A public library has a total of 450 books distributed across three sections: Fiction, Non-fiction, and Reference. If a book...

1. TRANSLATE the problem information

• Given information:
  • Total books: 450
  • \(\mathrm{P(Fiction\ book)} = 0.32\)
  • \(\mathrm{P(Non\text{-}fiction\ book)} = 0.38\)
• What we need to find: Number of books in Reference section

2. INFER the mathematical relationship

• Since Fiction, Non-fiction, and Reference are the only three sections, they represent all possible outcomes
• These sections are mutually exclusive (a book can't be in multiple sections)
• Therefore: \(\mathrm{P(Fiction)} + \mathrm{P(Non\text{-}fiction)} + \mathrm{P(Reference)} = 1\)

3. SIMPLIFY to find the missing probability

• Substitute known values: \(0.32 + 0.38 + \mathrm{P(Reference)} = 1\)
• Combine: \(0.70 + \mathrm{P(Reference)} = 1\)
• Solve: \(\mathrm{P(Reference)} = 1 - 0.70 = 0.30\)

4. SIMPLIFY to find the actual number of books

• Number of Reference books = \(\mathrm{P(Reference)} \times \mathrm{Total\ books}\)
• Number of Reference books = \(0.30 \times 450 = 135\)

Answer: B (135)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to use the complement rule to find the missing probability. Instead, they might try to work backwards from the answer choices or assume they don't have enough information to solve the problem. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to use the complement rule but make arithmetic errors. For example, they might calculate \(1 - 0.32 - 0.38 = 0.40\) instead of 0.30, leading to \(0.40 \times 450 = 180\) books. Since 180 isn't among the choices, this causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that probabilities of mutually exclusive, exhaustive events must sum to 1. The key insight is recognizing that you have enough information to find the missing probability using the complement rule, then converting that probability back to an actual count.