The number w is 110% greater than the number z. The number z is 55% less than 50. What is...

1. TRANSLATE the second condition first

• Given: "z is 55% less than 50"
• This means: \(\mathrm{z = 50 - (55\% \text{ of } 50)}\)
• Calculate: \(\mathrm{z = 50 - 0.55(50)}\)
  \(\mathrm{z = 50 - 27.5}\)
  \(\mathrm{z = 22.5}\)

2. TRANSLATE the first condition

• Given: "w is 110% greater than z"
• This means: \(\mathrm{w = z + (110\% \text{ of } z)}\)
  \(\mathrm{w = z + 1.10z}\)
  \(\mathrm{w = 2.10z}\)
• Key insight: "110% greater than" means the original amount PLUS 110% more

3. SIMPLIFY by substitution

• Substitute \(\mathrm{z = 22.5}\) into \(\mathrm{w = 2.10z}\)
• \(\mathrm{w = 2.10(22.5)}\)
  \(\mathrm{w = 47.25}\)

Answer: \(\mathrm{47.25}\) (also acceptable as \(\mathrm{189/4}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "110% greater than z" as meaning "110% of z"

They think \(\mathrm{w = 1.10z}\) instead of \(\mathrm{w = 2.10z}\). This fundamental misunderstanding of percentage language leads them to calculate \(\mathrm{w = 1.10(22.5) = 24.75}\), which doesn't match any typical answer choice and causes confusion and guessing.

Second Most Common Error:

Weak TRANSLATE skill: Students misinterpret "55% less than 50" as "55% of 50"

They calculate \(\mathrm{z = 0.55(50) = 27.5}\) instead of \(\mathrm{z = 50 - 27.5 = 22.5}\). Even if they correctly handle the "110% greater than" part, they end up with \(\mathrm{w = 2.10(27.5) = 57.75}\), leading to an incorrect final answer.

The Bottom Line:

The key challenge is correctly translating percentage language - "X% greater than" means 100% + X% of the original, while "X% less than" means 100% - X% of the original. Students who rush through the translation step often confuse these constructions with simpler "X% of" calculations.