The averge of x and 3y is 12, the average of 2x and 3z is 21. What is the average of x,y and z ?

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.
The average of \( x \) and \( 3y \) is 12. This can be expressed mathematically as:
\[ \frac{x + 3y}{2} = 12 \]
Multiplying both sides by 2 gives:
\[ x + 3y = 24 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation.
The average of \( 2x \) and \( 3z \) is 21. This can be expressed as:
\[ \frac{2x + 3z}{2} = 21 \]
Multiplying both sides by 2 gives:
\[ 2x + 3z = 42 \quad \text{(Equation 2)} \]

### Step 3: Add both equations.
Now, we will add Equation 1 and Equation 2:
\[ (x + 3y) + (2x + 3z) = 24 + 42 \]
This simplifies to:
\[ 3x + 3y + 3z = 66 \]

### Step 4: Factor out the common term.
We can factor out 3 from the left side:
\[ 3(x + y + z) = 66 \]

### Step 5: Solve for \( x + y + z \).
Dividing both sides by 3 gives:
\[ x + y + z = 22 \]

### Step 6: Find the average of \( x, y, z \).
To find the average of \( x, y, z \), we divide the sum by 3:
\[ \text{Average} = \frac{x + y + z}{3} = \frac{22}{3} \]

Thus, the average of \( x, y, z \) is:
\[ \frac{22}{3} \]

### Final Answer:
The average of \( x, y, z \) is \( \frac{22}{3} \).

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