A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget...

1. TRANSLATE the problem information

• Given information:
  • 81 chairs needed (same model)
  • Total budget = $14,000 (this INCLUDES 7% sales tax)
  • Need to find maximum price per chair BEFORE sales tax
• Let \(\mathrm{p}\) = price per chair before sales tax

2. INFER the mathematical relationship

• Since the budget includes sales tax, we need to account for the 7% tax
• Total cost with tax = (cost before tax) × (1 + tax rate)
• Total cost with tax = \(\mathrm{81p \times 1.07}\)
• This total cannot exceed our budget of $14,000

3. TRANSLATE this constraint into an inequality

• Budget constraint: \(\mathrm{81p(1.07) \leq 14,000}\)
• The "≤" symbol means "at most" - we cannot spend more than $14,000

4. SIMPLIFY to solve for the maximum price per chair

• Divide both sides by 81(1.07):

\(\mathrm{p \leq 14,000 \div (81 \times 1.07)}\)

\(\mathrm{p \leq 14,000 \div 86.67}\) (use calculator)

\(\mathrm{p \leq 161.53}\)

5. APPLY CONSTRAINTS to select the final answer

• The maximum price per chair before sales tax is $161.53

Answer: B. $161.53

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand what "including sales tax" means and treat the $14,000 as the pre-tax amount.

They calculate: \(\mathrm{14,000 \div 81 = \$172.84}\) and think this is the answer, completely ignoring the sales tax component.

This leads them to select Choice C ($172.84) - which is actually the maximum total price per chair including tax, not the pre-tax price.

Second Most Common Error:

Poor INFER skill: Students recognize that sales tax is involved but incorrectly think they need to add 7% to the budget rather than recognizing the budget already includes the tax.

They might calculate: \(\mathrm{(14,000 \times 1.07) \div 81 = \$184.94}\), thinking they need to account for tax on top of the budget.

This may lead them to select Choice D ($184.94).

The Bottom Line:

The key challenge is correctly interpreting "includes sales tax" - students must recognize this means the $14,000 is the final amount (post-tax), so they need to work backwards to find the pre-tax price that would result in this total after adding 7% tax.