If y divided by (x minus z) equals (y plus x) divided by z, and also equals x divided by y, that is y/(x - z) = (y + x)/z = x/y, then what is the ratio z : y : x?

Introduction / Context:
This algebraic ratio question involves three expressions all equal to the same common value. The given condition y/(x - z) = (y + x)/z = x/y links the three variables x, y and z. The aim is to determine the ratio z : y : x. This is more challenging than direct ratio problems because it requires using algebraic manipulation and reasoning with a common parameter to express each variable in terms of another.

Given Data / Assumptions:
- y/(x - z) = (y + x)/z = x/y.
- All denominators are assumed to be non zero so the expressions are valid.
- The equality holds for some positive common value.
- We are asked to find the ratio z : y : x in simplest integer form.

Concept / Approach:
Let the common value of all three expressions be k. Then we have three equations: x / y = k, so x = k y. Also, (y + x) / z = k so y + x = k z, and y / (x - z) = k so y = k (x - z). Substituting x in terms of y into the other equations allows us to solve for z in terms of y, and then we can express the ratio z : y : x in whole numbers by eliminating k.

Step-by-Step Solution:
Step 1: Let x / y = k, so x = k y. Step 2: From (y + x) / z = k, we have y + x = k z. Step 3: Substitute x = k y into Step 2 to get y + k y = k z, so y(1 + k) = k z. Step 4: Hence z = y(1 + k) / k. Step 5: From y / (x - z) = k, we have y = k (x - z). Step 6: Substitute x = k y and z from Step 4 into y = k (x - z). Step 7: This gives y = k (k y - y(1 + k) / k) which simplifies to y = k^2 y - (1 + k) y. Step 8: Rearranging gives y(k^2 - 1) = y(1 + k), so k^2 - 1 = 1 + k. Step 9: Solve k^2 - k - 2 = 0, which factors as (k - 2)(k + 1) = 0, so k = 2 (positive solution). Step 10: With k = 2, x = 2y and z = y(1 + 2) / 2 = 3y/2. Step 11: Write z : y : x = (3y/2) : y : 2y = 3 : 2 : 4.

Verification / Alternative check:
Take a convenient value such as y = 2 units. Then x = 2y = 4, and z = 3y/2 = 3. Substitute into the original expressions: y/(x - z) = 2/(4 - 3) = 2, (y + x)/z = (2 + 4)/3 = 6/3 = 2, and x/y = 4/2 = 2. All three equal 2, confirming that the ratio z : y : x = 3 : 2 : 4 satisfies the given condition.

Why Other Options Are Wrong:
1 : 2 : 3, 4 : 3 : 2 and 2 : 3 : 4 do not satisfy the system when substituted back into the expressions. For each of these, if you pick a value for y and compute x and z according to the ratio, at least one of the three fractions y/(x - z), (y + x)/z and x/y will differ from the others. Therefore, they cannot represent a consistent solution to the given equation.

Common Pitfalls:
A common mistake is to assume that each ratio can be treated independently without considering the shared common value. Another pitfall is algebraic manipulation errors when substituting and simplifying expressions involving k, especially when clearing denominators. Learners may also incorrectly cancel terms or forget that y, x and z must remain positive to interpret a meaningful ratio.

Final Answer:
The required ratio of the three variables is z : y : x = 3 : 2 : 4.