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To solve this, we need to find two things:

1. The **ratio of the length** of a tile in the model to the length of an actual tile.
2. The **ratio of the area** of a tile in the model to the area of an actual tile.

Let's go through each part step by step.

---

### Step 1: Ratio of Lengths
- **Model Tile Length:** \( \frac{1}{3} \) inches
- **Actual Tile Length:** \( \frac{1}{4} \) feet

Since these lengths are in different units (inches and feet), let’s convert them to the same unit. Converting feet to inches:

\[
\frac{1}{4} \text{ feet} = \frac{1}{4} \times 12 = 3 \text{ inches}
\]

Now, the ratio of the model length to the actual length is:

\[
\frac{\text{Model Length}}{\text{Actual Length}} = \frac{\frac{1}{3}}{3} = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}
\]

So, the ratio of the length of a tile in the model to the length of an actual tile is **1:9**.

---

### Step 2: Ratio of Areas
To find the ratio of the areas, we need the area of each tile in both the model and the actual size.

1. **Model Tile Area:**
   \[
   \text{Model Length} \times \text{Model Width} = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} \text{ square inches}
   \]

2. **Actual Tile Area:**
   - **Actual Width:** \( \frac{1}{8} \) feet (convert to inches)
   \[
   \frac{1}{8} \text{ feet} = \frac{1}{8} \times 12 = 1.5 \text{ inches}
   \]
   - **Area Calculation:**
   \[
   \text{Actual Length} \times \text{Actual Width} = 3 \times 1.5 = 4.5 \text{ square inches}
   \]

3. **Area Ratio:**
   \[
   \frac{\text{Model Area}}{\text{Actual Area}} = \frac{\frac{1}{18}}{4.5} = \frac{1}{18} \times \frac{1}{4.5} = \frac{1}{81}
   \]

Thus, the ratio of the area of a tile in the model to the area of an actual tile is **1:81**.

---

Would you like further details or have any questions?

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### Related Questions:
1. How can unit conversions affect the accuracy of ratios?
2. What would be the ratio if both measurements were taken in feet?
3. How does the scale factor for length impact the scale factor for area?
4. What would be the area ratio if the model used a different length and width?
5. How do we calculate ratios for volumes if the model were in 3D?

#### Tip:
When working with ratios in geometry, always ensure measurements are in the same units before calculating.

An architect makes a model of a new house. The model shows a tile patio in the backyard. In the model, each tile has length 1/3 in. and width 1/6 in. The actual tiles have length 1/4 ft and width 1/8 ft. What is the ratio of the length of a tile in the model to the length of an actual tile? What is the ratio of the area of a tile in the model to the area of an actual tile?

Calculate the ratio of the length and area of a tile in a model to an actual tile. This math problem explores ratios, unit conversions, and area calculations, suitable for Grade 7 students.

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