Question

Difficulty: Very hardAnalogy and Classification

Each of the following items presents a numerical analogy in the form A:B::C:XA : B :: C : X, where the missing term XX is derived using a multi-step arithmetic rule (n3kn^3 - k, n3nn^3 - n, n2+nn^2 + n, or n2nn^2 - n). Evaluate the value of XX for each analogy item, and arrange the items in ascending order based on their solved value of XX (from smallest value to largest value).

  1. 1Analogy 4: 11:110::13:X411 : 110 :: 13 : X_4
  2. 2Analogy 3: 12:156::14:X312 : 156 :: 14 : X_3
  3. 3Analogy 2: 6:210::8:X26 : 210 :: 8 : X_2
  4. 4Analogy 1: 7:340::9:X17 : 340 :: 9 : X_1

Answer

The correct ascending order based on the value of XX is Analogy 4 (X4=156X_4 = 156), Analogy 3 (X3=210X_3 = 210), Analogy 2 (X2=504X_2 = 504), and Analogy 1 (X1=726X_1 = 726).
Solving each analogy yields X4=156X_4 = 156, X3=210X_3 = 210, X2=504X_2 = 504, and X1=726X_1 = 726. Ordering these from least to greatest places Analogy 4 first, followed by Analogy 3, Analogy 2, and Analogy 1.

Step-by-Step Solution

1
Solve for X1X_1 in Analogy 1 (7:340::9:X17 : 340 :: 9 : X_1)
X1=726X_1 = 726
Identify the relationship 733=3407^3 - 3 = 340. Consequently, X1=933=7293=726X_1 = 9^3 - 3 = 729 - 3 = 726.
2
Solve for X2X_2 in Analogy 2 (6:210::8:X26 : 210 :: 8 : X_2)
X2=504X_2 = 504
Identify the relationship 636=2106^3 - 6 = 210. Consequently, X2=838=5128=504X_2 = 8^3 - 8 = 512 - 8 = 504.
3
Solve for X3X_3 in Analogy 3 (12:156::14:X312 : 156 :: 14 : X_3)
X3=210X_3 = 210
Identify the relationship 122+12=15612^2 + 12 = 156. Consequently, X3=142+14=196+14=210X_3 = 14^2 + 14 = 196 + 14 = 210.
4
Solve for X4X_4 in Analogy 4 (11:110::13:X411 : 110 :: 13 : X_4)
X4=156X_4 = 156
Identify the relationship 11211=11011^2 - 11 = 110. Consequently, X4=13213=16913=156X_4 = 13^2 - 13 = 169 - 13 = 156.
5
Arrange the calculated values of XX in ascending numerical order
156<210<504<726156 < 210 < 504 < 726
Comparing the values gives X4(156)<X3(210)<X2(504)<X1(726)X_4 (156) < X_3 (210) < X_2 (504) < X_1 (726).

Key Concept

Numerical Analogy and Exponential Pattern Recognition
Rate this question