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Let's break down and solve the problem step by step.

### Information Given:
- There are white and black shapes.
- Some shapes are circles, the rest are squares.
- Ratios:
  - The ratio of white shapes to black shapes is **3:7**.
  - The ratio of white circles to white squares is **4:5**.
  - The ratio of black circles to black squares is **2:5**.

We are asked to find what fraction of all the shapes are circles.

### Step 1: Define Variables
Let the total number of white shapes be \( W \), and the total number of black shapes be \( B \). From the ratio, we know:
\[
W : B = 3:7
\]
So we can write \( W = 3x \) and \( B = 7x \) for some constant \( x \).

### Step 2: Calculate White Circles and White Squares
The ratio of white circles to white squares is 4:5. Let the number of white circles be \( C_w \) and the number of white squares be \( S_w \). Thus, we know:
\[
C_w : S_w = 4:5
\]
Since the total number of white shapes is \( W = 3x \), we can write:
\[
C_w = \frac{4}{9} \times 3x = \frac{12x}{9} = \frac{4x}{3}
\]
\[
S_w = \frac{5}{9} \times 3x = \frac{15x}{9} = \frac{5x}{3}
\]

### Step 3: Calculate Black Circles and Black Squares
The ratio of black circles to black squares is 2:5. Let the number of black circles be \( C_b \) and the number of black squares be \( S_b \). Thus, we know:
\[
C_b : S_b = 2:5
\]
Since the total number of black shapes is \( B = 7x \), we can write:
\[
C_b = \frac{2}{7} \times 7x = 2x
\]
\[
S_b = \frac{5}{7} \times 7x = 5x
\]

### Step 4: Total Number of Circles
The total number of circles is the sum of white circles and black circles:
\[
\text{Total Circles} = C_w + C_b = \frac{4x}{3} + 2x = \frac{4x}{3} + \frac{6x}{3} = \frac{10x}{3}
\]

### Step 5: Total Number of Shapes
The total number of shapes is the sum of white and black shapes:
\[
\text{Total Shapes} = W + B = 3x + 7x = 10x
\]

### Step 6: Fraction of Shapes that are Circles
Now, the fraction of shapes that are circles is:
\[
\frac{\text{Total Circles}}{\text{Total Shapes}} = \frac{\frac{10x}{3}}{10x} = \frac{10x}{3 \times 10x} = \frac{1}{3}
\]

### Final Answer:
The fraction of all the shapes that are circles is **\(\frac{1}{3}\)**.

Let me know if you need further clarification or a detailed breakdown of any step!

#### 5 Related Questions:
1. What would the fraction be if the ratio of white to black shapes was 1:1?
2. How would the problem change if there were twice as many black circles as white circles?
3. What is the total number of squares in terms of \( x \)?
4. Can you express the total number of black shapes if the ratio of white shapes to black shapes changes to 5:4?
5. How would the fraction of circles change if the ratio of white circles to white squares was 1:1?

#### Tip:
Always check the total number of elements and ratios when working with ratios to ensure that each part aligns properly with the total.

White shapes and black shapes are used in a game. Some of the shapes are circles. All the other shapes are squares. The ratio of the number of white shapes to the number of black shapes is 3:7. The ratio of the number of white circles to the number of white squares is 4:5. The ratio of the number of black circles to the number of black squares is 2:5. Work out what fraction of all the shapes are circles.

In this problem, you will learn how to find the fraction of shapes that are circles given the ratios of white to black shapes, white circles to white squares, and black circles to black squares. Suitable for students in Grades 8-10.

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