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's denote the number of white balls as \( W \) and the number of black balls as \( B \).

Step 1: Relating metallic balls to total balls

From the information given:

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

We are told that the number of metallic white balls is equal to the number of metallic black balls, so:

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

Step 2: Total number of balls

\p>The total number of balls is given as \( W + B = 450 \), which is:

 

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

Step 3: Substituting equation (i) into equation (ii)

From equation (i), we can express \( W \) in terms of \( B \):

\[ W = \frac{5}{4}B \quad \text{... (iii)} \] Substituting this 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: Finding the number of white balls

Since \( B = 200 \), we can substitute this back into equation (iii) to find \( W \): \[ W = \frac{5}{4} \times 200 = 250 \]

Step 5: Counting metallic balls

Now, using the percentage of metallic balls: \[ \text{Metallic white balls} = 0.40 \times 250 = 100 \] \[ \text{Metallic black balls} = 0.50 \times 200 = 100 \]

Step 6: Counting non-metallic balls

The number of non-metallic balls is calculated as: \[ \text{Non-metallic white balls} = 250 - 100 = 150 \] \[ \text{Non-metallic black balls} = 200 - 100 = 100 \]

Step 7: Total number of non-metallic balls

The total number of non-metallic balls is: \[ 150 + 100 = 250 \]

Final Answer:

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

Answer 2

Step 1: Let Variables

Let the number of white balls = \( x \) 
Let the number of black balls = \( y \) 
Total number of balls: \[ x + y = 450 \tag{1} \]

Step 2: Metallic Ball Fractions

• Metallic black balls = \( 0.5y \)
• Metallic white balls = \( 0.4x \)

Step 3: Metallic Balls Are Equal

Given: \[ 0.4x = 0.5y \Rightarrow 4x = 5y \Rightarrow 4x - 5y = 0 \tag{2} \]

Step 4: Solve Equations (1) and (2)

From Eq. (1): \( y = 450 - x \) 
Substitute into (2): \[ 4x - 5(450 - x) = 0 \Rightarrow 4x - 2250 + 5x = 0 \Rightarrow 9x = 2250 \Rightarrow x = 250 \Rightarrow y = 450 - 250 = 200 \]

Step 5: Calculate Non-Metallic Balls

• Non-metallic white balls = \( 0.6x = 0.6 \times 250 = 150 \)
• Non-metallic black balls = \( 0.5y = 0.5 \times 200 = 100 \)

Total non-metallic balls: \[ 150 + 100 = \boxed{250} \]

 Final Answer: 250