Of the 20 members in a local group of Mensa International, 40% are philosophers and the rest are not. Among...

Phase 1: Owning the Dataset

Visualization

Let's create a table to track PhD status across both groups:

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

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.