Select the option that is related to the third term in the same way as the…
2023
Select the option that is related to the third term in the same way as the second term is related to the first term and the sixth term is related to the fifth term.
5 : 127 :: 9 : ? :: 7 : 247
Answer: C. 407 — Concept: In a number analogy of the form a : b, one arithmetic rule must carry the left member of a pair to its right member, and the very same rule must work…
- A.
470
- B.
417
- C.
407
- D.
471
Attempted by 1 students.
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Correct answer: C
Concept: In a number analogy of the form a : b, one arithmetic rule must carry the left member of a pair to its right member, and the very same rule must work for every pair the question supplies. A candidate rule may be accepted only when it reproduces all of the supplied pairs; reproducing a single pair is never enough.
Application: This question supplies two complete pairs, 5 : 127 and 7 : 247. Test the rule n → 5n2 + 2, that is, five times the square of the number, plus two, against both of them.
For n = 5: 52 = 25, then 25 × 5 = 125, and 125 + 2 = 127, which reproduces the first supplied pair.
For n = 7: 72 = 49, then 49 × 5 = 245, and 245 + 2 = 247, which reproduces the second supplied pair.
The rule survives both checks, so apply it to the third term. For n = 9: 92 = 81, then 81 × 5 = 405, and 405 + 2 = 407.
Cross-check: Two readings reproduce the first supplied pair 5 : 127 without being the operative rule: the cube reading n3 + 2 and the linear reading 25n + 2. Compare all three candidate rules across both supplied pairs.
Candidate rule | Value at n = 5 | Value at n = 7 |
|---|---|---|
5n2 + 2 | 127 | 247 |
n3 + 2 | 127 | 345 |
25n + 2 | 127 | 177 |
Both trap readings break on the second supplied pair: the cube reading gives 73 + 2 = 345 and the linear reading gives 25 × 7 + 2 = 177, while the question supplies 7 : 247, so both are discarded. Only n → 5n2 + 2 reproduces both supplied pairs, and it gives 5 × 92 + 2 = 407 for the missing term.