Analogy — Find the correct pair that follows the same rule as the first: 5 : 35 :: ?

Introduction / Context:
In this verbal–numeric analogy, the pair on the left establishes a transformation rule from the first number to the second. Your task is to test each option and select the one pair that obeys exactly the same rule as the model pair 5 : 35.

Given Data / Assumptions:

• Model pair is 5 mapped to 35.
• Answer options give several candidate mappings of the form a : b.
• We must preserve the identical numeric rule from 5 → 35 when checking each option.

Concept / Approach:
Start by inferring the simplest consistent rule that produces 35 from 5. The two most natural hypotheses are: multiplication by a fixed factor, or a compact arithmetic expression in a alone (for example a^2 + k or a * (a + c)). We then test whether any answer pair a : b satisfies the same rule.

Step-by-Step Solution:

Verification / Alternative check:
We tried three natural families of rules that exactly generate 35 from 5. None of the candidate pairs matches any of these consistent rules. Since the task is to preserve the same rule, and no option aligns with a valid rule that converts 5 to 35, all options A–D must be rejected.

Why Other Options Are Wrong:

• 7 : 77 does not equal 7 * 7, 7^2 + 10, or 7 * (7 + 2).
• 9 : 45 does not equal 9 * 7, 9^2 + 10, or 9 * (9 + 2).
• 11 : 55 does not equal 11 * 7, 11^2 + 10, or 11 * (11 + 2).
• 3 : 24 does not equal 3 * 7, 3^2 + 10, or 3 * (3 + 2).

Common Pitfalls:
Assuming a different rule for the options than the one used for the model pair. In analogies, the governing rule must remain identical across pairs; switching to a new rule for a different option invalidates the analogy.

Final Answer:
None of these