Average of eight numbers is 12. If each number is increased by 2, then average of new numbers is:

Options

A. 13
B. 12
C. 15
D. 14

Answer

To solve the problem step by step, we will follow the mathematical reasoning based on the information given.

### Step 1: Understand the average of the original numbers
The average of eight numbers is given as 12. The formula for average is:

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

In this case, we have:

\[ 12 = \frac{\text{Sum of 8 numbers}}{8} \]

### Step 2: Calculate the sum of the original numbers
To find the sum of the eight numbers, we can rearrange the formula:

\[ \text{Sum of 8 numbers} = \text{Average} \times \text{Total numbers} \]

Substituting the values we have:

\[ \text{Sum of 8 numbers} = 12 \times 8 = 96 \]

### Step 3: Increase each number by 2
Now, if each of the eight numbers is increased by 2, the new numbers can be represented as:

\[ x_1 + 2, x_2 + 2, x_3 + 2, x_4 + 2, x_5 + 2, x_6 + 2, x_7 + 2, x_8 + 2 \]

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

\[ \text{Sum of new numbers} = (x_1 + 2) + (x_2 + 2) + (x_3 + 2) + (x_4 + 2) + (x_5 + 2) + (x_6 + 2) + (x_7 + 2) + (x_8 + 2) \]

This simplifies to:

\[ \text{Sum of new numbers} = (x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8) + (2 \times 8) \]

Substituting the sum of the original numbers:

\[ \text{Sum of new numbers} = 96 + 16 = 112 \]

### Step 5: Calculate the average of the new numbers
Now, we can find the average of the new numbers using the average formula again:

\[ \text{Average of new numbers} = \frac{\text{Sum of new numbers}}{\text{Total numbers}} = \frac{112}{8} \]

Calculating this gives:

\[ \text{Average of new numbers} = 14 \]

### Final Answer
The average of the new numbers is **14**.

---