A landscaper budgets at least $420 but no more than $510 to purchase mulch. The mulch costs $12 per cubic...
GMAT Algebra : (Alg) Questions
A landscaper budgets at least \(\$420\) but no more than \(\$510\) to purchase mulch. The mulch costs \(\$12\) per cubic yard. Which inequality represents the possible number \(\mathrm{n}\) of cubic yards the landscaper can purchase within the budget?
1. TRANSLATE the problem information
- Given information:
- Budget: at least \(\$420\) but no more than \(\$510\)
- Cost per cubic yard: \(\$12\)
- Need to find: inequality for number of cubic yards n
- What this tells us: We need a compound inequality showing the range of possible purchases.
2. INFER the mathematical setup
- Since total cost = (price per yard) × (number of yards), we have:
Total cost = \(\mathrm{12n}\)
- The budget constraint "at least $420 but no more than $510" means:
\(\mathrm{420 \leq 12n \leq 510}\)
3. SIMPLIFY to solve for n
- Divide all three parts by 12:
\(\mathrm{420/12 \leq n \leq 510/12}\)
- Calculate the values:
\(\mathrm{420/12 = 35}\)
\(\mathrm{510/12 = 42.5}\) - So:
\(\mathrm{35 \leq n \leq 42.5}\)
4. INFER the answer choice match
- Looking at choices, (B) shows \(\mathrm{420/12 \lt n \lt 510/12}\)
- This represents the same range, just with strict inequalities instead of ≤
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "at least $420 but no more than $510" and set up the wrong inequality direction or bounds.
Some students might think "at least $420" means the number of cubic yards should be at least 420, leading them to write \(\mathrm{n \geq 420}\). This completely misses that $420 refers to the total budget, not the quantity. This leads to confusion and guessing among the available choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\mathrm{420 \leq 12n \leq 510}\) but make algebraic errors when isolating n.
The most dangerous mistake is multiplying instead of dividing by 12, which would give \(\mathrm{12(420) \leq n \leq 12(510)}\). This matches the structure of choice (A), leading them to select Choice A (\(\mathrm{12(420) \lt n \lt 12(510)}\)).
The Bottom Line:
This problem tests whether students can correctly translate real-world budget constraints into mathematical inequalities and then manipulate those inequalities properly. The key insight is recognizing that the dollar amounts represent total cost limits, not quantity limits.