A company budgeted $50 for each employee attending a training session. The relationship between the number of employees who attended,...

1. TRANSLATE the problem information

• Given information:
  • Budget: $50 per employee
  • Equation: \(50\mathrm{e} - \mathrm{s} = 20\)
  • \(\mathrm{e}\) = number of employees attending
  • \(\mathrm{s}\) = total amount actually spent
  • Need to interpret what "20" means

2. INFER the approach

• The key insight is that we need to understand what this equation tells us about the relationship between budgeted and actual spending
• Let's rearrange the equation to see this relationship more clearly

3. SIMPLIFY by rearranging the equation

Starting with: \(50\mathrm{e} - \mathrm{s} = 20\)

Add \(\mathrm{s}\) to both sides: \(50\mathrm{e} = \mathrm{s} + 20\)

Subtract 20 from both sides: \(50\mathrm{e} - 20 = \mathrm{s}\)

Or written as: \(\mathrm{s} = 50\mathrm{e} - 20\)

4. INFER what this rearranged equation means

• The term \(50\mathrm{e}\) represents what they budgeted (50 dollars per employee × number of employees)
• The actual spending \(\mathrm{s}\) equals this budget minus 20
• This means they spent $20 less than they budgeted

Answer: D - The company spent $20 less than it budgeted for the training

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often focus on individual numbers rather than understanding the equation's structure and what it represents in context.

They might see "20" in the equation and think it directly represents employees (choice A), total spending (choice B), or total budget (choice C), without considering what the entire equation structure tells us about the spending relationship.

This may lead them to select Choice A (20 employees attended), Choice B (company spent $20), or Choice C (company budgeted $20).

The Bottom Line:

This problem requires students to move beyond just identifying numbers in an equation to understanding what the equation's structure reveals about real-world relationships. The key is recognizing that rearranging the equation shows us how actual spending compares to budgeted spending.