The equation 5r - 2.5ℓ = 80 represents the net displacement (in units) when a person takes r steps to...

1. TRANSLATE the problem information

• Given information:
  • The equation \(5\mathrm{r} - 2.5\mathrm{ℓ} = 80\) represents net displacement
  • Each right step (r) moves 5 units right
  • Each left step (ℓ) moves 2.5 units left
  • The person takes 16 steps to the right, so \(\mathrm{r} = 16\)
  • We want a net displacement of 80 units to the right

• What this tells us: We need to substitute \(\mathrm{r} = 16\) into the equation and solve for \(\mathrm{ℓ}\)

2. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{r} = 16\) into \(5\mathrm{r} - 2.5\mathrm{ℓ} = 80\):
  \(5(16) - 2.5\mathrm{ℓ} = 80\)
  \(80 - 2.5\mathrm{ℓ} = 80\)

3. SIMPLIFY to isolate the variable

• Subtract 80 from both sides:
  \(80 - 80 - 2.5\mathrm{ℓ} = 80 - 80\)
  \(-2.5\mathrm{ℓ} = 0\)

• Divide both sides by -2.5:
  \(\mathrm{ℓ} = 0\)

Answer: A (0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misunderstand what the equation represents or confuse which variable corresponds to which direction. They might think they need to solve for r instead of ℓ, or not realize that \(\mathrm{r} = 16\) should be substituted into the given equation.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make sign errors during algebraic manipulation. For example, when they get to \(80 - 2.5\mathrm{ℓ} = 80\), they might incorrectly think this means \(2.5\mathrm{ℓ} = 0\) (forgetting about the negative sign), or make arithmetic errors when dividing by -2.5.

This may lead them to select Choice C (40) if they accidentally compute something like \(80/2\) instead of recognizing that \(\mathrm{ℓ} = 0\).

The Bottom Line:

This problem tests whether students can translate contextual information into algebraic substitution and then carefully execute the algebraic steps. The key insight is recognizing that when 16 right steps already give exactly 80 units displacement, no left steps are needed.