Question

A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black balls. If 40% of the white balls and 50% of the black balls are metallic, then the number of non-metallic balls in the box is

Answer 1

Let \( W \) represent the count of white balls and \( B \) represent the count of black balls.

Step 1: Relationship between metallic and total balls

Given the information:

• Metallic white balls constitute \( 0.40W \).
• Metallic black balls constitute \( 0.50B \).

It is stated that the quantity of metallic white balls equals the quantity of metallic black balls:

\[ 0.40W = 0.50B \quad \text{... (i)} \]

Step 2: Total ball count

The aggregate number of balls is provided as \( W + B = 450 \).

\[ W + B = 450 \quad \text{... (ii)} \]

Step 3: Substitution of equation (i) into equation (ii)

Rearranging equation (i) to express \( W \) in terms of \( B \):

\[ W = \frac{5}{4}B \quad \text{... (iii)} \] Substituting this expression into equation (ii): \[ \left(\frac{5}{4}\right)B + B = 450 \] \[ \frac{5B + 4B}{4} = 450 \quad \Rightarrow \quad \frac{9B}{4} = 450 \] \[ 9B = 1800 \quad \Rightarrow \quad B = 200 \]

Step 4: Calculation of white balls

With \( B = 200 \), substitute this value back into equation (iii) to determine \( W \): \[ W = \frac{5}{4} \times 200 = 250 \]

Step 5: Quantifying metallic balls

Utilizing the percentages for metallic balls:

\[ \text{Metallic white balls} = 0.40 \times 250 = 100 \]

\[ \text{Metallic black balls} = 0.50 \times 200 = 100 \]

Step 6: Quantifying non-metallic balls

The count of non-metallic balls is computed as:

\[ \text{Non-metallic white balls} = 250 - 100 = 150 \]

\[ \text{Non-metallic black balls} = 200 - 100 = 100 \]

Step 7: Total non-metallic balls

The combined quantity of non-metallic balls is:

\[ 150 + 100 = 250 \]

Final Answer:

The box contains \( \boxed{250} \) non-metallic balls.