The average of six numbers is x and the average of three of these is y. If the average of the remaining three is z then,what is relation between x, y and z?

Answer

To find the relationship between the averages \( x \), \( y \), and \( z \) given in the problem, we can follow these steps:

### Step 1: Understand the given averages
We know:
- The average of six numbers is \( x \).
- The average of three of these numbers is \( y \).
- The average of the remaining three numbers is \( z \).

### Step 2: Express the total of the six numbers
The average of six numbers being \( x \) means:
\[ \text{Total of six numbers} = 6x \]

### Step 3: Express the total of the first three numbers
The average of the first three numbers being \( y \) means:
\[ \text{Total of first three numbers} = 3y \]

### Step 4: Express the total of the remaining three numbers
The average of the remaining three numbers being \( z \) means:
\[ \text{Total of remaining three numbers} = 3z \]

### Step 5: Set up the equation
Since the total of the six numbers is the sum of the totals of the two groups of three numbers, we can write:
\[ 6x = 3y + 3z \]

### Step 6: Simplify the equation
We can simplify this equation by dividing everything by 3:
\[ 2x = y + z \]

### Conclusion
Thus, the relationship between \( x \), \( y \), and \( z \) is:
\[ 2x = y + z \]