Fraction Analogy — “1/9 : 1/81 :: 3/13 : ?”. Detect the pattern that maps the first fraction to the second and apply it to 3/13.

Introduction / Context:
Fraction analogies often involve transformations on the numerator, denominator, or both. The pair “1/9 : 1/81” suggests a rule that holds the numerator fixed at 1 while transforming the denominator in a specific way. We must discover that transformation and then apply it to “3/13.”

Given Data / Assumptions:

• The first numerator is 1 and remains 1 in the mapped fraction.
• The denominator changes from 9 to 81.
• We seek a consistent transformation that we can apply when the left fraction is 3/13.

Concept / Approach:
Notice 81 = 9^2. A clean rule is: map a/b → 1/b^2 (fixing the numerator at 1 and squaring the denominator). This reproduces 1/9 → 1/81 exactly. Applying the same rule to 3/13 gives 1/13^2 = 1/169.

Step-by-Step Solution:

Verification / Alternative check:
Alternative interpretations like dividing by 9 (1/9 → 1/81 because (1/9)/9 = 1/81) fail for 3/13 because (3/13)/9 = 3/117 (not among standard simplest forms and generally not offered). The denominator-square rule works perfectly and yields an offered option.

Why Other Options Are Wrong:

Common Pitfalls:
Overfitting a single arithmetic tweak (like division by a constant) when a clean structural transformation (denominator squared) more directly explains the example.

Final Answer:
1/169