Fractional power analogy: “1/9 : 1/81 :: 1/13 : ?” — observe the first pair and apply the same rule to find the missing fraction in the second pair.

Introduction / Context:
Analogy problems on fractions often hinge on recognizing exponent or power transformations. In the pair “1/9 : 1/81,” the denominator 9 becomes 81, which is 9^2. The same squaring rule should be consistently applied to the second pair.

Given Data / Assumptions:

• First pair: 1/9 maps to 1/81.
• Transformation: denominator is squared (9 → 9^2 = 81).
• Apply the same to 1/13: denominator 13 squared equals 169.

Concept / Approach:
Identify that 1/a → 1/a^2. This preserves the structure of a unit fraction while altering the denominator by the same power operation.

Step-by-Step Solution:
1) Recognize 81 = 9^2 in the exemplar pair.2) Apply the same to the new denominator: 13^2 = 169.3) Therefore 1/13 maps to 1/169.

Verification / Alternative check:
You can verify quickly by taking square roots backward: sqrt(81) = 9 matches the original denominator; likewise sqrt(169) = 13, confirming the consistent rule.

Why Other Options Are Wrong:

• 1/127, 1/125, 1/120, 1/196: None equals 1/13^2.

Common Pitfalls:
Confusing squaring the whole fraction (which would give 1/81 from 1/9) with squaring numerator/denominator separately in inconsistent ways. Here only the denominator pattern matters because the numerator stays 1.

Final Answer:
1/169