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Difficulty: MediumAnalogy and Classification

Evaluate the mathematical logic governing the number pairs in Group X. Match each pair with its analog in Group Y that follows the exact same logical transformation.

  • 5:1245 : 1247:3427 : 342
  • 9:909 : 9011:13211 : 132
  • 27:8127 : 8134:4934 : 49
  • 14:4314 : 4319:5819 : 58

Answer

The correct pairings align identical underlying mathematical transformations: 5:1245:124 matches 7:3427:342 (cubes minus one), 9:909:90 matches 11:13211:132 (squares plus the number), 27:8127:81 matches 34:4934:49 (square of digit sum), and 14:4314:43 matches 19:5819:58 (multiplied by three plus one).
Each pair from Group X must be matched with a pair from Group Y that demonstrates the identical algebraic or numeric transformation rule. Solving these requires looking past simple differences and testing for squares, cubes, consecutive products, and digit manipulations.

Step-by-Step Solution

1
Analyze the first pair in Group X (5:1245 : 124) to determine its underlying logic.
The number 124124 is exactly 11 less than 125125 (535^3), revealing the rule n:n3−1n : n^3 - 1. Locating the same rule in Group Y yields 7:3427 : 342 since 73−1=3427^3 - 1 = 342.
Identifying the structural pattern involving powers allows for accurate cross-referencing with the target group.
2
Examine the second pair (9:909 : 90).
The number 9090 is the product of 99 and 1010, indicating the rule n:n(n+1)n : n(n+1). In Group Y, 11:13211 : 132 perfectly matches this logic because 11×12=13211 \times 12 = 132.
Testing consecutive multipliers is a standard method for identifying patterns in numeric classification.
3
Evaluate the third pair (27:8127 : 81).
While 8181 is a direct multiple (27×327 \times 3), no pair in Group Y shares this specific multiplier. Alternatively, summing the digits of 2727 gives 2+7=92+7=9, and 92=819^2=81. Applying this rule to Group Y isolates 34:4934 : 49, since (3+4)2=72=49(3+4)^2 = 7^2 = 49.
When standard arithmetic operations fail to yield a matching pair, analyzing digit-level operations is required.
4
Determine the rule for the final pair (14:4314 : 43).
The number 4343 is obtained via the operation 14×3+114 \times 3 + 1. The corresponding pair in Group Y is 19:5819 : 58, since 19×3+1=57+1=5819 \times 3 + 1 = 57 + 1 = 58.
Establishing a standard linear transformation finalizes the correct matching process.

Key Concept

Identifying multi-step mathematical operations, power series, and digit-based transformations to classify and match logical sequences.
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Topics

Analogy and Classification

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