If the average of a, b, c and d is the average of b and c, then which one of the following is necessarily true?

Options

A. `a+d=b+c`
B. `a+b=c+d`
C. `a-d=b-c`
D. `(a+d)/(4)=(b+c)/(2)`

Explanation

To solve the problem, we need to analyze the given condition: the average of \( a, b, c, \) and \( d \) is equal to the average of \( b \) and \( c \).

### Step-by-Step Solution:

1. **Write the formula for the average of \( a, b, c, \) and \( d \)**:
\[ \text{Average of } a, b, c, d = \frac{a + b + c + d}{4} \]

2. **Write the formula for the average of \( b \) and \( c \)**:
\[ \text{Average of } b, c = \frac{b + c}{2} \]

3. **Set the two averages equal to each other**:
\[ \frac{a + b + c + d}{4} = \frac{b + c}{2} \]

4. **Clear the denominators by multiplying both sides by 4**:
\[ a + b + c + d = 2(b + c) \]

5. **Expand the right side**:
\[ a + b + c + d = 2b + 2c \]

6. **Rearrange the equation to isolate \( a + d \)**:
\[ a + d = 2b + 2c - b - c \]
\[ a + d = b + c \]

7. **Conclusion**:
The relationship derived from the condition is:
\[ a + d = b + c \]
Therefore, it is necessarily true that \( a + d \) is equal to \( b + c \).