Select the option that is related to the third number in the same way as the…

Concept: In a number analogy a : b :: c : ?, the second number must be produced from the first by one fixed arithmetic relation, and that very same relation must then produce the answer from the third number. A relation qualifies only if it is a property of the numbers themselves — one that holds exactly, with no remainder and no dependence on how the numbers happen to be written down. The relation of this family used here is: the first number is one less than the square of the second — equivalently, the second number is the square root of (first number + 1).

Application:

1. Confirm the relation on the given pair: 80 + 1 = 81 = 92, so 9 is the square root of (80 + 1).

2. Apply the identical relation to the third number: 120 + 1 = 121 = 112, so the number that completes the analogy is 11.

Cross-check: Reverse the relation as an independent check: 112 − 1 = 121 − 1 = 120, exactly the third number given. The identity holds in both directions and for both pairs, with no approximation.

Why the other numbers fail:

• 12 is just 120 with its trailing zero stripped off (120 ÷ 10). The same step on the given pair gives 80 ÷ 10 = 8, not 9, so it does not even reproduce the pair you were given.

• 10 does not satisfy the relation: 102 − 1 = 99, not 120.

• 13 does not satisfy the relation either: 132 − 1 = 168, not 120. It arises from a different reading, addressed next.

Why not 13: 120 ÷ 10 + 1 does also equal 13, and those same two steps genuinely do turn 80 into 9 — this is a fair observation and it deserves a real answer. To be clear, the rule 'divide by 10, then add 1' is a valid exact rule; it is not mathematically wrong. The point is that it is not the relation this analogy is built on, and there is concrete evidence for that. First, the two readings are not really in competition on the evidence given: the square root of (a + 1) and a ÷ 10 + 1 are equal at exactly one positive value, a = 80, and at no other number at all. The example pair is precisely that single point of coincidence, which is the only reason the divide-by-ten reading survives it. Second, the third term settles it. The third number is 120, and 120 + 1 = 121 = 112 lands exactly on a perfect square. Under the square-minus-one reading that exactness is the whole design; under the divide-by-ten reading it would be sheer luck. Third, x : x2 − 1 is a standard, widely used number-analogy family, and the pair 9 : 80 appears on exactly this identity in published reasoning material, whereas 'divide by ten, add one' is not an established analogy relation. Taken together, the relation being tested is the exact square-minus-one identity, so the completion is 11 rather than 13.

Correct answer: 11