The average of a, b, c, d, and e is 28. If the average of a, c, and e is 24, what is the average of b and d?

Options

A. `31`
B. `32`
C. `33`
D. `34`

Answer

To solve the problem step by step, we will follow the definitions of averages and manipulate the equations accordingly.

### Step 1: Understanding the average of a, b, c, d, and e
The average of five numbers \( a, b, c, d, e \) is given as 28. The formula for average is:

\[ \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \]

So, we can write:

\[ \frac{a + b + c + d + e}{5} = 28 \]

### Step 2: Calculate the sum of a, b, c, d, and e
To find the sum of \( a, b, c, d, e \), we multiply both sides of the equation by 5:

\[ a + b + c + d + e = 28 \times 5 \]

Calculating the right side:

\[ a + b + c + d + e = 140 \]

Let's call this **Equation 1**.

### Step 3: Understanding the average of a, c, and e
Next, we know that the average of \( a, c, e \) is 24. Using the same average formula:

\[ \frac{a + c + e}{3} = 24 \]

### Step 4: Calculate the sum of a, c, and e
To find the sum of \( a, c, e \), we multiply both sides of the equation by 3:

\[ a + c + e = 24 \times 3 \]

Calculating the right side:

\[ a + c + e = 72 \]

Let's call this **Equation 2**.

### Step 5: Finding the sum of b and d
Now, we want to find the sum of \( b \) and \( d \). We can do this by subtracting Equation 2 from Equation 1:

\[ (a + b + c + d + e) - (a + c + e) = 140 - 72 \]

This simplifies to:

\[ b + d = 68 \]

### Step 6: Finding the average of b and d
Now, we need to find the average of \( b \) and \( d \):

\[ \text{Average of } b \text{ and } d = \frac{b + d}{2} \]

Substituting the value we found for \( b + d \):

\[ \text{Average of } b \text{ and } d = \frac{68}{2} = 34 \]

### Final Answer
The average of \( b \) and \( d \) is **34**.

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