Question:40% of a number equals 80. What is 60% of the same number?

1. TRANSLATE the problem information

• Given information:
  • 40% of a number equals 80
  • We need to find 60% of the same number

• In mathematical notation:
  • Let x = the unknown number
  • \(0.4\mathrm{x} = 80\)
  • Find: \(0.6\mathrm{x} = ?\)

2. INFER the solution strategy

• We can't directly find 60% of the number because we don't know what the number is yet
• Strategy: First solve for the unknown number, then calculate 60% of it
• This is a two-step approach

3. SIMPLIFY to find the unknown number

• Starting with: \(0.4\mathrm{x} = 80\)
• Divide both sides by 0.4: \(\mathrm{x} = 80 \div 0.4\)
• \(\mathrm{x} = 200\)

4. SIMPLIFY to find 60% of the number

• Now we know the number is 200
• Calculate: \(60\% \text{ of } 200 = 0.6 \times 200 = 120\)

Answer: C (120)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle with converting the word problem into the correct mathematical setup, particularly with decimal representations of percentages.

Some students might write "\(40\mathrm{x} = 80\)" instead of "\(0.4\mathrm{x} = 80\)", treating the percentage as a whole number rather than converting it to a decimal. This leads to \(\mathrm{x} = 2\) instead of \(\mathrm{x} = 200\), and then \(60\% \text{ of } 2 = 1.2\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when dividing by decimals.

When calculating \(80 \div 0.4\), students might incorrectly get 20 (thinking \(80 \div 4 = 20\)) instead of 200. This would lead them to calculate \(60\% \text{ of } 20 = 12\), which again doesn't match the choices, causing them to second-guess their approach and potentially select Choice A (48) through confused reasoning.

The Bottom Line:

This problem requires careful attention to decimal conversions and methodical two-step thinking. Students who rush through the percentage-to-decimal conversion or the decimal arithmetic often find themselves with answers that don't match the choices, leading to frustration and guessing rather than systematic problem-solving.