A city's total expense budget for one year was x million dollars. The city budgeted y million dollars for departmental...

1. TRANSLATE the problem information

• Given information:
  • Total expense budget: \(\mathrm{x}\) million dollars
  • Departmental expenses: \(\mathrm{y}\) million dollars
  • All other expenses: \(\mathrm{201}\) million dollars

2. INFER the mathematical relationship

• The total budget must equal the sum of all its parts
• This means: Total = Departmental expenses + Other expenses
• So: \(\mathrm{x = y + 201}\)

3. SIMPLIFY to match the answer format

• Starting with: \(\mathrm{x = y + 201}\)
• Subtract \(\mathrm{y}\) from both sides: \(\mathrm{x - y = 201}\)
• This matches choice B

Answer: B. \(\mathrm{x - y = 201}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret which quantity represents the total and which represent the parts.

They might think that since \(\mathrm{x}\) is the "total budget" and \(\mathrm{y}\) is "departmental expenses," they need to add them together, leading to the equation \(\mathrm{x + y = 201}\). They don't recognize that \(\mathrm{y}\) is actually a component of \(\mathrm{x}\), not separate from it.

This may lead them to select Choice A (\(\mathrm{x + y = 201}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly set up \(\mathrm{x = y + 201}\) but then get confused about which variable should be isolated.

They might think they need \(\mathrm{y}\) by itself and rearrange to get \(\mathrm{y = x - 201}\), then mistakenly flip this to match an answer choice, potentially selecting \(\mathrm{y - x = 201}\).

This may lead them to select Choice D (\(\mathrm{y - x = 201}\)).

The Bottom Line:

Success on this problem requires clearly identifying the part-whole relationship: the total budget \(\mathrm{x}\) contains both the departmental expenses \(\mathrm{y}\) and the other expenses \(\mathrm{201}\), so \(\mathrm{x = y + 201}\). Many students struggle because they don't carefully track which quantities are parts of the total versus separate from it.