Difficulty: Medium
Correct Answer: x : 9 = 4 : y
Explanation:
Introduction / Context:
Proportional equations can encode multiplicative relationships compactly. Here we are told that x·y = 36, and we must identify which given proportion necessarily follows from that fact for any positive x and y satisfying the product constraint.
Given Data / Assumptions:
Concept / Approach:
Translate each option into an equality of fractions and cross multiply to see what condition it encodes. The correct choice will be the one that is equivalent to x·y = 36 for all positive solutions, without adding extra restrictions.
Step-by-Step Solution:
Option A: x : 9 = 4 : y ⇒ x/9 = 4/y ⇒ x*y = 36 (after cross multiplication x*y = 9*4 ). This is identically the given constraint. Option B: 9 : x = 4 : y ⇒ 9/x = 4/y ⇒ 9y = 4x ⇒ With y = 36/x, we get 9*(36/x) = 4x ⇒ 324/x = 4x ⇒ x^2 = 81 ⇒ x = 9 (or −9), which adds an extra restriction and is not always true. Option C: x : 17 = y : 7 ⇒ x/17 = y/7 ⇒ 7x = 17y; using y = 36/x typically fails except for one special pair. Option D: x : 6 = y : 6 ⇒ x/6 = y/6 ⇒ x = y, which forces x = y = 6 under xy = 36, not general. Option E: x : 4 = 9 : y ⇒ x/4 = 9/y ⇒ x*y = 36 (same as A). Though algebraically equivalent, the question expects a single best match; Option A is the standard presentation given the set.
Verification / Alternative check:
Pick x = 12, y = 3. Then A gives 12 : 9 = 4 : 3 → 4/3 = 4/3 (true). B gives 9/12 = 4/3 → 3/4 = 4/3 (false). C gives 12/17 = 3/7 (false). D gives 12/6 = 3/6 → 2 = 0.5 (false). E gives 12/4 = 9/3 → 3 = 3 (true but redundant with A; choose A).
Why Other Options Are Wrong:
They imply extra constraints beyond xy = 36 (equality of x and y, or a specific value of x), or are simply inconsistent with many valid pairs (x, y) satisfying the product.
Common Pitfalls:
Not translating proportions to fractional equalities and neglecting to cross multiply; always test with a generic pair satisfying the product.
Final Answer:
x : 9 = 4 : y
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