Let x and y be positive numbers related by x = y - (50 / y). If the value of y is doubled in this equation, how will the value of x change?

Introduction / Context:
This problem tests your understanding of how changing one variable affects another when the two are related through a non linear expression. The relationship x = y - (50 / y) involves both y and its reciprocal. You are asked to predict what happens to x when y is doubled, without substituting any particular numerical value for y. This type of question is common in algebra and data sufficiency style problems.

Given Data / Assumptions:

- x and y are both greater than 0. - They satisfy the equation x = y - (50 / y). - We consider what happens when y is replaced by 2y. - We must compare the new value of x with its original value.

Concept / Approach:
Instead of guessing or plugging in many sample values, we can derive a general expression for the new value of x after doubling y. Then we compare it algebraically to the original x. We will find the difference and see whether it is positive, zero, or negative, and then also compare the new x with 2x to determine whether the increase is more than double. Working symbolically keeps the reasoning valid for all positive y instead of just specific numerical examples.

Step-by-Step Solution:
Step 1: Original relation is x = y - 50 / y. Step 2: After doubling y, the new value of y is 2y. Step 3: Substitute 2y into the expression to get the new x, call it x'. Step 4: Compute x': x' = 2y - 50 / (2y) = 2y - 25 / y. Step 5: Write the original x alongside: x = y - 50 / y. Step 6: Find the change in x by subtracting: x' - x = (2y - 25 / y) - (y - 50 / y). Step 7: Simplify: x' - x = 2y - 25 / y - y + 50 / y = y + 25 / y. Step 8: Since y is positive, both y and 25 / y are positive, so y + 25 / y is greater than 0. Step 9: Therefore x' - x > 0, which means x increases when y is doubled. Step 10: Now compare x' with 2x. Compute 2x = 2y - 100 / y. Step 11: Compare x' and 2x: x' - 2x = (2y - 25 / y) - (2y - 100 / y) = 75 / y. Step 12: Since y is positive, 75 / y is greater than 0, so x' > 2x; the new x is more than double the old x.

Verification / Alternative check:
To build intuition, try a specific positive value like y = 5. Then x = 5 - 50 / 5 = 5 - 10 = -5. Doubling y to 10 gives x' = 10 - 50 / 10 = 10 - 5 = 5. Here x changed from -5 to 5, which is an increase of 10, clearly more than double. Trying another value, say y = 10, confirms the pattern: x = 10 - 5 = 5, while x' = 20 - 2.5 = 17.5, which is again more than twice 5. This matches the algebraic result that x' > 2x for all positive y.

Why Other Options Are Wrong:
The value of x clearly does not decrease, because x' - x = y + 25 / y is positive, so option A is incorrect. It does not stay the same since the difference is not zero. Option C claiming an increase four fold would require x' = 4x, which is not supported by the general formula. Only option D, that x increases to more than double, matches the inequality x' > 2x proved above.

Common Pitfalls:
A common mistake is to think that doubling y will simply double x, because students expect a linear relationship. Here the presence of the reciprocal term 50 / y makes the relationship non linear. Another error is to test only one example and then generalize without algebraic proof; depending on the chosen value, this may give limited insight. Systematically deriving x' and comparing it with x and 2x is the most reliable method.

Final Answer:
When y is doubled, the value of x increases to more than double its original value, which corresponds to option D.