If `(x -y +z) : (y - z + 2w) : (2x + z-w) =2: 3:5,` then `(3x + 3z - 2w): w=?`

Options

A. `7:1`
B. `4:3`
C. `9:1`
D. `8:1`

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equations based on the given ratios.
We have the ratios:
\[ (x - y + z) : (y - z + 2w) : (2x + z - w) = 2 : 3 : 5 \]
Let’s denote:
\[ x - y + z = 2k, \quad y - z + 2w = 3k, \quad 2x + z - w = 5k \]

### Step 2: Write the equations based on the ratios.
From the ratios, we can write:
1. \( x - y + z = 2k \) (Equation 1)
2. \( y - z + 2w = 3k \) (Equation 2)
3. \( 2x + z - w = 5k \) (Equation 3)

### Step 3: Rearrange the equations.
Now, we will rearrange each equation to express \( x \), \( y \), \( z \), and \( w \) in terms of \( k \).

From Equation 1:
\[ x = 2k + y - z \quad (1) \]

From Equation 2:
\[ y = 3k + z - 2w \quad (2) \]

From Equation 3:
\[ 2x = 5k - z + w \quad (3) \]

### Step 4: Substitute and simplify.
Now, we will substitute \( y \) from Equation (2) into Equation (1):
\[ x = 2k + (3k + z - 2w) - z \]
This simplifies to:
\[ x = 5k - 2w \]

Now substitute \( x \) into Equation (3):
\[ 2(5k - 2w) + z - w = 5k \]
This simplifies to:
\[ 10k - 4w + z - w = 5k \]
Rearranging gives:
\[ z = 5k - 10k + 5w = -5k + 5w \]

### Step 5: Substitute \( z \) back into the equations.
Now we substitute \( z \) back into the expression for \( x \):
\[ x = 5k - 2w \]

### Step 6: Find \( 3x + 3z - 2w \).
Now we need to find \( 3x + 3z - 2w \):
\[ 3x + 3z - 2w = 3(5k - 2w) + 3(-5k + 5w) - 2w \]
This simplifies to:
\[ = 15k - 6w - 15k + 15w - 2w \]
Combining like terms gives:
\[ = 7w \]

### Step 7: Find the ratio \( (3x + 3z - 2w) : w \).
Now we can find the ratio:
\[ (3x + 3z - 2w) : w = 7w : w = 7 : 1 \]

### Final Answer:
Thus, the value of \( (3x + 3z - 2w) : w \) is \( 7 : 1 \).

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