In a video game, a player collects silver coins and gold coins to earn points. The total points from S...

1. TRANSLATE the question

• We need to find: "How many more points is a gold coin worth than a silver coin?"
• This means: difference between gold coin value and silver coin value

2. INFER what the equation tells us

• Given equation: \(2\mathrm{S} + 20\mathrm{G} = 500\)
• The coefficient of S is 2 → each silver coin is worth 2 points
• The coefficient of G is 20 → each gold coin is worth 20 points
• Key insight: We don't need to solve for S and G. The coefficients directly give us the point values!

3. Calculate the difference

• Gold coin value: 20 points
• Silver coin value: 2 points
• Difference: \(20 - 2 = 18\) points

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that coefficients represent point values per coin. Instead, they think they need to solve the equation for S and G.

They might try to find specific values like "if \(\mathrm{S} = 50\) and \(\mathrm{G} = 20\), then..." but get confused because there are infinitely many solutions. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "how many more points" means and try to find the total points (500) or calculate some other relationship.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students understand that coefficients in linear equations represent rates or values per unit. The key insight is recognizing that you don't need to solve the equation - the coefficients themselves answer the question directly.