The equation 84 = 6m + 4n represents the total cost in dollars of buying m premium items at $6...

1. TRANSLATE the problem setup

• Given information:
  • Total cost equation: \(84 = 6\mathrm{m} + 4\mathrm{n}\)
  • Value of one variable: \(\mathrm{m} = 8\)
  • Need to find: \(\mathrm{n}\)

• What this tells us: We have a linear equation with two variables, but since we know one variable's value, we can solve for the other.

2. INFER the solution approach

• Since we know \(\mathrm{m} = 8\), substitute this value directly into the equation
• This will give us a simple equation with only \(\mathrm{n}\) as the unknown

3. SIMPLIFY through substitution and algebra

• Substitute \(\mathrm{m} = 8\):
  \(84 = 6(8) + 4\mathrm{n}\)
• Calculate \(6 \times 8 = 48\):
  \(84 = 48 + 4\mathrm{n}\)
• Subtract 48 from both sides:
  \(84 - 48 = 4\mathrm{n}\)
  \(36 = 4\mathrm{n}\)
• Divide both sides by 4:
  \(\mathrm{n} = 9\)

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Arithmetic errors during the calculation steps

Students might calculate \(6 \times 8 = 42\) instead of 48, leading to:
\(84 = 42 + 4\mathrm{n} \rightarrow 42 = 4\mathrm{n} \rightarrow \mathrm{n} = 10.5\)

Or they might incorrectly compute \(84 - 48 = 34\), leading to:
\(34 = 4\mathrm{n} \rightarrow \mathrm{n} = 8.5\)

These arithmetic mistakes produce non-integer answers that don't make sense in the context of buying whole items.

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding which variable to substitute

Some students might try to solve for \(\mathrm{m}\) instead of \(\mathrm{n}\), despite \(\mathrm{m}\) already being given, or become confused about what the equation represents. This leads to confusion and guessing rather than systematic solution.

The Bottom Line:

This problem tests basic algebraic substitution skills. The key challenge is maintaining accuracy through multiple arithmetic steps while keeping track of the algebraic process.