In a recent local election, 360 registered voters cast a ballot for the incumbent mayor. These 360 votes represent 80%...

1. TRANSLATE the problem information

• Given information:
  • 360 votes were cast for the incumbent mayor
  • These 360 votes represent 80% of the total votes cast for mayor
  • Need to find: total number of votes cast for mayor
• What this tells us: We have a part (360) and know what percentage it represents (80%), so we can find the whole.

2. INFER the solution approach

• Key insight: When you know a part and its percentage, you can find the whole using: \(\mathrm{part} \div \mathrm{percentage} = \mathrm{whole}\)
• We need to set up an equation where 360 equals 80% of some total number

3. TRANSLATE into mathematical notation

• Let \(\mathrm{T}\) = total number of votes cast for mayor
• Set up the equation: \(360 = 80\% \text{ of } \mathrm{T}\)
• Convert percentage: \(360 = 0.80 \times \mathrm{T}\)

4. SIMPLIFY to solve for T

• Divide both sides by 0.80: \(\mathrm{T} = 360 \div 0.80\)
• Alternative approach using fractions: \(80\% = \frac{4}{5}\), so \(\mathrm{T} = 360 \div \frac{4}{5} = 360 \times \frac{5}{4}\)
• Calculate:
  \(\mathrm{T} = 360 \times 5 \div 4\)
  \(\mathrm{T} = 1800 \div 4\)
  \(\mathrm{T} = 450\)

Answer: C) 450

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the relationship and think they need to find 80% of 360, rather than finding the number that 360 represents 80% of.

They calculate: \(360 \times 0.80 = 288\)

This leads them to select Choice A (288)

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to divide but use the wrong operation, calculating \(360 \div 80\) instead of \(360 \div 0.80\).

They get: \(360 \div 80 = 4.5\), which doesn't match any answer choice, leading to confusion and guessing.

The Bottom Line:

The key challenge is recognizing that when you're told "X represents Y% of the total," you need to find what total makes this statement true, not what Y% of X equals. This requires careful translation of the English into the correct mathematical relationship.