A bag contains 18 balls out of which x balls are red.
(i) If one ball is drawn at random from the bag what is the probability that it is not red?
(ii) If two more red balls are put in the bag, the probability of drawing a red ball will be `(9)/(8)` times the probability of drawing a red ball in the first case. Find the value of x.

Answer

To solve the problem step-by-step, we will break down the two parts of the question.

### Step 1: Determine the Probability of Not Drawing a Red Ball

1. **Identify Total Balls and Red Balls**:
- Total number of balls in the bag = 18
- Number of red balls = x

2. **Calculate the Probability of Drawing a Red Ball**:
- Probability of drawing a red ball = (Number of red balls) / (Total number of balls) = x / 18

3. **Calculate the Probability of Not Drawing a Red Ball**:
- Probability of not drawing a red ball = 1 - Probability of drawing a red ball
- Therefore, Probability of not drawing a red ball = 1 - (x / 18) = (18 - x) / 18

### Step 2: Determine the New Probability After Adding Red Balls

1. **Update the Number of Red Balls and Total Balls**:
- After adding 2 red balls, the new number of red balls = x + 2
- The new total number of balls = 18 + 2 = 20

2. **Calculate the New Probability of Drawing a Red Ball**:
- New probability of drawing a red ball = (New number of red balls) / (New total number of balls) = (x + 2) / 20

3. **Set Up the Equation Based on the Given Condition**:
- According to the problem, the new probability of drawing a red ball is (9/8) times the original probability of drawing a red ball.
- Therefore, we can write the equation:
\[ \frac{x + 2}{20} = \frac{9}{8} \cdot \frac{x}{18} \]

### Step 3: Solve the Equation

1. **Cross-Multiply to Eliminate the Fractions**:
- Cross-multiplying gives:
\[ 8(x + 2) = 9 \cdot \frac{20x}{18} \]

2. **Simplify the Right Side**:
- Simplifying gives:
\[ 8(x + 2) = \frac{180x}{18} = 10x \]

3. **Expand and Rearrange the Equation**:
- Expanding the left side:
\[ 8x + 16 = 10x \]
- Rearranging gives:
\[ 10x - 8x = 16 \implies 2x = 16 \implies x = 8 \]

### Conclusion
The value of x, which represents the number of red balls in the bag, is **8**.

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