The sum of the average of three consecutive even numbers and three consecutive odd numbers is 31 If the largest odd number is 21 . What is the smallest even number?

Options

A. 8
B. 12
C. 6
D. None of these

Answer

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

### Step 1: Define the Variables
Let the smallest even number be \( x \). Then the three consecutive even numbers can be represented as:
- \( x \) (the smallest even number)
- \( x + 2 \) (the next even number)
- \( x + 4 \) (the largest even number)

Let the largest odd number be \( y + 4 \) (since the largest odd number is given as 21). Therefore, the three consecutive odd numbers can be represented as:
- \( y \) (the smallest odd number)
- \( y + 2 \) (the next odd number)
- \( y + 4 \) (the largest odd number)

### Step 2: Calculate the Averages
The average of the three consecutive even numbers is:
\[ \text{Average of even numbers} = \frac{x + (x + 2) + (x + 4)}{3} = \frac{3x + 6}{3} = x + 2 \]

The average of the three consecutive odd numbers is:
\[ \text{Average of odd numbers} = \frac{y + (y + 2) + (y + 4)}{3} = \frac{3y + 6}{3} = y + 2 \]

### Step 3: Set Up the Equation
According to the problem, the sum of these averages is equal to 31:
\[ (x + 2) + (y + 2) = 31 \]
Simplifying this gives:
\[ x + y + 4 = 31 \]
Thus, we can rearrange it to find:
\[ x + y = 27 \]

### Step 4: Use the Information about the Odd Number
We know that the largest odd number is 21, which can be expressed as:
\[ y + 4 = 21 \]
From this, we can find \( y \):
\[ y = 21 - 4 = 17 \]

### Step 5: Substitute \( y \) Back into the Equation
Now we substitute \( y = 17 \) into the equation \( x + y = 27 \):
\[ x + 17 = 27 \]
Solving for \( x \):
\[ x = 27 - 17 = 10 \]

### Step 6: Conclusion
The smallest even number is:
\[ \text{Smallest even number} = x = 10 \]