The number a is 70% less than the positive number b. The number c is 60% greater than a. The...

1. TRANSLATE the percentage relationships

• Given information:
  • a is 70% less than positive number b
  • c is 60% greater than a
  • Need to find: c is how many times b?

• What this tells us in mathematical terms:
  • '70% less than b' means \(\mathrm{a = b - 0.70b = 0.30b}\)
  • '60% greater than a' means \(\mathrm{c = a + 0.60a = 1.60a}\)

2. INFER the solution strategy

• We have a in terms of b, and c in terms of a
• To find c in terms of b, we need to use substitution
• Replace the 'a' in the second equation with '0.30b'

3. SIMPLIFY through substitution

• Start with: \(\mathrm{c = 1.60a}\)
• Substitute \(\mathrm{a = 0.30b}\): \(\mathrm{c = 1.60(0.30b)}\)
• Multiply: \(\mathrm{c = 0.48b}\)

Answer: \(\mathrm{0.48}\) (which equals \(\mathrm{12/25}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting '70% less than b' as meaning \(\mathrm{a = 0.70b}\) instead of \(\mathrm{a = 0.30b}\).

Students often think '70% less' means 'take 70%' rather than 'subtract 70% and keep 30%.' This leads to \(\mathrm{a = 0.70b}\), then \(\mathrm{c = 1.60(0.70b) = 1.12b}\). This causes confusion since none of the typical answer choices would be 1.12, leading to guessing.

Second Most Common Error:

Poor INFER reasoning: Attempting to work backwards from c to b instead of using forward substitution.

Some students try to express b in terms of a (\(\mathrm{b = a/0.30}\)), then substitute into the c equation, creating unnecessarily complex fractions and calculation errors. This leads to confusion and often abandoning the systematic approach.

The Bottom Line:

This problem tests your ability to carefully translate percentage language into algebra and then systematically use substitution. The key insight is recognizing that 'X% less' means you keep (100-X)% of the original amount.