The average of 6 consecutive even numbers is 25. If the next even number is also considered, then what is the new average?

Options

A. 27
B. 25
C. 26
D. 28

Answer

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

### Step 1: Understand the Problem
We are given that the average of 6 consecutive even numbers is 25. We need to find the new average when the next even number is included.

### Step 2: Calculate the Sum of the 6 Even Numbers
The average of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers. Therefore, if the average of 6 numbers is 25, we can find the total sum of these numbers:
\[ \text{Sum} = \text{Average} \times \text{Count} = 25 \times 6 = 150 \]

### Step 3: Identify the 6 Consecutive Even Numbers
Let’s denote the first even number as \( x \). The 6 consecutive even numbers can be represented as:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)
- \( x + 10 \)

The sum of these numbers is:
\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) = 6x + 30 \]
Setting this equal to the sum we calculated:
\[ 6x + 30 = 150 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):
\[ 6x = 150 - 30 \]
\[ 6x = 120 \]
\[ x = 20 \]

### Step 5: List the 6 Even Numbers
Now that we have \( x = 20 \), the 6 consecutive even numbers are:
- 20
- 22
- 24
- 26
- 28
- 30

### Step 6: Include the Next Even Number
The next even number after 30 is 32. Now we have 7 even numbers:
- 20, 22, 24, 26, 28, 30, 32

### Step 7: Calculate the New Average
To find the new average, we first calculate the sum of these 7 numbers:
\[ \text{New Sum} = 20 + 22 + 24 + 26 + 28 + 30 + 32 = 182 \]

Now, we find the new average:
\[ \text{New Average} = \frac{\text{New Sum}}{\text{Count}} = \frac{182}{7} = 26 \]

### Final Answer
The new average of the 7 even numbers is **26**.

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