A town parks director has specified that for every 3 aspen trees in the downtown park, there should be exactly...

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Phase 1: Understanding the Dataset

Let's organize the current tree counts and required ratios:

Current Trees:

Director's Specifications:

• Ratio 1: For every 3 aspens → exactly 2 birches AND 4 firs
• Ratio 2: For every 1 larch → exactly 3 spruces AND 2 aspens

Phase 2: Analyzing the Requirements

Let's check if current counts meet the specifications:

For Ratio 1 (Aspen:Birch:Fir = \(3:2:4\)):

• With 12 aspens, we need:
• Birches: \(12 \div 3 \times 2 = 8\) ✓ (matches current)
• Firs: \(12 \div 3 \times 4 = 16\) (need 16, have 15)

For Ratio 2 (Larch:Spruce:Aspen = \(1:3:2\)):

• With 8 larches, we need:
• Spruces: \(8 \times 3 = 24\) (need 24, have 18)
• Aspens: \(8 \times 2 = 16\) (need 16, have 12)

Phase 3: Finding the Solution

Key insight: Aspens appear in BOTH ratios, creating a constraint we must satisfy.

From Ratio 2: Number of aspens = 2 × number of larches

Since we currently have 12 aspens, the number of larches should be: \(12 \div 2 = 6\)

Let's verify what happens if we adjust to 6 larches:

• With 6 larches, we need:
• Spruces: \(6 \times 3 = 18\) ✓ (matches current)
• Aspens: \(6 \times 2 = 12\) ✓ (matches current)

And with 12 aspens maintained:

• Birches: 8 ✓ (already correct)
• Firs: 16 (need to increase from 15)

Phase 4: Solution

To meet the director's specifications:

• Reduced: Larch (from 8 to 6)
• Increased: Fir (from 15 to 16)

This creates a perfectly balanced park where:

• 12 aspens : 8 birches : 16 firs (ratio \(3:2:4\)) ✓
• 6 larches : 18 spruces : 12 aspens (ratio \(1:3:2\)) ✓