The result of increasing the quantity x by 1,800% is 684. What is the value of x?

1. TRANSLATE the problem information

• Given information:
  • Starting with some quantity \(\mathrm{x}\)
  • Increasing \(\mathrm{x}\) by 1,800%
  • The result of this increase is 684
• What this tells us: We need to find the original value \(\mathrm{x}\)

2. TRANSLATE the percentage increase language

• "Increasing \(\mathrm{x}\) by 1,800%" means:
  • Keep the original: \(\mathrm{x}\)
  • Add the increase: 1,800% of \(\mathrm{x}\) = \(\mathrm{(1{,}800/100) \times x}\) = \(\mathrm{18x}\)
  • Total result: \(\mathrm{x + 18x = 19x}\)
• So our equation is: \(\mathrm{19x = 684}\)

3. SIMPLIFY to solve for \(\mathrm{x}\)

• Divide both sides by 19:
  \(\mathrm{x = 684 \div 19 = 36}\)

Answer: D. 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "increase by 1,800%" and think the final result IS 1,800% of \(\mathrm{x}\), rather than \(\mathrm{x}\) PLUS 1,800% of \(\mathrm{x}\).

This leads them to set up: \(\mathrm{18x = 684}\), which gives \(\mathrm{x = 684 \div 18 = 38}\).

This may lead them to select Choice C (38).

Second Most Common Error:

Conceptual confusion about percentage calculations: Students might think "increase by 1,800%" means multiply by 1,800 (forgetting the percentage conversion), leading to impossible equations or getting confused about the setup.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is correctly translating percentage increase language. "Increase by P%" means "original + P% of original," not just "P% of original."