If `(x-y+z):(y-z+2w):(2x+z-w)=3:4:7`, then `(2x+2z-2w):w=?`

Options

A. `7:1`
B. `4:3`
C. `4:1`
D. `8:1`

Answer

To solve the problem step by step, we start with the given ratio:

\[ (x - y + z) : (y - z + 2w) : (2x + z - w) = 3 : 4 : 7 \]

### Step 1: Assign variables based on the ratio
We can express the three parts of the ratio in terms of a variable \( k \):
- \( x - y + z = 3k \) (Equation 1)
- \( y - z + 2w = 4k \) (Equation 2)
- \( 2x + z - w = 7k \) (Equation 3)

### Step 2: Add Equation 1 and Equation 2
Now, we add Equation 1 and Equation 2:
\[ (x - y + z) + (y - z + 2w) = 3k + 4k \]
This simplifies to:
\[ x + 2w = 7k \]
(Equation 4)

### Step 3: Set Equation 3 equal to Equation 4
Since both Equation 4 and Equation 3 equal \( 7k \), we can set them equal to each other:
\[ 2x + z - w = x + 2w \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 2x - x + z = 2w + w \]
This simplifies to:
\[ x + z = 3w \]
(Equation 5)

### Step 5: Find \( 2x + 2z - 2w \)
Now, we need to find the ratio of \( (2x + 2z - 2w) : w \).
We can express \( 2x + 2z - 2w \) in terms of \( w \):
\[ 2x + 2z - 2w = 2(x + z) - 2w \]
Substituting from Equation 5:
\[ = 2(3w) - 2w = 6w - 2w = 4w \]

### Step 6: Form the ratio
Now we can form the ratio:
\[ (2x + 2z - 2w) : w = 4w : w \]
This simplifies to:
\[ 4 : 1 \]

### Final Answer
Thus, the ratio \( (2x + 2z - 2w) : w \) is \( 4 : 1 \).

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