A supplier sells bundles of cables. Each bundle contains at least 18 cables but no more than 22. The supplier...

1. TRANSLATE the problem information

• Given information:
  • Each bundle contains at least 18 cables but no more than 22 cables
  • Total cables sold = 320
  • Need to find possible number of bundles

• What this tells us: We need to find values of \(n\) (number of bundles) where the total cables (320) could realistically come from \(n\) bundles with 18-22 cables each.

2. INFER the approach

• Key insight: The total cables must fall between the minimum possible (if all bundles had 18 cables) and maximum possible (if all bundles had 22 cables)
• Strategy: Set up inequalities using both the minimum and maximum constraints

3. TRANSLATE constraints into inequalities

• If \(n\) = number of bundles:
  • Minimum total cables: \(18n\)
  • Maximum total cables: \(22n\)
  • Actual total cables: \(320\)

• This gives us: \(18n \leq 320 \leq 22n\)

4. SIMPLIFY each inequality

• From \(18n \leq 320\):
  \(n \leq \frac{320}{18} = 17.78...\) (use calculator)

• From \(320 \leq 22n\):
  \(n \geq \frac{320}{22} = 14.54...\) (use calculator)

• Combined: \(14.54... \leq n \leq 17.78...\)

5. APPLY CONSTRAINTS to find valid solutions

• Since number of bundles must be a whole number: \(n\) can be 15, 16, or 17
• Checking answer choices: Only 16 appears in our list

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often set up only one inequality, typically \(320 \div 18\) or \(320 \div 22\), missing that both minimum AND maximum constraints create boundaries.

For example, they might calculate \(320 \div 20 = 16\) (assuming exactly 20 cables per bundle) and immediately select Choice C without verifying this falls within the valid range. While this happens to be correct, this approach could easily lead to wrong answers in similar problems.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students correctly find the decimal range \(14.54... \leq n \leq 17.78...\) but fail to recognize that \(n\) must be a whole number, or they don't systematically check which answer choices fall within their calculated range.

This leads to confusion and guessing among the provided choices.

The Bottom Line:

This problem requires students to work with compound inequalities and understand that real-world constraints (whole number of bundles, range of cables per bundle) create boundaries that must all be satisfied simultaneously.