A researcher collected a total of 500 samples of river water from various locations along a certain river and tested...

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

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:

1. Both X and Z: Total samples containing BOTH Microorganism X AND Microorganism Z
2. 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.