The average of 126 numbers is 951. If each number is multipled by 0.2 and added to 3.6, the average of the new set of the numbers is :

Options

A. 193.8
B. 28.8
C. 479.1
D. CND

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Sum of the Original Numbers
Given that the average of 126 numbers is 951, we can find the sum of these numbers using the formula for average:

\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Total numbers}} \]

Rearranging this gives us:

\[ \text{Sum of numbers} = \text{Average} \times \text{Total numbers} \]

Substituting the values:

\[ \text{Sum of numbers} = 951 \times 126 \]

### Step 2: Calculate the Sum of the Original Numbers
Now, we perform the multiplication:

\[ \text{Sum of numbers} = 951 \times 126 = 119826 \]

### Step 3: Determine the New Numbers
Each number in the original set is transformed by multiplying by 0.2 and then adding 3.6. Therefore, the new numbers can be expressed as:

\[ \text{New number} = 0.2 \times x_i + 3.6 \]

for each original number \(x_i\).

### Step 4: Calculate the Sum of the New Numbers
The sum of the new numbers can be calculated as follows:

\[ \text{Sum of new numbers} = \sum_{i=1}^{126} (0.2 \times x_i + 3.6) \]

This can be separated into two parts:

\[ \text{Sum of new numbers} = 0.2 \times \sum_{i=1}^{126} x_i + \sum_{i=1}^{126} 3.6 \]

The second sum is simply \(3.6\) added \(126\) times:

\[ \sum_{i=1}^{126} 3.6 = 126 \times 3.6 \]

### Step 5: Substitute the Sum of Original Numbers
Now, substituting the sum of the original numbers:

\[ \text{Sum of new numbers} = 0.2 \times 119826 + 126 \times 3.6 \]

Calculating \(126 \times 3.6\):

\[ 126 \times 3.6 = 453.6 \]

### Step 6: Calculate the Total Sum of New Numbers
Now we can calculate the total sum of new numbers:

\[ \text{Sum of new numbers} = 0.2 \times 119826 + 453.6 \]

Calculating \(0.2 \times 119826\):

\[ 0.2 \times 119826 = 23965.2 \]

Adding these together:

\[ \text{Sum of new numbers} = 23965.2 + 453.6 = 24318.8 \]

### Step 7: Calculate the Average of the New Numbers
Finally, to find the average of the new numbers, we divide the sum of the new numbers by the total number of numbers (which remains 126):

\[ \text{Average of new numbers} = \frac{\text{Sum of new numbers}}{126} = \frac{24318.8}{126} \]

Calculating this gives:

\[ \text{Average of new numbers} = 193.8 \]

### Final Answer
Thus, the average of the new set of numbers is:

\[ \boxed{193.8} \]