In a sample, 80% of the items are faulty. There are 88 faulty items in the sample. How many total...

1. TRANSLATE the problem information

• Given information:
  • 80% of items are faulty
  • There are 88 faulty items
  • Need to find: total number of items

• What this tells us: The number of faulty items (88) represents 80% of the total sample

2. INFER the approach

• Since we know a percentage and its corresponding value, we can work backwards to find the total
• Strategy: Set up an equation where 80% of the total equals 88

3. TRANSLATE into mathematical form

• Let x = total number of items
• '80% of the total' = \(\mathrm{0.8x}\)
• This equals 88, so: \(\mathrm{0.8x = 88}\)

4. SIMPLIFY to find the answer

• Divide both sides by 0.8: \(\mathrm{x = 88 \div 0.8}\)
• Calculate: \(\mathrm{x = 110}\) (use calculator)

Answer: 110

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the relationship and think 88 represents the total, then try to find 80% of 88.

They calculate: \(\mathrm{88 \times 0.8 = 70.4}\) and get confused because this doesn't match any reasonable answer format. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the correct equation \(\mathrm{(0.8x = 88)}\) but make arithmetic errors with decimal division.

Common mistake: \(\mathrm{88 \div 0.8 = 11}\) (incorrectly thinking they're dividing by 8 instead of 0.8). This leads them to guess or attempt other incorrect calculations.

The Bottom Line:

This problem requires clear understanding of the relationship between a part and the whole in percentage problems. Students must recognize that they're given the part (88) and the percentage it represents (80%), then work backwards to find the whole.