A corporation uses a model of diminishing returns to make predictions about the expected returns on research investment. For this...
GMAT Two Part Analysis : (TPA) Questions
A corporation uses a model of diminishing returns to make predictions about the expected returns on research investment. For this model, in order to produce an x % increase in annual profits in subsequent years, the corporation must invest y % of annual profits into research, where \(\mathrm{y} = 2\mathrm{x}^2\).
Select two different numbers that are jointly compatible with the information provided and could be the values for \(\mathrm{x}\) and for \(\mathrm{y}\). Make only two selections, one in each column.
Phase 1: Owning the Dataset
Understanding the Model
We have a diminishing returns model where:
- x = percentage increase in annual profits
- y = percentage of annual profits invested in research
- Relationship: \(\mathrm{y = 2x^2}\)
Visualization
Let's create a simple table to organize our thinking:
| Variable | Represents | Formula Role |
| x | % profit increase | Input |
| y | % invested in research | Output = \(\mathrm{2x^2}\) |
Concrete Example
If a company wants a 10% profit increase (\(\mathrm{x = 10}\)), they must invest:
\(\mathrm{y = 2(10)^2 = 2(100) = 200\%}\) of annual profits
This shows the "diminishing returns" - higher profit goals require disproportionately higher research investment.
Phase 2: Understanding the Question
What We're Looking For
We need to find:
- Statement 1: A value for x (profit increase percentage)
- Statement 2: A value for y (research investment percentage)
These values must:
- Satisfy the equation \(\mathrm{y = 2x^2}\)
- Both be present in our answer choices: [1, 3, 5, 20, 50, 80]
Key Insight
We're looking for a pair (x, y) where plugging x into our formula gives us a y value that's also in our answer choices.
Phase 3: Finding the Answer
Systematic Check
Let's test each possible x value to see if it produces a y value in our choices:
If \(\mathrm{x = 1}\):
\(\mathrm{y = 2(1)^2 = 2(1) = 2}\)
Is 2 in our choices? No, continue.
If \(\mathrm{x = 3}\):
\(\mathrm{y = 2(3)^2 = 2(9) = 18}\)
Is 18 in our choices? No, continue.
If \(\mathrm{x = 5}\):
\(\mathrm{y = 2(5)^2 = 2(25) = 50}\)
Is 50 in our choices? Yes! ✓
Stop here - we found our answer.
Phase 4: Solution
Final Answer
- Statement 1 (x): 5
- Statement 2 (y): 50
Verification
With x = 5% profit increase, the company must invest y = 50% of annual profits into research. This satisfies our equation \(\mathrm{y = 2x^2}\) and both values are in our answer choices.