210 is p% greater than 30. What is the value of p?

1. TRANSLATE the problem information

• Given information:
  • 210 is p% greater than 30
  • Need to find the value of p

• What this tells us: We need to set up an equation using the percent increase formula

2. TRANSLATE the percent increase relationship

• When something is 'p% greater than' a value, it means:
  \(\mathrm{New\ value = Original\ value \times (1 + p/100)}\)

• So '210 is p% greater than 30' becomes:
  \(\mathrm{210 = 30 \times (1 + p/100)}\)

3. SIMPLIFY by solving the equation step by step

• Divide both sides by 30:
  \(\mathrm{210/30 = 1 + p/100}\)
  \(\mathrm{7 = 1 + p/100}\)

• Subtract 1 from both sides:
  \(\mathrm{6 = p/100}\)

• Multiply both sides by 100:
  \(\mathrm{p = 600}\)

Answer: 600

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'p% greater than' as simple addition rather than multiplication.

They might think: '210 is p% greater than 30' means \(\mathrm{210 = 30 + p}\), leading them to calculate \(\mathrm{p = 210 - 30 = 180}\). This fundamental misunderstanding of percent increase language prevents them from setting up the correct equation entirely.

Second Most Common Error:

Conceptual confusion about percent increase: Students might correctly recognize they need multiplication but set up the wrong relationship.

They could write \(\mathrm{210 = 30 \times (p/100)}\) instead of \(\mathrm{210 = 30 \times (1 + p/100)}\), forgetting that percent increase means the new value includes the original amount PLUS the increase. This leads to \(\mathrm{p = 700}\), which seems reasonable but is incorrect.

The Bottom Line:

This problem hinges entirely on correctly translating the phrase 'p% greater than' into mathematical notation. Students who master this translation step find the algebra straightforward, while those who misinterpret it cannot recover regardless of their algebraic skills.