5G + 45R = 380. At a school fair, students can win colored tokens that are worth a different number...

1. TRANSLATE the problem setup

• Given information:
  • Equation: \(5\mathrm{G} + 45\mathrm{R} = 380\)
  • \(\mathrm{G}\) = number of green tokens won
  • \(\mathrm{R}\) = number of red tokens won
  • Total value = 380 points
• Question asks: How many more points is a red token worth than a green token?

2. INFER what the equation tells us about token values

• In the equation \(5\mathrm{G} + 45\mathrm{R} = 380\), each term represents the total points from that color
• Since \(5\mathrm{G}\) represents "total points from green tokens" and \(\mathrm{G}\) is the number of green tokens, each green token must be worth 5 points
• Similarly, since \(45\mathrm{R}\) represents "total points from red tokens" and \(\mathrm{R}\) is the number of red tokens, each red token must be worth 45 points

3. SIMPLIFY to find the difference

• Red token value: 45 points
• Green token value: 5 points
• Difference: \(45 - 5 = 40\) points

Answer: 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see the equation \(5\mathrm{G} + 45\mathrm{R} = 380\) and immediately try to solve for \(\mathrm{G}\) and \(\mathrm{R}\) by using algebraic techniques like substitution or elimination. They don't recognize that the problem isn't asking for the number of tokens, but rather the point values per token.

This leads to confusion because they have one equation with two unknowns, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the coefficients represent. They might think 5 and 45 are just random numbers in the equation rather than the point values per token.

This conceptual confusion prevents them from extracting the key information needed to answer the question, leading them to attempt unnecessary algebraic manipulations.

The Bottom Line:

This problem tests whether students can interpret the real-world meaning of coefficients in a linear equation context, rather than just manipulate the equation algebraically. The key insight is recognizing that when tokens have different point values, the coefficient tells you the value per token.