Complete the number analogy: 12 : 144 :: 20 : ?

Introduction / Context:
This problem is a straightforward number analogy based on squaring. The pair 12 : 144 shows a clear mathematical relationship. You must identify that relationship and then pick from the options the pair that matches the same pattern. It tests recognition of basic arithmetic operations, especially squares.

Given Data / Assumptions:
- The first pair is 12 and 144.
- 144 is a well known perfect square of 12.
- The correct option must show another pair where the second number is the square of the first number.
- Options are 10 : 40, 15 : 125, 20 : 400, and 22 : 464.

Concept / Approach:
Check for common power relationships. Since 144 equals 12 squared, the relationship appears to be of the form b = a^2. Therefore, the correct pair should have the second number as the square of the first. We will check each option until we find the one that satisfies this rule exactly.

Step-by-Step Solution:
Step 1: Confirm the pattern for the initial pair. Compute 12^2 = 144, so the rule is "square the first number to get the second." Step 2: Check option A 10 : 40. Compute 10^2 = 100, not 40, so this option does not fit. Step 3: Check option B 15 : 125. Compute 15^2 = 225, which is different from 125, so this option fails too. Step 4: Check option C 20 : 400. Compute 20^2 = 400. This matches perfectly, so this pair satisfies the same rule as the given one. Step 5: Check option D 22 : 464 only for completeness. Compute 22^2 = 484, not 464, so it does not fit the pattern. Step 6: Therefore, the pair 20 : 400 is the correct completion of the analogy.

Verification / Alternative check:
The rule b = a^2 is simple and consistent. It holds exactly for 12 and 144 and for 20 and 400. None of the other options match this relationship. They might be close in value, but analogy questions require exact equality, not approximate or near matches. Since there is no simpler or more natural pattern that explains all pairs, squaring the first number is confirmed as the intended rule.

Why Other Options Are Wrong:
Option A "10 : 40" would match a rule like multiplying by 4, but 12 is not multiplied by 4 to get 144, so this is inconsistent.
Option B "15 : 125" resembles the cube relation 5^3 = 125, but the first pair is based on a square, not a cube.
Option D "22 : 464" does not fit any simple square based rule; 22^2 is 484, not 464.

Common Pitfalls:
Learners sometimes see familiar numbers like 125 and guess a cube based rule without confirming whether it matches the original pair. Another mistake is to accept any simple multiplication, such as multiply by 4 or 5, when the actual relationship is more specific, like squaring. Always verify the exact operation on the first pair before applying it to options.

Final Answer:
The correct completed pair is 20 : 400.