Equations from Ball Counts

Let V, Y, R, and G represent the number of Violet, Yellow, Red, and Green balls, respectively. The problem statements translate to the following equations:

• (i) \( Y + 2V = 50 \)
• (ii) \( V + G = 50 \)
• (iii) \( Y + R = 50 \)
• (iv) \( V + 2R = 50 \)

Solving the System of Equations

We can solve this system of linear equations step-by-step:

1. From equation (iii), express Y in terms of R: \( Y = 50 - R \).
2. Substitute this expression for Y into equation (i): \( (50 - R) + 2V = 50 \).
3. Simplify the equation: \( -R + 2V = 0 \), which means \( R = 2V \).
4. Substitute \( R = 2V \) into equation (iv): \( V + 2(2V) = 50 \).
5. Solve for V: \( V + 4V = 50 \Rightarrow 5V = 50 \Rightarrow V = 10 \).
6. Calculate R using \( R = 2V \): \( R = 2 \times 10 = 20 \).
7. Calculate Y using equation (iii): \( Y + 20 = 50 \Rightarrow Y = 30 \).
8. Calculate G using equation (ii): \( 10 + G = 50 \Rightarrow G = 40 \).

The number of balls are:

• Violet (V): 10
• Yellow (Y): 30
• Red (R): 20
• Green (G): 40

Calculating Ball Proportions

First, find the total number of balls:

Total balls = \( V + Y + R + G = 10 + 30 + 20 + 40 = 100 \).

Now, calculate the proportion of each color:

• Violet: \( \frac{10}{100} = 0.1 \) (10%)
• Yellow: \( \frac{30}{100} = 0.3 \) (30%)
• Red: \( \frac{20}{100} = 0.2 \) (20%)
• Green: \( \frac{40}{100} = 0.4 \) (40%)

Matching Proportions to Pie Chart

The correct pie chart must represent these proportions: 10% Violet, 30% Yellow, 20% Red, and 40% Green.

Comparing these proportions to the options, the pie chart provided in Option 1 represents these values accurately.

The image URL for the correct pie chart is: