The given equation is 2x = 3y = 4z, and we need to determine the ratio of x, y, and z.

Step 1: Solve for x, y, and z
Since 2x = 3y = 4z, we can equate any two of these expressions and solve for x, y, and z.

Let's equate 2x and 3y:
2x = 3y
Dividing both sides by 2:
x = (3/2)y

Now, let's equate 3y and 4z:
3y = 4z
Dividing both sides by 3:
y = (4/3)z

Step 2: Express the variables in terms of a common variable
To find the ratio of x, y, and z, we need to express them in terms of a common variable.

Let's express y in terms of z:
y = (4/3)z

Now, let's substitute this value of y into the equation x = (3/2)y:
x = (3/2)(4/3)z
x = 2z

Step 3: Determine the ratio of x, y, and z
Now that we have expressed x, y, and z in terms of z, we can determine the ratio.

Let's take z as a common variable and express x and y in terms of z:
x = 2z
y = (4/3)z

The ratio of x, y, and z is:
x : y : z = 2z : (4/3)z : z

Simplifying the expression:
x : y : z = 6z : 4z : 3z

Step 4: Simplify the ratio
To simplify the ratio, we can divide all the terms by the greatest common divisor, which is z.

Dividing each term by z:
x : y : z = 6 : 4 : 3

Therefore, the ratio of x : y : z is 6 : 4 : 3.

Conclusion
The ratio of x : y : z is 6 : 4 : 3, which means that for every 6 units of x, there are 4 units of y and 3 units of z.