Question

A bag contains 14 balls of which \(x\) are white. If 6 more white balls are added to the bag, the probability of drawing a white ball is \( \frac{1}{2} \). Then the value of \( x = \)

• A. 7
• B. 4
• C. 8
• D. none of these

Hint:

1. Initial state: Total = 14, White = \(x\). 2. After adding 6 white balls: New White = \(x+6\). New Total = \(14+6 = 20\). 3. Probability of drawing a white ball now = (New White) / (New Total). Given this probability is \(1/2\). 4. Equation: \( \frac{x+6}{20} = \frac{1}{2} \). 5. Solve for \(x\): \(2(x+6) = 20 \times 1\) \(2x + 12 = 20\) \(2x = 20 - 12 = 8\) \(x = 4\).

Answer 1

The Correct Option is B

Concept: Probability = (Number of favorable outcomes) / (Total number of possible outcomes). Step 1: Describe the initial state of the bag Initially:
Total number of balls = 14.
Number of white balls = \(x\).
Number of non-white balls = \(14 - x\). Step 2: Describe the state of the bag after adding more white balls 6 more white balls are added to the bag. After adding:
New number of white balls = \(x + 6\).
New total number of balls = Initial total + Added balls = \(14 + 6 = 20\). Step 3: Set up the probability equation for drawing a white ball after adding more The probability of drawing a white ball from the modified bag is given as \( \frac{1}{2} \). Using the probability formula: P(drawing a white ball) = \(\frac{\text{New number of white balls}}{\text{New total number of balls}}\) \[ \frac{x+6}{20} = \frac{1}{2} \] Step 4: Solve the equation for \(x\) \[ \frac{x+6}{20} = \frac{1}{2} \] To solve for \(x\), we can cross-multiply or multiply both sides by 20. Multiplying both sides by 20: \[ x+6 = \frac{1}{2} \times 20 \] \[ x+6 = 10 \] Subtract 6 from both sides: \[ x = 10 - 6 \] \[ x = 4 \] Step 5: Check the answer (optional) If \(x=4\): Initially: 4 white balls, total 14 balls. After adding 6 white balls: Number of white balls = \(4+6 = 10\). Total number of balls = \(14+6 = 20\). Probability of drawing a white ball = \(\frac{10}{20} = \frac{1}{2}\). This matches the given information. Therefore, the value of \(x\) is 4.