If the average of x, y, and z is 32 and the average of y and z is 27. What is the average of x and 2x?

Options

A. `42`
B. `45`
C. `48`
D. `63`

Answer

To solve the problem step by step, we will use the information given about the averages of x, y, and z.

### Step 1: Set up the equations based on the averages given.
We know that:
1. The average of x, y, and z is 32.
2. The average of y and z is 27.

From the first statement, we can write the equation:
\[ \frac{x + y + z}{3} = 32 \]
Multiplying both sides by 3 gives:
\[ x + y + z = 96 \quad \text{(Equation 1)} \]

From the second statement, we can write the equation:
\[ \frac{y + z}{2} = 27 \]
Multiplying both sides by 2 gives:
\[ y + z = 54 \quad \text{(Equation 2)} \]

### Step 2: Substitute Equation 2 into Equation 1.
Now we will substitute the value of \(y + z\) from Equation 2 into Equation 1:
\[ x + (y + z) = 96 \]
Substituting \(y + z = 54\) into the equation gives:
\[ x + 54 = 96 \]

### Step 3: Solve for x.
Now, we will isolate x:
\[ x = 96 - 54 \]
Calculating this gives:
\[ x = 42 \]

### Step 4: Find the average of x and 2x.
Now we need to find the average of \(x\) and \(2x\):
\[ \text{Average} = \frac{x + 2x}{2} \]
This simplifies to:
\[ \text{Average} = \frac{3x}{2} \]

### Step 5: Substitute the value of x into the average formula.
Now we substitute \(x = 42\) into the average formula:
\[ \text{Average} = \frac{3 \times 42}{2} \]
Calculating this gives:
\[ \text{Average} = \frac{126}{2} = 63 \]

### Final Answer:
Thus, the average of \(x\) and \(2x\) is:
\[ \boxed{63} \]