPLIFY using order of operations

• Step 1: Handle what's inside parentheses first
  • Combine like terms: \(\mathrm{7r - 4r = 3r}\)
  • Now we have: \(\mathrm{5(3r)}\)

• Step 2: Complete the multiplication
  • \(\mathrm{5 \times 3r = 15r}\)

Answer: C (\(\mathrm{15r}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students correctly combine like terms inside parentheses (\(\mathrm{7r - 4r = 3r}\)) but then forget to multiply by the 5 outside the parentheses.

They see \(\mathrm{5(7r - 4r)}\), simplify to get \(\mathrm{3r}\), and stop there without completing the final multiplication step.

This leads them to select Choice A (\(\mathrm{3r}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread the expression as \(\mathrm{5 + 7r - 4r}\) instead of \(\mathrm{5(7r - 4r)}\).

They think: "5 plus 7r minus 4r equals \(\mathrm{5 + 3r = 8r}\)."

This leads them to select Choice B (\(\mathrm{8r}\)).

Third Most Common Error:

Flawed SIMPLIFY approach: Students attempt to distribute the 5 but only apply it to the first term, calculating \(\mathrm{5 \times 7r - 4r = 35r - 4r = 31r}\).

This shows confusion about how the distributive property works with the entire expression in parentheses.

This leads them to select Choice D (\(\mathrm{31r}\)).

The Bottom Line:

This problem tests whether students can maintain focus through a two-step simplification process while correctly applying order of operations. The key is remembering that parentheses create a "package" that must be completely simplified before moving to the next operation.