Sofia is in charge of ordering ingredients for a restaurant. Only one of the meals on the restaurant's menu features...
GMAT Two Part Analysis : (TPA) Questions
Sofia is in charge of ordering ingredients for a restaurant. Only one of the meals on the restaurant's menu features shrimp, and each time that meal is ordered, the restaurant uses exactly 5 shrimp. Sofia will purchase shrimp at a price of 12.50 euros per kilogram, with approximately 42 shrimp per kilogram on average.
Sofia determined that if a total of \(\mathrm{n}\) shrimp meals are ordered at the restaurant per day on average, then a good approximation for the average daily cost to the restaurant, in euros, for the shrimp served can be found by \(\frac{\mathrm{n} \times \mathrm{p}}{\mathrm{q}}\). Select a value for \(\mathrm{p}\) and a value for g that would together create the closest approximation from among the options given for the average daily cost, in euros, to the restaurant for the shrimp served. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Understanding the Shrimp Cost Structure
Let's organize the given information:
- Each shrimp meal uses exactly 5 shrimp
- Shrimp price: 12.50 euros per kilogram
- Average shrimp per kilogram: 42 shrimp
- n = number of shrimp meals ordered per day
Visualization
Let's trace the cost calculation flow:
\(\mathrm{n}\) meals → \(5\mathrm{n}\) shrimp → \(\frac{5\mathrm{n}}{42}\) kg → \(\frac{5\mathrm{n}}{42} \times 12.50\) euros
Phase 2: Understanding the Question
Breaking Down the Formula
The question states we need to approximate the average daily cost using: \(\frac{\mathrm{n} \times \mathrm{p}}{\mathrm{q}}\)
Let's calculate the actual daily cost first:
- If n meals are ordered, we need 5n shrimp total
- Number of kilograms needed: \(5\mathrm{n} \div 42\) kg
- Cost in euros: \(\frac{5\mathrm{n}}{42} \times 12.50\)
Simplifying:
\(\mathrm{Cost} = \frac{5\mathrm{n} \times 12.50}{42} = \frac{62.5\mathrm{n}}{42}\)
Key Insight
For the formula \(\frac{\mathrm{n} \times \mathrm{p}}{\mathrm{q}}\) to equal our actual cost:
\(\frac{\mathrm{n} \times \mathrm{p}}{\mathrm{q}} = \frac{62.5\mathrm{n}}{42}\)
This means: \(\frac{\mathrm{p}}{\mathrm{q}} = \frac{62.5}{42} ≈ 1.488\)
We need to find values of p and q from the choices [2, 3, 5, 7, 8] that make \(\frac{\mathrm{p}}{\mathrm{q}}\) closest to 1.488.
Phase 3: Finding the Answer
Systematic Check of Ratios
Let's check key combinations that might be close to 1.488:
- \(\frac{2}{2} = 1.000\)
- \(\frac{3}{2} = 1.500\) ✓ (very close!)
- \(\frac{5}{3} = 1.667\)
- \(\frac{7}{5} = 1.400\)
- \(\frac{8}{5} = 1.600\)
Comparing differences from 1.488:
- \(\frac{3}{2}\): \(|1.500 - 1.488| = 0.012\) ← smallest difference
- \(\frac{7}{5}\): \(|1.400 - 1.488| = 0.088\)
- \(\frac{8}{5}\): \(|1.600 - 1.488| = 0.112\)
Verification
With p = 3 and q = 2:
- Formula becomes: \(\frac{\mathrm{n} \times 3}{2} = 1.5\mathrm{n}\)
- Actual cost: \(\frac{62.5\mathrm{n}}{42} ≈ 1.488\mathrm{n}\)
- This is indeed the closest approximation!
Phase 4: Solution
Final Answer:
- p = 3
- q = 2
These values create the ratio \(\frac{3}{2} = 1.5\), which is the closest approximation to the actual cost coefficient of 1.488.