In a survey of a group of people, 240 preferred brand A. This number represents 80% of the total number...

1. TRANSLATE the problem information

• Given information:
  • 240 people preferred brand A
  • This 240 represents \(80\%\) of the total people surveyed
  • We need to find the total number surveyed

2. INFER the mathematical approach

• When we know a part and what percentage it represents, we find the whole by dividing the part by the percentage
• We need: \(\mathrm{Total = Part \div Percentage}\)

3. TRANSLATE into equation form

• \(80\%\) of total = 240
• \(0.8 \times \mathrm{total} = 240\)

4. SIMPLIFY to solve for total

• \(\mathrm{total} = 240 \div 0.8\)
• \(\mathrm{total} = 240 \div (4/5)\)
• \(\mathrm{total} = 240 \times (5/4)\)
• \(\mathrm{total} = (240 \times 5) \div 4 = 1200 \div 4 = 300\)

Answer: C) 300

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think "80% of something" means they should multiply by 0.8, so they calculate \(240 \times 0.8 = 192\).

They're confusing "finding 80% of a number" with "a number that IS 80% of something else." This backwards thinking about the part-whole relationship is very common in percentage problems.

This leads them to select Choice A (192).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread and think 240 is already the total number surveyed, missing that 240 represents only the portion who preferred brand A.

This causes them to select Choice B (240).

The Bottom Line:

The key insight is recognizing the direction of the percentage relationship - you're given the part (240) and need to work backwards to find the whole, not forward to find a part.