For a party, 50 dinner rolls are needed. Dinner rolls are sold in packages of 12. What is the minimum...

1. TRANSLATE the problem information

• Given information:
  • Need: 50 dinner rolls for party
  • Package size: 12 rolls per package
  • Find: Minimum number of packages to buy

• What this tells us: We need enough packages so that total rolls \(\geq 50\)

2. INFER the mathematical approach

• Since we need 'at least' 50 rolls, this creates an inequality situation
• Let \(\mathrm{p}\) = number of packages to buy
• Total rolls from \(\mathrm{p}\) packages = \(12\mathrm{p}\)
• We need: \(12\mathrm{p} \geq 50\)

3. SIMPLIFY the inequality

• Divide both sides by 12: \(\mathrm{p} \geq 50/12\)
• Calculate: \(\mathrm{p} \geq 4.166...\) (use calculator for the division)

4. APPLY CONSTRAINTS to find the final answer

• Since we can only buy whole packages (not partial packages)
• We must round UP to the next whole number
• Therefore: \(\mathrm{p} = 5\) packages

• Verification: \(5 \times 12 = 60\) rolls (more than enough)
• Check: \(4 \times 12 = 48\) rolls (not enough!)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students correctly solve the inequality to get \(\mathrm{p} \geq 4.17\), but then round DOWN to 4 packages instead of rounding UP to 5.

They think '4.17 is closer to 4 than to 5, so the answer is 4.' However, 4 packages only gives 48 rolls, which falls short of the required 50. This leads them to incorrectly conclude the answer is 4.

Second Most Common Error:

Poor TRANSLATE execution: Students misread the problem and set up an equation (\(12\mathrm{p} = 50\)) instead of an inequality (\(12\mathrm{p} \geq 50\)).

This leads them to calculate \(\mathrm{p} = 50/12 = 4.17\), then get confused about what to do with the decimal. Some might round to 4, others might try to give 4.17 as an answer, neither of which addresses the real-world constraint that packages must be whole numbers.

The Bottom Line:

This problem tests whether students understand that 'minimum needed' situations require inequalities and that real-world constraints (whole packages only) sometimes force us to exceed the mathematical minimum to meet practical requirements.