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To find the ratio of the area of the shaded part to the unshaded part, let's break the problem down step by step:

### Given Information:
- There are three circles with diameters of \( 2r \), \( 4r \), and \( 6r \).
- The diagram shows the larger circle enclosing a smaller circle with a shaded region in between.
  
### Steps to Solve:

1. **Find the Radii:**
   - The smallest circle has a diameter of \( 2r \), so its radius is \( r_1 = r \).
   - The middle circle has a diameter of \( 4r \), so its radius is \( r_2 = 2r \).
   - The largest circle has a diameter of \( 6r \), so its radius is \( r_3 = 3r \).

2. **Calculate the Areas of the Circles:**
   The area \( A \) of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius.

   - **Smallest Circle (inner circle, radius = \( r \)):**
     \[
     A_{\text{small}} = \pi r^2
     \]

   - **Middle Circle (shaded region, radius = \( 2r \)):**
     \[
     A_{\text{middle}} = \pi (2r)^2 = 4\pi r^2
     \]

   - **Largest Circle (outer circle, radius = \( 3r \)):**
     \[
     A_{\text{large}} = \pi (3r)^2 = 9\pi r^2
     \]

3. **Calculate the Shaded Area:**
   The shaded area is the region between the middle circle and the small circle:
   \[
   \text{Shaded Area} = A_{\text{middle}} - A_{\text{small}} = 4\pi r^2 - \pi r^2 = 3\pi r^2
   \]

4. **Calculate the Unshaded Area:**
   The unshaded area is the region between the largest circle and the middle circle:
   \[
   \text{Unshaded Area} = A_{\text{large}} - A_{\text{middle}} = 9\pi r^2 - 4\pi r^2 = 5\pi r^2
   \]

5. **Find the Ratio of Shaded to Unshaded Areas:**
   Now, find the ratio of the shaded area to the unshaded area:
   \[
   \text{Ratio} = \frac{\text{Shaded Area}}{\text{Unshaded Area}} = \frac{3\pi r^2}{5\pi r^2} = \frac{3}{5}
   \]

### Final Answer:
The ratio of the area of the shaded part to the unshaded part is \( \frac{3}{5} \).

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Would you like further explanation or clarification on any of these steps? 

Here are 5 related questions for more practice:
1. How do you find the area of a circle when given its circumference?
2. What is the ratio of the areas of two circles with diameters in a 3:1 ratio?
3. If the radius of the smallest circle is doubled, how does the shaded-to-unshaded ratio change?
4. What would the area of the shaded part be if the diameter of the largest circle were increased to \( 8r \)?
5. How can you apply the same method to find the ratio of the volumes of spheres?

**Tip:** The area of a circle grows quadratically with the radius. This means that small changes in radius result in larger changes in area!

The diagram shows three circles with diameters 2r, 4r, and 6r. Find the ratio of the area of the shaded part to the unshaded part.

Learn how to find the ratio of the shaded part to the unshaded part in a diagram showing three concentric circles with diameters 2r, 4r, and 6r. This problem covers key geometry concepts and the formula for the area of circles. Suitable for students in Grades 8-10.

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