The number a is 110% greater than the number b. The number b is 90% less than 47. What is...

1. TRANSLATE the problem information

• Given information:
  • The number a is 110% greater than b
  • The number b is 90% less than 47
  • Need to find the value of a

2. INFER the solving strategy

• Since a depends on b, we must find b first
• We'll work with the given value (47) to find b, then use b to find a

3. TRANSLATE "90% less than 47"

• "90% less than 47" means: \(\mathrm{b = 47 - (0.90 \times 47)}\)
• This simplifies to: \(\mathrm{b = 47(1 - 0.90) = 47(0.10) = 4.7}\)

4. TRANSLATE "110% greater than b"

• "110% greater than b" means: \(\mathrm{a = b + (1.10 \times b)}\)
• This simplifies to: \(\mathrm{a = b(1 + 1.10) = 2.1b}\)

5. SIMPLIFY by substitution

• Substitute \(\mathrm{b = 4.7}\) into \(\mathrm{a = 2.1b}\)
• \(\mathrm{a = 2.1 \times 4.7 = 9.87}\) (use calculator)

Answer: 9.87

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "110% greater than b" as simply multiplying by 1.10 instead of 2.10

Students think "110% greater" means "multiply by 1.10" when it actually means "multiply by (1 + 1.10) = 2.10". They calculate \(\mathrm{a = 1.10 \times 4.7 = 5.17}\) instead of the correct 9.87.

This leads to confusion when their answer doesn't match any reasonable expectation.

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting "90% less than 47" as multiplying by 0.90 instead of 0.10

Students calculate \(\mathrm{b = 47 \times 0.90 = 42.3}\) instead of \(\mathrm{b = 47 \times 0.10 = 4.7}\). Even if they correctly handle the "110% greater" part afterward, they get \(\mathrm{a = 2.1 \times 42.3 = 88.83}\), which is wildly different from the correct answer.

This causes them to doubt their entire approach and potentially guess.

The Bottom Line:

Percentage language is tricky! "X% greater than" means multiply by \(\mathrm{(1 + X/100)}\), while "X% less than" means multiply by \(\mathrm{(1 - X/100)}\). Getting these translations wrong at the start guarantees an incorrect final answer.