In this number pair analogy, “3 is to 9 as ______ is to ______”. Choose the pair from the options that follows the same square relationship as 3 and 9.

Introduction / Context:
This number analogy is based on the operation of squaring. The pair “3 : 9” relates 3 to its square, because 3² = 9. We must find which of the given pairs exhibits the same relationship between the first number and the second. Recognising simple exponent patterns like squares is crucial in basic quantitative reasoning.

Given Data / Assumptions:

• Example pair: 3 → 9, where 9 = 3². • Options: 4 : 11, 5 : 31, 7 : 57, 7 : 49. • We assume the relationship is “second number = first number squared” unless evidence suggests otherwise.

Concept / Approach:
We test each option by squaring the first number and checking if the result matches the second number. For the correct analogy, the second term in the pair must be exactly equal to the square of the first term. Numbers like 11, 31, and 57 are not squares of small integers, while 49 is a well known square of 7, which suggests it may be the correct answer.

Step-by-Step Solution:
Step 1: Confirm the pattern in the example. 3² = 3 * 3 = 9; so the rule is “square the first number”. Step 2: Check each option. Option A: 4² = 16, not 11, so 4 : 11 does not follow the pattern. Option B: 5² = 25, not 31, so 5 : 31 is incorrect. Option C: 7² = 49, not 57, so 7 : 57 is also incorrect. Option D: 7² = 49, which matches exactly the second number in 7 : 49. Step 3: Therefore, 7 : 49 is the only pair that displays the same square relationship as 3 : 9.

Verification / Alternative check:
We can express the analogy as “a : a²”. The example is 3 : 3². For the correct option, we need some integer a such that the pair is a : a². Among the options, only a = 7 produces a² that is present as the second term (49). This reaffirms that 7 : 49 is the correct matching pair.

Why Other Options Are Wrong:
• 4 : 11 – 11 is not 4² (16), so the pattern fails. • 5 : 31 – 31 is not 5² (25); it also is not close to a square. • 7 : 57 – 57 is not 7² (49); there is no simple square relation.

Common Pitfalls:
Some candidates look for differences or other arithmetic operations instead of first checking the obvious possibility of a square. When a small integer is matched with a slightly larger number like 9, 16, or 25, always test for squared relationships. Recognising that 49 is 7² immediately points to 7 : 49 as the correct answer.

Final Answer:
The pair that correctly completes the analogy is 7 : 49.