21 is 21% of what number?

1. TRANSLATE the problem information

• Given information:
  • 21 is 21% of some unknown number
  • Need to find that unknown number
• This tells us we have: (percentage) × (unknown whole) = (part)

2. TRANSLATE into mathematical notation

• Let \(\mathrm{x}\) = the unknown number we're looking for
• "21 is 21% of what number?" becomes: \(\mathrm{21 = 21\% \times x}\)
• Convert percentage to decimal: \(\mathrm{21\% = 0.21}\)
• Our equation: \(\mathrm{21 = 0.21x}\)

3. SIMPLIFY by solving for x

• Divide both sides by 0.21: \(\mathrm{x = 21 \div 0.21}\)
• Calculate: \(\mathrm{x = 21 \div 0.21 = 100}\) (use calculator)
• Alternative:
  \(\mathrm{x = 21 \div (21/100)}\)
  \(\mathrm{x = 21 \times (100/21)}\)
  \(\mathrm{x = 100}\)

4. Verify the answer

• Check: \(\mathrm{21\%}\) of \(\mathrm{100 = 0.21 \times 100 = 21}\) ✓

Answer: D. 100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the relationship and set up the equation backwards, thinking "what is 21% of 21?" instead of "21 is 21% of what number?"

They might set up:
\(\mathrm{x = 21\% \times 21}\)
\(\mathrm{x = 0.21 \times 21}\)
\(\mathrm{x = 4.41}\)

This doesn't match any answer choice and leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about percentages: Students might think 21% means 21 (forgetting to convert to decimal or fraction), leading to the equation: \(\mathrm{21 = 21 \times x}\), so \(\mathrm{x = 1}\).

This may lead them to select Choice B (1).

The Bottom Line:

The key challenge is correctly interpreting the English phrase structure. "A is B% of what number?" means A = B% × (unknown), not (unknown) = B% × A. Students must carefully identify what they're solving for versus what they're given.