A random sample of 50 people from a town with a population of 14,878 were asked to name their favorite...

1. TRANSLATE the problem information

• Given information:
  • Sample size: 50 people
  • Chocolate lovers in sample: 7 people
  • Total town population: 14,878 people
  • Need to find: Expected chocolate lovers in entire town

2. INFER the mathematical approach

• Key insight: Since this was a random sample, the proportion of chocolate lovers in the sample should be the same as in the entire population
• This means we can set up a proportion: Sample ratio = Population ratio
• Strategy: Set up the proportion and solve for the unknown

3. TRANSLATE into mathematical notation

• Sample proportion: \(\frac{7}{50}\)
• Population proportion: \(\frac{x}{14,878}\) (where x is what we're looking for)
• Proportion equation: \(\frac{7}{50} = \frac{x}{14,878}\)

4. SIMPLIFY by solving the proportion

• Cross multiply: \(7 \times 14,878 = 50 \times x\)
• Calculate: \(104,146 = 50x\)
• Divide both sides by 50: \(x = 104,146 \div 50\) (use calculator)
• Result: \(x = 2,082.92\)

5. APPLY CONSTRAINTS to select final answer

• Look at answer choices: A. 350, B. 2,100, C. 7,500, D. 10,500
• Our calculated value 2,082.92 is closest to 2,100

Answer: B. 2,100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a proportion problem and instead try to use the sample data directly or perform incorrect calculations.

For example, they might multiply \(7 \times \frac{14,878}{50}\) incorrectly, or try to scale up by doing \(7 \times 14,878 = 104,146\), leading them to select Choice D (10,500) as the closest large number.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the proportion incorrectly, perhaps writing it as \(\frac{50}{7} = \frac{14,878}{x}\) or mixing up which numbers go where.

This leads to incorrect cross multiplication like \(50 \times 14,878 = 7 \times x\), giving \(x = 106,271\), which doesn't match any answer choice and causes confusion and guessing.

The Bottom Line:

This problem requires recognizing that random sampling creates proportional relationships between sample and population. Students who miss this connection often resort to inappropriate arithmetic operations instead of systematic proportion solving.