Translate a verbal condition into algebra: If the square of the sum of two numbers equals 4 times their product, find the ratio of the two numbers.

Introduction / Context:Verbal-to-algebra translation is a core skill. Here, the phrase “the square of the sum equals four times the product” suggests a direct identity that simplifies immediately.

Given Data / Assumptions:

• Let the numbers be x and y.
• (x + y)^2 = 4xy.
• Find the ratio x : y (with x, y real and not both zero).

Concept / Approach:Expand the left-hand side and bring all terms to one line. Recognize a perfect square on the left that forces equality only when x and y are equal.

Step-by-Step Solution:

Verification / Alternative check:Pick x = y = 5 (any equal pair). Then (x + y)^2 = 10^2 = 100 and 4xy = 4 * 25 = 100; condition holds, confirming the ratio.

Why Other Options Are Wrong:Ratios like 2 : 1, 1 : 3, 1 : 2, 3 : 1 would make (x − y)^2 > 0 and violate the equality.

Common Pitfalls:Confusing (x + y)^2 with x^2 + y^2 (forgetting the 2xy term), or not recognizing the perfect-square structure after rearranging.

Final Answer:1 : 1