The number a is 70% less than the positive number b. The number c is 80% greater than a. The...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
The number \(\mathrm{a}\) is \(\mathrm{70\%}\) less than the positive number \(\mathrm{b}\). The number \(\mathrm{c}\) is \(\mathrm{80\%}\) greater than \(\mathrm{a}\). The number \(\mathrm{c}\) is how many times \(\mathrm{b}\)?
1. TRANSLATE the percentage relationships
- Given information:
- a is 70% less than positive number b
- c is 80% greater than a
- Need to find: c is how many times b?
- What this tells us:
- '70% less than b' means \(\mathrm{a = b - 0.70b = 0.30b}\)
- '80% greater than a' means \(\mathrm{c = a + 0.80a = 1.80a}\)
2. INFER the solution strategy
- We have c expressed in terms of a, but we need c in terms of b
- Since we know \(\mathrm{a = 0.30b}\), we can substitute this into our expression for c
- This will give us c directly in terms of b
3. SIMPLIFY using substitution
- Start with: \(\mathrm{c = 1.80a}\)
- Substitute \(\mathrm{a = 0.30b}\): \(\mathrm{c = 1.80(0.30b)}\)
- Multiply: \(\mathrm{c = 0.54b}\)
Answer: \(\mathrm{0.54}\) or \(\mathrm{27/50}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Misinterpreting '70% less than b'
Students often think '70% less than b' means \(\mathrm{a = 0.70b}\), when it actually means \(\mathrm{a = 0.30b}\) (we keep 30% of the original). This fundamental translation error carries through the entire problem.
With \(\mathrm{a = 0.70b}\) instead of \(\mathrm{a = 0.30b}\):
- \(\mathrm{c = 1.80a = 1.80(0.70b) = 1.26b}\)
This leads to an incorrect answer of \(\mathrm{1.26}\), which doesn't match either given form.
Second Most Common Error:
Poor SIMPLIFY execution: Arithmetic errors in the final calculation
Students correctly set up \(\mathrm{c = 1.80(0.30b)}\) but make errors when multiplying:
- Some calculate \(\mathrm{1.80 \times 0.30 = 0.54}\) incorrectly
- Others might confuse the decimal placement
This causes confusion and may lead to guessing among the answer choices.
The Bottom Line:
This problem tests whether students can accurately translate percentage language into algebra and then systematically work through substitution. The key insight is recognizing that 'X% less than Y' means keeping (100-X)% of Y, not taking X% of Y.