At a particular track meet, the ratio of coaches to athletes is 1:26. If there are x coaches at the...

1. TRANSLATE the problem information

• Given information:
  • Ratio of coaches to athletes is \(\mathrm{1:26}\)
  • There are x coaches at the track meet
  • Need to find: expression for number of athletes

• What this tells us: For every 1 coach, there are 26 athletes

2. INFER the approach

• Key insight: Ratios must stay constant - if one part changes, the other must change proportionally
• Strategy: Use the scaling property of ratios

3. INFER the scaling relationship

• Original ratio: \(\mathrm{1\ coach : 26\ athletes}\)
• Current situation: \(\mathrm{x\ coaches : ?\ athletes}\)
• Since coaches increased from 1 to x (multiplied by x), athletes must increase from 26 to \(\mathrm{26×x}\) to maintain the same ratio

4. Apply the scaling

• Number of athletes = \(\mathrm{26 × x = 26x}\)

Answer: B (\(\mathrm{26x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret ratios as addition relationships rather than multiplication relationships.

They think: "If the ratio is 1 to 26, and there are x coaches, then there are x plus 26 more athletes." This fundamental misunderstanding of what ratios represent leads them to select Choice C (\(\mathrm{x + 26}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify it's a ratio problem but get confused about which direction the ratio goes.

They think: "If coaches to athletes is \(\mathrm{1:26}\), then maybe athletes to coaches would be \(\mathrm{x:26}\), so athletes = \(\mathrm{x/26}\)." This backward thinking about the ratio relationship may lead them to select Choice A (\(\mathrm{x/26}\)).

The Bottom Line:

Ratio problems require understanding that ratios represent multiplicative relationships, not additive ones, and that scaling one part of a ratio requires scaling the other part by the same factor.