At a state fair, attendees can win tokens that are worth a different number of points depending on the shape....

1. TRANSLATE the given equation

• Given equation: \(80\mathrm{S} + 90\mathrm{C} = 1,120\)
• What this means:
  • \(\mathrm{S}\) = number of square tokens won
  • \(\mathrm{C}\) = number of circle tokens won
  • 1,120 = total points earned
  • \(80\mathrm{S}\) = total points from all square tokens
  • \(90\mathrm{C}\) = total points from all circle tokens

2. INFER what the coefficients represent

• If \(80\mathrm{S}\) equals the total points from \(\mathrm{S}\) square tokens, then:
  • Each square token must be worth 80 points
  • (Because: points per token × number of tokens = total points)
• If \(90\mathrm{C}\) equals the total points from \(\mathrm{C}\) circle tokens, then:
  • Each circle token must be worth 90 points

3. SIMPLIFY to find the difference

• Circle token value: 90 points
• Square token value: 80 points
• Difference: \(90 - 80 = 10\) points

Answer: D. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret what the equation represents, thinking they need to solve for \(\mathrm{S}\) and \(\mathrm{C}\) rather than recognizing that the coefficients directly tell us the point values.

Instead of seeing that 80 is the points per square token, they might think they need to find specific values of \(\mathrm{S}\) and \(\mathrm{C}\) first. This leads to unnecessary complexity and may cause them to select Choice B (90) or Choice C (80) - mistakenly choosing one of the individual token values instead of their difference.

The Bottom Line:

This problem tests whether students can interpret the meaning of coefficients in a real-world linear equation context, rather than their ability to solve the equation algebraically.