A farm consists of an 8-acre wheat field and a 15-acre corn field. The total harvest from both fields is...

1. TRANSLATE the equation components

• Given equation: \(8\mathrm{w} + 15\mathrm{c} = 2{,}300\)
• What each number represents:
  • \(8\) = acres in wheat field
  • \(15\) = acres in corn field
  • \(2{,}300\) = total bushels harvested from both fields

2. INFER the variable meanings from equation structure

• The equation structure is: \(\text{(wheat acres)} \times \mathrm{w} + \text{(corn acres)} \times \mathrm{c} = \text{total bushels}\)
• For this multiplication to make sense dimensionally:
  • \(\text{acres} \times \mathrm{w}\) must equal bushels
  • Therefore \(\mathrm{w}\) must be "bushels per acre"
• Since \(8\) represents wheat field acres, \(\mathrm{w}\) must be bushels per acre for wheat

3. Verify the interpretation makes sense

• If \(\mathrm{w}\) = bushels per acre in wheat field, then:
  • \(8\mathrm{w}\) = total bushels from wheat field
  • \(15\mathrm{c}\) = total bushels from corn field
  • \(8\mathrm{w} + 15\mathrm{c}\) = total bushels from both fields ✓

Answer: A. The average number of bushels per acre in the wheat field

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus on what they want to find rather than analyzing the equation structure. They see "wheat field" and "\(\mathrm{w}\)" and immediately think \(\mathrm{w}\) must represent the total bushels from wheat.

This leads them to reason: "\(\mathrm{w}\) is for wheat, so \(\mathrm{w}\) must be the total wheat bushels," without considering that \(8\mathrm{w}\) (not just \(\mathrm{w}\)) would represent the total.

This may lead them to select Choice C (The total number of bushels harvested from the wheat field)

Second Most Common Error:

Poor dimensional analysis: Students don't check whether their interpretation makes the equation mathematically consistent. They might pick an answer that "sounds right" without verifying the units work out.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

Success requires understanding that in rate × quantity equations, you must identify which terms represent quantities (like acres) and which represent rates (like bushels per acre). The variable meanings emerge from making the equation dimensionally consistent.