If `2a+b=7 and b+2c=23`, what is the average of a, b, and c?

Options

A. `5`
B. `7.5`
C. `15`
D. `12.25`

Answer

To solve the problem step by step, we will start with the given equations and work towards finding the average of \( a \), \( b \), and \( c \).

### Step 1: Write down the equations
We have two equations:
1. \( 2a + b = 7 \) (Equation 1)
2. \( b + 2c = 23 \) (Equation 2)

### Step 2: Add the two equations
We will add Equation 1 and Equation 2 together:
\[ (2a + b) + (b + 2c) = 7 + 23 \]
This simplifies to:
\[ 2a + 2b + 2c = 30 \]

### Step 3: Simplify the equation
We can factor out the 2 from the left-hand side:
\[ 2(a + b + c) = 30 \]
Now, divide both sides by 2:
\[ a + b + c = 15 \quad \text{(Equation 3)} \]

### Step 4: Calculate the average
The average of \( a \), \( b \), and \( c \) is given by the formula:
\[ \text{Average} = \frac{a + b + c}{3} \]
Substituting the value from Equation 3:
\[ \text{Average} = \frac{15}{3} = 5 \]

### Conclusion
Thus, the average of \( a \), \( b \), and \( c \) is \( 5 \).

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