A researcher collected a total of 500 samples of river water from various locations along a certain river and tested...
GMAT Two Part Analysis : (TPA) Questions
A researcher collected a total of 500 samples of river water from various locations along a certain river and tested the samples for the presence of Microorganisms X, Y, and Z, with the following results:
- Exactly 175 samples contained Microorganism X but not Microorganism Y, and exactly \(4\%\) of those contained Microorganism Z.
- Exactly 145 samples contained Microorganism Y but not Microorganism X, and exactly \(20\%\) of those contained Microorganism Z.
- Exactly 60 samples contained both Microorganism X and Microorganism Y, and exactly \(15\%\) of those contained Microorganism Z.
Based on the information provided, select for Both X and Z the total number of samples that contained both Microorganism X and Microorganism Z, and select for Both Y and Z the total number of samples that contained both Microorganism Y and Microorganism Z. Make only two selections, one in each column.
Solution
Let's visualize this problem to make it crystal clear...
Phase 1: Owning the Dataset
Visualization Selection
This is a set membership problem with overlapping categories. We'll use a structured table to track our three distinct groups and their relationships with Microorganism Z.
Create Clear Visual Representation
| Sample Group | Number of Samples | Contains Z | Samples with Z |
| X only (not Y) | 175 | 4% | \(175 \times 0.04 = 7\) |
| Y only (not X) | 145 | 20% | \(145 \times 0.20 = 29\) |
| Both X and Y | 60 | 15% | \(60 \times 0.15 = 9\) |
Total samples accounted for: \(175 + 145 + 60 = 380\) samples
(Note: \(500 - 380 = 120\) samples contain neither X nor Y)
Phase 2: Understanding the Question
Simplify Complex Statements
We need to find:
- Both X and Z: Total samples containing BOTH Microorganism X AND Microorganism Z
- Both Y and Z: Total samples containing BOTH Microorganism Y AND Microorganism Z
Key Insights Before Answer Choices
- Samples with "X and Z" can come from two sources:
- The "X only" group (those 7 samples)
- The "Both X and Y" group (those 9 samples)
- Similarly, samples with "Y and Z" can come from:
- The "Y only" group (those 29 samples)
- The "Both X and Y" group (those 9 samples)
Phase 3: Finding the Answer
Systematic Calculation Process
For Both X and Z:
- From "X only" group: 7 samples have both X and Z
- From "Both X and Y" group: 9 samples have both X and Z
- Total with both X and Z: \(7 + 9 = 16\) samples ✓
For Both Y and Z:
- From "Y only" group: 29 samples have both Y and Z
- From "Both X and Y" group: 9 samples have both Y and Z
- Total with both Y and Z: \(29 + 9 = 38\) samples ✓
Stop here - we found our answers.
Phase 4: Solution
Final Answer
- Both X and Z: 16
- Both Y and Z: 38
These values directly emerge from our systematic analysis. The samples in the "Both X and Y" group that contain Z contribute to both totals, which is why they appear in both calculations.