In a set of prime and composite numbers, the composite numbers are twice the number of prime numbers and the average of all the number of the set is 9. If the number of prime numbers and composite numbers are exchanged then the average of the set of numbers is increased by 2. If during the exchange of the numbers the average of the prime numbers and composite numbers individually remained constant, then the ratio of the average of composite numbers to the average of prime numbers (initially) was :

Options

A. a. `(7)/(13)`
B. b. `(13)/(7)`
C. c. `9//11`
D. d. none of these

Answer

To solve the problem step by step, let's define the variables and equations based on the information provided.

### Step 1: Define Variables
Let:
- \( n \) = number of prime numbers
- \( 2n \) = number of composite numbers (since composite numbers are twice the number of prime numbers)

### Step 2: Calculate Total Numbers
The total number of numbers in the set is:
\[ n + 2n = 3n \]

### Step 3: Set Up the Average Equation
We know that the average of all numbers in the set is 9. Therefore, we can set up the equation:
\[ \frac{P + 2C}{3n} = 9 \]
where \( P \) is the total sum of prime numbers and \( C \) is the total sum of composite numbers. This leads to:
\[ P + 2C = 27 \quad \text{(Equation 1)} \]

### Step 4: Average After Exchange
When the numbers are exchanged, the average increases by 2, making the new average 11:
\[ \frac{2P + C}{3n} = 11 \]
This leads to:
\[ 2P + C = 33 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
We now have two equations:
1. \( P + 2C = 27 \)
2. \( 2P + C = 33 \)

We can solve these equations simultaneously.

From Equation 1, we can express \( P \) in terms of \( C \):
\[ P = 27 - 2C \]

Substituting this expression for \( P \) into Equation 2:
\[ 2(27 - 2C) + C = 33 \]
Expanding this gives:
\[ 54 - 4C + C = 33 \]
Combining like terms:
\[ 54 - 3C = 33 \]
Solving for \( C \):
\[ -3C = 33 - 54 \]
\[ -3C = -21 \]
\[ C = 7 \]

### Step 6: Find \( P \)
Now substituting \( C = 7 \) back into Equation 1 to find \( P \):
\[ P + 2(7) = 27 \]
\[ P + 14 = 27 \]
\[ P = 27 - 14 = 13 \]

### Step 7: Calculate Averages
Now we can calculate the averages:
- Average of prime numbers \( = \frac{P}{n} = \frac{13}{n} \)
- Average of composite numbers \( = \frac{C}{2n} = \frac{7}{2n} \)

### Step 8: Find the Ratio of Averages
The ratio of the average of composite numbers to the average of prime numbers is:
\[ \text{Ratio} = \frac{\frac{7}{2n}}{\frac{13}{n}} = \frac{7}{2n} \times \frac{n}{13} = \frac{7}{26} \]

### Final Answer
Thus, the ratio of the average of composite numbers to the average of prime numbers is:
\[ \frac{7}{13} \]