1. **State the problem:** We need to find the length of Yui's zen garden given that the perimeter is 32 meters and the length is 3 times the width. 2. **Write the formula for the perimeter of a rectangle:** $$P = 2(\text{length} + \text{width})$$ 3. **Define variables:** Let the width be $w$ meters. Then the length is $3w$ meters. 4. **Substitute into the perimeter formula:** $$32 = 2(3w + w)$$ 5. **Simplify inside the parentheses:** $$32 = 2(4w)$$ 6. **Multiply:** $$32 = 8w$$ 7. **Solve for $w$ by dividing both sides by 8:** $$32 = \cancel{8}w \div \cancel{8} \Rightarrow w = 4$$ 8. **Find the length:** $$\text{length} = 3w = 3 \times 4 = 12$$ 9. **Answer:** The length of Yui's zen garden is **12 meters**.