In a box there are12 blue ,X red balls & green balls , probability of choosing one red ball from the given box is `(1)/(3) ` .Then find the sum of red and blue balls in the box?

Options

A. 40
B. 24
C. 28
D. 36

Explanation

To solve the problem step by step, we will use the information given about the number of balls and the probability of selecting a red ball.

### Step 1: Identify the number of balls
We know:
- Number of blue balls = 12
- Number of red balls = X (unknown)
- Number of green balls = 20

### Step 2: Calculate the total number of balls
The total number of balls in the box can be calculated as:
\[ \text{Total balls} = \text{Blue balls} + \text{Red balls} + \text{Green balls} \]
\[ \text{Total balls} = 12 + X + 20 \]
\[ \text{Total balls} = X + 32 \]

### Step 3: Set up the probability equation
The probability of choosing one red ball is given as \( \frac{1}{3} \). The probability can be expressed as:
\[ \text{Probability of red ball} = \frac{\text{Number of red balls}}{\text{Total number of balls}} \]
Substituting the values we have:
\[ \frac{X}{X + 32} = \frac{1}{3} \]

### Step 4: Cross-multiply to solve for X
Cross-multiplying gives us:
\[ 3X = 1 \cdot (X + 32) \]
This simplifies to:
\[ 3X = X + 32 \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation to isolate X:
\[ 3X - X = 32 \]
\[ 2X = 32 \]

### Step 6: Solve for X
Dividing both sides by 2 gives:
\[ X = 16 \]

### Step 7: Find the sum of red and blue balls
Now that we have the value of X (the number of red balls), we can find the sum of the red and blue balls:
- Number of red balls = 16
- Number of blue balls = 12

Thus, the sum is:
\[ \text{Sum of red and blue balls} = 16 + 12 = 28 \]

### Final Answer
The sum of the red and blue balls in the box is **28**.

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