A juice company fills two bottle sizes: each small bottle contains 300 milliliters of juice, and each large bottle contains...
GMAT Algebra : (Alg) Questions
A juice company fills two bottle sizes: each small bottle contains 300 milliliters of juice, and each large bottle contains 500 milliliters of juice. The company fills s small bottles and b large bottles using exactly 14,000 milliliters of juice. Which equation best represents this relationship?
- \(300\mathrm{s} + 500\mathrm{b} = 14{,}000\)
- \(\frac{\mathrm{s}}{300} + \frac{\mathrm{b}}{500} = 14{,}000\)
- \(300\mathrm{s} + 500\mathrm{b} = 14\)
- \(300\mathrm{s} + 500\mathrm{b} = 140\)
1. TRANSLATE the problem information
- Given information:
- Small bottles: \(300\text{ mL}\) each, \(s\) bottles made
- Large bottles: \(500\text{ mL}\) each, \(b\) bottles made
- Total juice used: exactly \(14{,}000\text{ mL}\)
- What this tells us: We need an equation where the total volume from both bottle types equals \(14{,}000\text{ mL}\)
2. INFER the relationship structure
- Each small bottle contributes \(300\text{ mL}\) to the total
- Each large bottle contributes \(500\text{ mL}\) to the total
- Total volume = contributions from all small bottles + contributions from all large bottles
3. Build the equation step by step
- Volume from small bottles: \(300\text{ mL per bottle} \times s\text{ bottles} = 300s\text{ mL}\)
- Volume from large bottles: \(500\text{ mL per bottle} \times b\text{ bottles} = 500b\text{ mL}\)
- Total equation: \(300s + 500b = 14{,}000\)
Answer: (A) \(300s + 500b = 14{,}000\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the direction of the multiplication relationship and write \(\frac{s}{300} + \frac{b}{500} = 14{,}000\)
They incorrectly think "s bottles divided by 300 mL each" instead of "s bottles times 300 mL each." This creates an equation that adds fractions representing "bottles per milliliter" rather than total milliliters, which doesn't match the physical situation.
This leads them to select Choice B \(\left(\frac{s}{300} + \frac{b}{500} = 14{,}000\right)\)
Second Most Common Error:
Poor TRANSLATE reasoning with units: Students correctly set up \(300s + 500b\) but use the wrong total value, either converting \(14{,}000\text{ mL}\) incorrectly or misreading the problem
They might think \(14{,}000\text{ mL}\) means 14 or 140 in some other unit, leading to equations like \(300s + 500b = 14\) or \(300s + 500b = 140\).
This may lead them to select Choice C \((300s + 500b = 14)\) or Choice D \((300s + 500b = 140)\)
The Bottom Line:
This problem tests whether students can correctly translate a rate-based word problem into a linear equation, particularly understanding that "rate × quantity = contribution" and all contributions sum to the total.