Of the 20 members in a local group of Mensa International, 40% are philosophers and the rest are not. Among...
GMAT Two Part Analysis : (TPA) Questions
Of the 20 members in a local group of Mensa International, \(40\%\) are philosophers and the rest are not. Among those who are not philosophers, the ratio of those who hold a PhD to those who do not hold a PhD is \(1:3\). Among those who are philosophers, the ratio of those who hold a PhD to those who do not hold a PhD is \(7:1\).
For all 20 members, the ratio of those who hold a PhD to those who do not is \(\mathrm{x} : \mathrm{y}\), where x and y have no common factor that is greater than 1. Select for x and for y values that are jointly consistent with the information provided. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visualization
Let's create a table to track PhD status across both groups:
| Group | Total Members | PhD Holders | Non-PhD Holders |
|---|---|---|---|
| Philosophers | ? | ? | ? |
| Non-Philosophers | ? | ? | ? |
| Total | 20 | ? | ? |
Given Information
- Total members: 20
- \(40\%\) are philosophers \(\rightarrow\) 8 philosophers
- \(60\%\) are non-philosophers \(\rightarrow\) 12 non-philosophers
- Non-philosophers: PhD to non-PhD ratio = \(1:3\)
- Philosophers: PhD to non-PhD ratio = \(7:1\)
Phase 2: Understanding the Question
We need to find the overall ratio \(x:y\) where:
- \(x\) = total PhD holders across all 20 members
- \(y\) = total non-PhD holders across all 20 members
- \(x\) and \(y\) must have no common factor greater than 1 (coprime)
Phase 3: Finding the Answer
Calculate Non-Philosophers' PhD Distribution
With 12 non-philosophers and ratio \(1:3\):
- Total parts = \(1 + 3 = 4\)
- PhD holders = \(12 \times \frac{1}{4} = 3\)
- Non-PhD holders = \(12 \times \frac{3}{4} = 9\)
Calculate Philosophers' PhD Distribution
With 8 philosophers and ratio \(7:1\):
- Total parts = \(7 + 1 = 8\)
- PhD holders = \(8 \times \frac{7}{8} = 7\)
- Non-PhD holders = \(8 \times \frac{1}{8} = 1\)
Update Our Table
| Group | Total Members | PhD Holders | Non-PhD Holders |
|---|---|---|---|
| Philosophers | 8 | 7 | 1 |
| Non-Philosophers | 12 | 3 | 9 |
| Total | 20 | 10 | 10 |
Find the Overall Ratio
- Total PhD holders: \(3 + 7 = 10\)
- Total non-PhD holders: \(9 + 1 = 10\)
- Ratio = \(10:10 = 1:1\)
Since \(\mathrm{gcd}(1,1) = 1\), the ratio is already in its simplest form.
Phase 4: Solution
From our calculations:
- \(x = 1\) (representing 10 PhD holders)
- \(y = 1\) (representing 10 non-PhD holders)
Both values are available in our answer choices.