The number a is 190% greater than the number b. The number b is 80% less than 24. What is...

1. TRANSLATE the problem information

• Given information:
  • The number b is 80% less than 24
  • The number a is 190% greater than the number b
  • Need to find the value of a

• What this tells us: We have a chain relationship where a depends on b, and b depends on 24

2. INFER the approach

• Since a depends on b, we must find b first
• We'll work backwards: 24 → b → a
• Key insight: "X% less than" and "X% greater than" have specific mathematical meanings

3. TRANSLATE and solve for b

• "b is 80% less than 24" means:
  \(\mathrm{b = 24 - (80\% \text{ of } 24)}\)
  \(\mathrm{b = 24 - 0.8(24)}\)
  \(\mathrm{b = 24 - 19.2 = 4.8}\)

4. TRANSLATE and solve for a

• "a is 190% greater than b" means:
  \(\mathrm{a = b + (190\% \text{ of } b)}\)
  \(\mathrm{a = 4.8 + 1.9(4.8)}\)
  \(\mathrm{a = 4.8 + 9.12}\) (use calculator)
  \(\mathrm{a = 13.92}\)

Answer: B. 13.92

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing "190% greater than" with "190% of"

Students might think "a is 190% greater than b" means \(\mathrm{a = 1.9b}\), when it actually means \(\mathrm{a = b + 1.9b = 2.9b}\). Similarly, they might interpret "80% less than 24" as \(\mathrm{0.8(24) = 19.2}\) instead of \(\mathrm{24 - 0.8(24) = 4.8}\).

This may lead them to select Choice A (9.12) or Choice D (36.48)

Second Most Common Error:

Poor INFER reasoning: Not recognizing the dependency relationship

Some students try to find a directly without first calculating b, or they work with the relationships in the wrong order, leading to confusion about which percentage applies to which number.

This causes them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests precise language interpretation more than complex calculations. The key challenge is correctly TRANSLATING percentage language into mathematical operations, then INFERRING the proper sequence of steps.