A proposal for a new library was included on an election ballot. A radio show stated that 3 times as...

1. TRANSLATE the problem information

• Given information:
  • Radio show: 3 times as many voted in favor as against
  • Social media: 15,000 more voted in favor than against
  • Find: number who voted against

• Let \(\mathrm{x}\) = number of people who voted against the proposal
• Then \(\mathrm{3x}\) = number of people who voted in favor

2. INFER the key relationship

• We have two pieces of information about the same election
• The difference between "in favor" and "against" can be calculated as: \(\mathrm{3x - x = 2x}\)
• This difference equals 15,000 according to social media

3. SIMPLIFY to solve the equation

• Set up: \(\mathrm{2x = 15,000}\)
• Divide both sides by 2: \(\mathrm{x = 7,500}\)
• Therefore, 7,500 people voted against the proposal

Answer: A. 7,500

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "3 times as many in favor as against" and incorrectly set up relationships like \(\mathrm{x + 3x}\) = total, or confuse which quantity should be \(\mathrm{3x}\).

They might think that if 15,000 more voted in favor, then 15,000 voted against, completely missing the multiplicative relationship. This may lead them to select Choice B (15,000).

Second Most Common Error:

Poor INFER reasoning: Students correctly translate both statements but fail to connect them logically. They don't recognize that both statements describe the same election data, so the "15,000 more" must equal the difference they can calculate from the ratio (\(\mathrm{2x}\)).

This leads to confusion about how to use both pieces of information together, causing them to get stuck and guess.

The Bottom Line:

This problem requires students to juggle two different ways of describing the same relationship - a ratio (3:1) and a numerical difference (15,000). The breakthrough comes from recognizing these describe the same scenario and can be connected through algebra.