If x/8 = 5, what is the value of 8/x?

1. INFER the solution strategy

• Given: \(\mathrm{x/8 = 5}\)
• Want to find: \(\mathrm{8/x}\)
• Key insight: We need to find the value of x first, then substitute it into the expression \(\mathrm{8/x}\)

2. SIMPLIFY the original equation to solve for x

• Multiply both sides by 8 to isolate x:
  \(\mathrm{x/8 × 8 = 5 × 8}\)
  \(\mathrm{x = 40}\)

3. SIMPLIFY the target expression by substitution

• Substitute \(\mathrm{x = 40}\) into \(\mathrm{8/x}\):
  \(\mathrm{8/x = 8/40}\)
• Reduce the fraction:
  \(\mathrm{8/40 = 1/5}\)
• Convert to decimal if needed:
  \(\mathrm{1/5 = 0.2}\)

Answer: 1/5 or 0.2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to manipulate the equation \(\mathrm{x/8 = 5}\) directly to get \(\mathrm{8/x}\) without first solving for x.

They might think "if \(\mathrm{x/8 = 5}\), then \(\mathrm{8/x = 1/5}\)" using reciprocal reasoning, but they get confused about whether this is valid or how to justify it systematically. While this intuition is actually correct, students often doubt themselves and make errors trying to work backwards from their guess.

This leads to confusion and guessing between answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly solve for \(\mathrm{x = 40}\) but make arithmetic errors when calculating \(\mathrm{8/40}\).

They might calculate \(\mathrm{8/40}\) as \(\mathrm{2/10}\) instead of \(\mathrm{1/5}\), or struggle with the decimal conversion, leading to incorrect numerical answers like 0.8 instead of 0.2.

This causes them to select wrong numerical values or get stuck with inconsistent decimal/fraction forms.

The Bottom Line:

This problem tests whether students can systematically work through a two-step process: solve for the variable, then evaluate the desired expression. The key challenge is recognizing that solving for x first is the most reliable path forward.