A community fundraiser sold two sizes of reusable water bottles and also collected a one-time venue fee of $125. The...
GMAT Algebra : (Alg) Questions
A community fundraiser sold two sizes of reusable water bottles and also collected a one-time venue fee of $125. The total amount collected last month was $2,457.35. The relationship is modeled by the equation \(9.85\mathrm{m} + 14.60\mathrm{n} + 125 = 2,457.35\), where \(\mathrm{m}\) is the number of smaller bottles sold and \(\mathrm{n}\) is the number of larger bottles sold. According to the equation, what is the price, in dollars, of each smaller bottle? Enter your answer as a decimal (to the nearest cent) or as a fraction in lowest terms.
1. TRANSLATE the equation components
- Given equation: \(9.85\mathrm{m} + 14.60\mathrm{n} + 125 = 2,457.35\)
- What each part means:
- \(\mathrm{m}\) = number of smaller bottles sold
- \(\mathrm{n}\) = number of larger bottles sold
- \(125\) = one-time venue fee
- \(2,457.35\) = total amount collected
2. INFER what the coefficients represent
- In a revenue model, the structure is: \((\mathrm{price} \times \mathrm{quantity}) + (\mathrm{price} \times \mathrm{quantity}) + \mathrm{fixed\ fee} = \mathrm{total}\)
- This means:
- \(9.85\mathrm{m}\) = (price per smaller bottle) × (number of smaller bottles)
- \(14.60\mathrm{n}\) = (price per larger bottle) × (number of larger bottles)
- Therefore, \(9.85\) must be the price per smaller bottle
3. SIMPLIFY to convert the decimal to a fraction
- \(9.85 = \frac{985}{100}\)
- Reduce to lowest terms: \(985 \div 5 = 197\), and \(100 \div 5 = 20\)
- So \(9.85 = \frac{197}{20}\)
Answer: \(9.85\) or \(\frac{197}{20}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that coefficients in revenue models represent unit prices.
Instead, they might try to solve for specific values of \(\mathrm{m}\) and \(\mathrm{n}\), thinking they need to find how many bottles were sold first. This leads to frustration because the equation has two unknowns and cannot be solved for unique values without additional information. This causes them to get stuck and abandon the systematic approach, leading to random guessing.
The Bottom Line:
The key insight is recognizing that this problem isn't asking you to solve the equation - it's asking you to interpret what the coefficient means in the context of a revenue model. The answer is hiding in plain sight within the equation structure itself.