The average of six numbers is X and the average of three of these is Y. the average of the remaining three is z, then

Options

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

Answer

To solve the problem step by step, we need to establish relationships between the averages of the numbers given in the question. Let's break it down:

### Step 1: Understand the 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: Express the sums in terms of averages
1. The sum of the six numbers can be expressed as:
\[ \text{Sum of six numbers} = 6X \]
2. The sum of the first three numbers (whose average is \( Y \)) can be expressed as:
\[ \text{Sum of first three numbers} = 3Y \]
3. The sum of the remaining three numbers (whose average is \( Z \)) can be expressed as:
\[ \text{Sum of remaining three numbers} = 3Z \]

### Step 3: Set up the equation
Since the sum of all 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 4: Simplify the equation
We can factor out the 3 from the right side:
\[ 6X = 3(Y + Z) \]
Now, divide both sides by 3:
\[ 2X = Y + Z \]

### Conclusion
From the equation \( 2X = Y + Z \), we can conclude that:
\[ X = \frac{Y + Z}{2} \]
This means that the correct option is:
- **Option B: \( 2X = Y + Z \)**