The average of six number is x and average of three of these is y. If the average of the remaining three is z,then

Options

A. `x=y+z`
B. `2x=y+z`
C. `x=2y+2z`
D. `x=y+2z`

Answer

To solve the problem step by step, we need to find the relationship between the averages of the six numbers, three of those numbers, and the remaining three numbers.

### Step 1: Understand the given averages
We are given:
- 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: Calculate the sum of the six numbers
The average of six numbers is given by:
\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of numbers}} \]
Thus, we can express the sum of the six numbers as:
\[ \text{Sum of six numbers} = 6 \times x = 6x \]

### Step 3: Calculate the sum of the first three numbers
The average of the first three numbers is \( y \). Therefore, the sum of these three numbers can be calculated as:
\[ \text{Sum of first three numbers} = 3 \times y = 3y \]

### Step 4: Calculate the sum of the remaining three numbers
The average of the remaining three numbers is \( z \). Thus, the sum of these three numbers is:
\[ \text{Sum of remaining three numbers} = 3 \times z = 3z \]

### Step 5: Set up the equation
Since the sum of the six numbers is equal to the sum of the first three numbers plus the sum of the remaining three numbers, we can write:
\[ 6x = 3y + 3z \]

### Step 6: Simplify the equation
To simplify the equation, we can divide every term by 3:
\[ 2x = y + z \]

### Conclusion
Thus, we arrive at the relationship:
\[ 2x = y + z \]