Step-by-step Solution:

Correct Answer: D (4:2:3)

To find the ratio $x:y:z$, we can create a system of equations from the given proportions and solve for the relationships between the variables.

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Step 1: Set up Equations from the Proportions

The given relation is:

$$ \frac{y}{x-z} = \frac{x+y}{z} = \frac{x}{y} $$

We can create two separate equations by pairing up the ratios:

1. From $\frac{x+y}{z} = \frac{x}{y}$, we can cross-multiply to get:
   $y(x+y) = zx \implies xy + y^2 = xz$
2. From $\frac{y}{x-z} = \frac{x}{y}$, we can cross-multiply to get:
   $y^2 = x(x-z) \implies y^2 = x^2 - xz$

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Step 2: Find the Relationship Between x and y

We now have two expressions for $xz$. Let's set them equal to each other:

$xy + y^2 = x^2 - y^2$

Rearrange this into a homogeneous quadratic equation:

$x^2 - xy - 2y^2 = 0$

To find the ratio of $x$ to $y$, we can divide the entire equation by $y^2$ (assuming $y \neq 0$):

$$ \left(\frac{x}{y}\right)^2 - \left(\frac{x}{y}\right) - 2 = 0 $$

Let $k = \frac{x}{y}$. The equation becomes $k^2 - k - 2 = 0$. Factoring this quadratic gives:

$(k-2)(k+1) = 0$

This yields two possible values for the ratio $k$: $k=2$ or $k=-1$. Assuming the variables are positive (as is common in ratio problems), we take the positive solution, $k=2$.

So, $\frac{x}{y} = 2$, which means $x=2y$.

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Step 3: Find z in Terms of y

Now that we know $x=2y$, we can use one of the original ratios to find $z$. The common value of the ratios is $k = \frac{x}{y} = 2$.

Let's use the ratio $\frac{x+y}{z} = k = 2$:

$$ \frac{x+y}{z} = 2 $$

Substitute $x=2y$:

$$ \frac{2y+y}{z} = 2 \implies \frac{3y}{z} = 2 $$

$3y = 2z \implies z = \frac{3}{2}y$

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Step 4: Determine the Final Ratio

We now have all three variables expressed in terms of $y$:

• $x = 2y$
• $y = y$
• $z = \frac{3}{2}y$

The ratio $x:y:z$ is therefore:

$$ 2y : y : \frac{3}{2}y $$

To express this with integers, we can divide by $y$ and then multiply all parts by 2:

$2 : 1 : \frac{3}{2} \implies 4 : 2 : 3$