The average of a, b, c, and d is p. If the average of a and c is q, what is the average of b and d in terms of p and q?

Options

A. `2p+q`
B. `2p-q`
C. `2q+q`
D. `2q-p`

Answer

To solve the problem step by step, we will derive the average of \( b \) and \( d \) in terms of \( p \) and \( q \).

### Step 1: Understand the definition of average
The average of a set of numbers is defined as the sum of the numbers divided by the count of the numbers.

### Step 2: Set up the equation for the average of \( a, b, c, \) and \( d \)
Given that the average of \( a, b, c, \) and \( d \) is \( p \), we can write:
\[ \frac{a + b + c + d}{4} = p \]
Multiplying both sides by 4 gives:
\[ a + b + c + d = 4p \tag{1} \]

### Step 3: Set up the equation for the average of \( a \) and \( c \)
We are also given that the average of \( a \) and \( c \) is \( q \). Thus, we can write:
\[ \frac{a + c}{2} = q \]
Multiplying both sides by 2 gives:
\[ a + c = 2q \tag{2} \]

### Step 4: Find the sum of \( b \) and \( d \)
To find the average of \( b \) and \( d \), we first need to find \( b + d \). We can use equations (1) and (2):
From equation (1):
\[ a + b + c + d = 4p \]
Substituting equation (2) into this:
\[ (2q) + b + d = 4p \]
Now, rearranging gives:
\[ b + d = 4p - 2q \tag{3} \]

### Step 5: Calculate the average of \( b \) and \( d \)
The average of \( b \) and \( d \) is given by:
\[ \text{Average of } b \text{ and } d = \frac{b + d}{2} \]
Substituting equation (3) into this:
\[ \text{Average of } b \text{ and } d = \frac{4p - 2q}{2} \]
Simplifying this gives:
\[ \text{Average of } b \text{ and } d = 2p - q \]

### Final Answer
Thus, the average of \( b \) and \( d \) in terms of \( p \) and \( q \) is:
\[ \boxed{2p - q} \]