Scientists collected fallen acorns that each housed a colony of the ant species P. ohioensis and analyzed each colony's structure....

1. INFER what the problem is asking

• Given information:
  • Equation: \(\mathrm{y = 0.67x + 2.6}\) (where \(\mathrm{x}\) = worker ants, \(\mathrm{y}\) = predicted larvae)
  • Domain constraint: \(\mathrm{20 \leq x \leq 110}\)
  • Specific colony has 58 worker ants
• What this tells us: We need to substitute \(\mathrm{x = 58}\) into the equation to predict the number of larvae

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 58}\) into \(\mathrm{y = 0.67x + 2.6}\):
  \(\mathrm{y = 0.67(58) + 2.6}\)
• Calculate step by step:
  • First: \(\mathrm{0.67 \times 58 = 38.86}\)
  • Then: \(\mathrm{38.86 + 2.6 = 41.46}\)

3. APPLY CONSTRAINTS to select the final answer

• Our calculated value: \(\mathrm{y = 41.46}\)
• Answer choices: A. 41, B. 61, C. 83, D. 190
• Since the problem asks for 'closest to,' we select 41

Answer: A. 41

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the calculation, particularly when multiplying \(\mathrm{0.67 \times 58}\). They might calculate this as 3.886 instead of 38.86 (decimal placement error), leading to \(\mathrm{y = 3.886 + 2.6 = 6.486}\), which doesn't match any answer choice well.

This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students might misunderstand what the equation represents or which variable corresponds to which quantity. They could mistakenly think they need to solve for x when \(\mathrm{y = 58}\), rather than finding y when \(\mathrm{x = 58}\).

This backwards approach would involve solving \(\mathrm{58 = 0.67x + 2.6}\), leading to a completely different calculation and potentially selecting Choice C (83) as an incorrect 'close' value.

The Bottom Line:

This problem requires careful attention to variable meaning and precise arithmetic execution. Students must recognize that linear equations work as prediction tools—plug in the input value to get the predicted output.