A bag contains balls of Red ,Black ,White ,and Brown colour .Probability of getting one Red ball from a bag full of balls is 2 /13 and number of Black balls in the bag is 5. If white ball is 30% less than Brown ball and 40% more than the black balls then find the number of Red balls

Options

A. A)5
B. B)4
C. C)6
D. D)10

Answer

To solve the problem step by step, we will follow the information given in the question and derive the required values systematically.

### Step 1: Define Variables
Let:
- \( R \) = Number of Red balls
- Number of Black balls = 5 (given)
- Let the number of White balls = \( W \)
- Let the number of Brown balls = \( B \)

### Step 2: Determine the Number of White Balls
According to the problem, the number of White balls is 40% more than the number of Black balls. Since there are 5 Black balls:
\[ W = 5 + 0.4 \times 5 = 5 + 2 = 7 \]
So, the number of White balls \( W = 7 \).

### Step 3: Determine the Number of Brown Balls
It is also given that the number of White balls is 30% less than the number of Brown balls. This can be expressed as:
\[ W = B - 0.3B = 0.7B \]
Substituting \( W = 7 \):
\[ 0.7B = 7 \]
To find \( B \), we solve for \( B \):
\[ B = \frac{7}{0.7} = 10 \]
So, the number of Brown balls \( B = 10 \).

### Step 4: Calculate the Total Number of Balls
Now we can calculate the total number of balls in the bag:
\[ \text{Total balls} = R + \text{(Number of Black balls)} + \text{(Number of White balls)} + \text{(Number of Brown balls)} \]
Substituting the known values:
\[ \text{Total balls} = R + 5 + 7 + 10 = R + 22 \]

### Step 5: Use the Probability Formula
The probability of drawing one Red ball from the bag is given as \( \frac{2}{13} \). The probability can also be expressed as:
\[ \text{Probability of Red ball} = \frac{R}{\text{Total balls}} = \frac{R}{R + 22} \]
Setting this equal to the given probability:
\[ \frac{R}{R + 22} = \frac{2}{13} \]

### Step 6: Cross-Multiply to Solve for \( R \)
Cross-multiplying gives:
\[ 13R = 2(R + 22) \]
Expanding the right side:
\[ 13R = 2R + 44 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 13R - 2R = 44 \]
\[ 11R = 44 \]

### Step 8: Solve for \( R \)
Dividing both sides by 11:
\[ R = \frac{44}{11} = 4 \]

### Final Answer
The number of Red balls \( R \) is **4**.

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