If `(y )/(x -z) = (y +x)/(z) = (x)/(y)` then `x :: y :z = ?`

Options

A. `3:2:4`
B. `4:3:2`
C. `4:2:3`
D. `1:2:3`

Answer

To solve the equation \( \frac{y}{x - z} = \frac{y + x}{z} = \frac{x}{y} \) and find the ratio \( x : y : z \), we can follow these steps:

### Step 1: Set the common ratio
Let \( k \) be the common ratio such that:
\[ \frac{y}{x - z} = k, \quad \frac{y + x}{z} = k, \quad \frac{x}{y} = k \]

### Step 2: Express \( y \), \( z \), and \( x \) in terms of \( k \)
From the first equation:
\[ y = k(x - z) \tag{1} \]

From the second equation:
\[ y + x = kz \implies y = kz - x \tag{2} \]

From the third equation:
\[ x = ky \tag{3} \]

### Step 3: Substitute \( y \) from equation (3) into equations (1) and (2)
Substituting \( y = ky \) into equation (1):
\[ ky = k(x - z) \implies y = x - z \tag{4} \]

Substituting \( y = ky \) into equation (2):
\[ ky + x = kz \implies ky + x = kz \tag{5} \]

### Step 4: Substitute \( y \) from equation (4) into equation (5)
From equation (4), we have \( y = x - z \):
\[ k(x - z) + x = kz \]
This simplifies to:
\[ kx - kz + x = kz \implies kx + x = 2kz \implies (k + 1)x = 2kz \tag{6} \]

### Step 5: Express \( z \) in terms of \( x \) and \( k \)
From equation (6):
\[ z = \frac{(k + 1)x}{2k} \tag{7} \]

### Step 6: Substitute \( z \) back into equation (4) to find \( y \)
Using equation (4):
\[ y = x - z = x - \frac{(k + 1)x}{2k} = x \left(1 - \frac{k + 1}{2k}\right) = x \left(\frac{2k - (k + 1)}{2k}\right) = x \left(\frac{k - 1}{2k}\right) \tag{8} \]

### Step 7: Now we have expressions for \( x \), \( y \), and \( z \)
From equations (3), (8), and (7):
- \( x = ky \)
- \( y = x \frac{k - 1}{2k} \)
- \( z = \frac{(k + 1)x}{2k} \)

### Step 8: Find the ratio \( x : y : z \)
Substituting \( k = 2 \) (as a trial):
- \( x = 2y \)
- \( y = \frac{1}{2}x \)
- \( z = \frac{3}{2}x \)

Thus, the ratio \( x : y : z \) becomes:
\[ x : y : z = 4 : 2 : 3 \]

### Final Answer
The ratio \( x : y : z = 4 : 2 : 3 \).