The sum of three consecutive even numbers is 14 less than one-fourth of 176. What is the middle number?

Options

A. 8
B. 10
C. 6
D. Data inadequate

Answer

To solve the problem step by step, we will first define the three consecutive even numbers and then set up the equation based on the information given in the question.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the first even number be \( x \). Since we are looking for three consecutive even numbers, the next two numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)

2. **Write the Sum of the Numbers**:
The sum of these three consecutive even numbers is:
\[ x + (x + 2) + (x + 4) = 3x + 6 \]

3. **Calculate One-Fourth of 176**:
We need to find one-fourth of 176:
\[ \frac{176}{4} = 44 \]

4. **Set Up the Equation**:
According to the problem, the sum of the three consecutive even numbers is 14 less than one-fourth of 176. Therefore, we can set up the equation:
\[ 3x + 6 = 44 - 14 \]

5. **Simplify the Right Side**:
Calculate \( 44 - 14 \):
\[ 44 - 14 = 30 \]
So, the equation now is:
\[ 3x + 6 = 30 \]

6. **Solve for \( x \)**:
Subtract 6 from both sides:
\[ 3x = 30 - 6 \]
\[ 3x = 24 \]
Now, divide by 3:
\[ x = \frac{24}{3} = 8 \]

7. **Find the Middle Number**:
The middle number of the three consecutive even numbers is:
\[ x + 2 = 8 + 2 = 10 \]

### Final Answer:
The middle number is **10**.