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Circles of diameter  inch and  inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

To find the ratio of the blue-painted area to the red-painted area, we need to compute the areas of the smaller and larger circles and then find the difference between them.

1. **Area of the Smaller Circle:**
   - Let the diameter of the smaller circle be \(d_1\).
   - The radius of the smaller circle \(r_1 = \frac{d_1}{2}\).
   - The area of the smaller circle \(A_{\text{red}} = \pi r_1^2\).

2. **Area of the Larger Circle:**
   - Let the diameter of the larger circle be \(d_2\).
   - The radius of the larger circle \(r_2 = \frac{d_2}{2}\).
   - The area of the larger circle \(A_{\text{large}} = \pi r_2^2\).

3. **Blue-Painted Area:**
   - The blue-painted area is the difference between the areas of the larger and smaller circles:
     \[
     A_{\text{blue}} = A_{\text{large}} - A_{\text{red}} = \pi r_2^2 - \pi r_1^2 = \pi (r_2^2 - r_1^2)
     \]

4. **Ratio of Blue-Painted Area to Red-Painted Area:**
   - The ratio of the blue-painted area to the red-painted area is:
     \[
     \text{Ratio} = \frac{A_{\text{blue}}}{A_{\text{red}}} = \frac{\pi (r_2^2 - r_1^2)}{\pi r_1^2} = \frac{r_2^2 - r_1^2}{r_1^2}
     \]

Since the specific diameters are not provided in the image, let's assume:
   - \(d_1 = 2\) inches (small circle's diameter)
   - \(d_2 = 4\) inches (large circle's diameter)

Then:
   - \(r_1 = \frac{2}{2} = 1\) inch
   - \(r_2 = \frac{4}{2} = 2\) inches

Using these values:
   \[
   \text{Ratio} = \frac{r_2^2 - r_1^2}{r_1^2} = \frac{2^2 - 1^2}{1^2} = \frac{4 - 1}{1} = \frac{3}{1} = 3
   \]

So, the ratio of the blue-painted area to the red-painted area is \(

Explore how to calculate the ratio of blue-painted area to red-painted area in circles with diameters of 2 inches and 4 inches. This problem involves applying the formula for circle area and understanding geometric relationships. Suitable for Grades 6-8 students.

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