The number a is 60% greater than the positive number b. The number c is 45% less than a. The...

1. TRANSLATE the percentage relationships into mathematical expressions

• Given information:
  • a is 60% greater than b
  • c is 45% less than a
  • Need to find: c is how many times b?

• TRANSLATE each relationship:
  • '60% greater than b' means \(\mathrm{a = b + 0.60b = 1.60b}\)
  • '45% less than a' means \(\mathrm{c = a - 0.45a = 0.55a}\)

2. INFER the solution strategy

• We have b → a → c, but need to find the direct relationship from c to b
• Strategy: Use substitution to eliminate the intermediate variable a
• This will give us c directly in terms of b

3. SIMPLIFY through substitution

• Start with: \(\mathrm{c = 0.55a}\)
• Substitute \(\mathrm{a = 1.60b}\): \(\mathrm{c = 0.55(1.60b)}\)
• Calculate: \(\mathrm{c = 0.88b}\) (use calculator for \(\mathrm{0.55 × 1.60}\))

Answer: 0.88 (or 22/25 or .88)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting '60% greater than b' as simply 60% of b

Students think '60% greater' means \(\mathrm{a = 0.60b}\) instead of \(\mathrm{a = 1.60b}\). They forget that 'greater than' means you add the percentage increase to the original amount (100% + 60% = 160% = 1.60).

This leads to \(\mathrm{a = 0.60b}\), then \(\mathrm{c = 0.55(0.60b) = 0.33b}\), which doesn't match any typical answer format and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making calculation errors in the final multiplication

Students correctly set up \(\mathrm{c = 0.55(1.60b)}\) but then calculate \(\mathrm{0.55 × 1.60}\) incorrectly, perhaps getting 0.80 or 0.85 instead of 0.88.

This leads to selecting an incorrect decimal value if multiple choice options are close.

The Bottom Line:

The key challenge is correctly interpreting percentage language - 'X% greater than' and 'Y% less than' have specific mathematical meanings that must be precisely translated before any calculation can succeed.