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: \(\mathrm{300\, mL}\) each, s bottles made
- Large bottles: \(\mathrm{500\, mL}\) each, b bottles made
- Total juice used: exactly \(\mathrm{14{,}000\, mL}\)
- What this tells us: We need an equation where the total volume from both bottle types equals \(\mathrm{14{,}000\, mL}\)
2. INFER the relationship structure
- Each small bottle contributes \(\mathrm{300\, mL}\) to the total
- Each large bottle contributes \(\mathrm{500\, mL}\) 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: \(\mathrm{300\, mL}\) per bottle × s bottles = \(\mathrm{300s\, mL}\)
- Volume from large bottles: \(\mathrm{500\, mL}\) per bottle × b bottles = \(\mathrm{500b\, mL}\)
- Total equation: \(\mathrm{300s + 500b = 14{,}000}\)
Answer: (A) \(\mathrm{300s + 500b = 14{,}000}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the direction of the multiplication relationship and write \(\mathrm{\frac{s}{300} + \frac{b}{500} = 14{,}000}\)
They incorrectly think "s bottles divided by \(\mathrm{300\, mL}\) each" instead of "s bottles times \(\mathrm{300\, mL}\) 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 (\(\mathrm{\frac{s}{300} + \frac{b}{500} = 14{,}000}\))
Second Most Common Error:
Poor TRANSLATE reasoning with units: Students correctly set up \(\mathrm{300s + 500b}\) but use the wrong total value, either converting \(\mathrm{14{,}000\, mL}\) incorrectly or misreading the problem
They might think \(\mathrm{14{,}000\, mL}\) means 14 or 140 in some other unit, leading to equations like \(\mathrm{300s + 500b = 14}\) or \(\mathrm{300s + 500b = 140}\).
This may lead them to select Choice C (\(\mathrm{300s + 500b = 14}\)) or Choice D (\(\mathrm{300s + 500b = 140}\))
The Bottom Line:
This problem tests whether students can correctly translate a rate-based word problem into a linear equation, particularly understanding that "\(\mathrm{rate \times quantity = contribution}\)" and all contributions sum to the total.