The average of 6 consecutive numbers is 12.5. If next 3 numbers are also included, then what will be the new average?

Options

A. 13
B. 13.5
C. 14
D. 14.5

Answer

To solve the problem step by step, let's follow the reasoning:

### Step 1: Understand the average of the first 6 consecutive numbers
The average of 6 consecutive numbers is given as 12.5.

### Step 2: Calculate the sum of the first 6 numbers
The average can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of numbers}} \]
Here, we have:
\[ 12.5 = \frac{\text{Sum of 6 numbers}}{6} \]
To find the sum of the 6 numbers, we can rearrange the formula:
\[ \text{Sum of 6 numbers} = 12.5 \times 6 = 75 \]

### Step 3: Identify the 6 consecutive numbers
Let the first of the 6 consecutive numbers be \( x \). Then the numbers can be represented as:
\[ x, x+1, x+2, x+3, x+4, x+5 \]
The sum of these numbers is:
\[ x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) = 6x + 15 \]
Setting this equal to the sum we calculated:
\[ 6x + 15 = 75 \]
Solving for \( x \):
\[ 6x = 75 - 15 = 60 \\ x = 10 \]
Thus, the 6 consecutive numbers are:
\[ 10, 11, 12, 13, 14, 15 \]

### Step 4: Calculate the next 3 consecutive numbers
The next 3 consecutive numbers after 10, 11, 12, 13, 14, 15 are:
\[ 16, 17, 18 \]

### Step 5: Calculate the sum of all 9 numbers
Now, we need to find the sum of all 9 numbers:
\[ \text{Sum of 9 numbers} = 75 + (16 + 17 + 18) \]
Calculating the sum of the next 3 numbers:
\[ 16 + 17 + 18 = 51 \]
So, the total sum is:
\[ 75 + 51 = 126 \]

### Step 6: Calculate the new average
Now, we find the new average by dividing the total sum by the total number of numbers (which is 9):
\[ \text{New Average} = \frac{126}{9} = 14 \]

### Final Answer
The new average when the next 3 numbers are included is **14**.

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