Understanding the Number Analogy Question

This question asks us to identify the relationship between the numbers in the given sets and then find which option set follows the same relationship. We are given two sets: (15, 13, 56) and (17, 11, 168). The key rule is that operations must be performed on the whole numbers as they are, not by breaking them into individual digits.

Analyzing the Given Number Sets

Let's look closely at the two example sets provided:

Set 1: (15, 13, 56)

Set 2: (17, 11, 168)

We need to find a pattern or a mathematical relationship that connects the first two numbers to the third number in both sets. Let's try some common operations.

Exploring Possible Relationships

Often, number analogy questions involve basic arithmetic operations, squares, cubes, or combinations of these. Let the three numbers in a set be represented by a, b, and c.

• Could it be a simple sum, difference, or product? \(15+13=28\), \(15-13=2\), \(15 \times 13=195\). None of these directly yield 56.
• What about differences or sums combined with multiplication? \(15-13=2\), and \(2 \times 28 = 56\). For the second set, \(17-11=6\), and \(6 \times 28 = 168\). This looks promising, but the multiplier 28 is constant, which is less common than a pattern derived from a and b.
• Let's consider squares. \(15^2 = 225\), \(13^2 = 169\). What is the difference? \(225 - 169 = 56\). This matches the third number in Set 1.
• Let's check if this pattern holds for Set 2: \(17^2 = 289\), \(11^2 = 121\). What is the difference? \(289 - 121 = 168\). This matches the third number in Set 2.

Identifying the Pattern

The relationship that holds true for both given sets is that the third number is the difference between the square of the first number and the square of the second number. Mathematically, if the set is (a, b, c), the relationship is:

\(a^2 - b^2 = c\)

This relationship is also known as the difference of squares, which can be factored as \((a-b)(a+b) = c\). Let's quickly check this alternative form:

• For Set 1: \((15-13)(15+13) = (2)(28) = 56\). This works.
• For Set 2: \((17-11)(17+11) = (6)(28) = 168\). This also works.

Both forms confirm the relationship.

Evaluating the Options using the Discovered Relationship

Now we will apply the relationship \(a^2 - b^2 = c\) to each of the given options to see which one fits.

• Option 1: (15, 11, 114)

  Here, \(a=15\) and \(b=11\). We calculate \(a^2 - b^2\):

  \(15^2 - 11^2 = 225 - 121 = 104\)

  The third number in the option is 114. Since \(104 \neq 114\), this option does not follow the relationship.

• Option 2: (23, 21, 78)

  Here, \(a=23\) and \(b=21\). We calculate \(a^2 - b^2\):

  \(23^2 - 21^2 = 529 - 441 = 88\)

  The third number in the option is 78. Since \(88 \neq 78\), this option does not follow the relationship.

• Option 3: (20, 14, 204)

  Here, \(a=20\) and \(b=14\). We calculate \(a^2 - b^2\):

  \(20^2 - 14^2 = 400 - 196 = 204\)

  The third number in the option is 204. Since \(204 = 204\), this option follows the relationship.

• Option 4: (22, 16, 238)

  Here, \(a=22\) and \(b=16\). We calculate \(a^2 - b^2\):

  \(22^2 - 16^2 = 484 - 256 = 228\)

  The third number in the option is 238. Since \(228 \neq 238\), this option does not follow the relationship.

Determining the Correct Option

Based on the analysis, only Option 3 (20, 14, 204) shares the same relationship as the numbers in the given sets (15, 13, 56) and (17, 11, 168). The relationship is \(a^2 - b^2 = c\).

Revision Table: Key Concepts in Number Relationships

Additional Information on Finding Number Patterns

Finding the relationship between numbers in reasoning questions often requires systematic testing of common patterns. Here are a few tips:

• Always examine the difference, sum, product, and quotient of the numbers.
• Consider squares and cubes of the numbers.
• Look for patterns involving the difference or sum of the numbers squared or cubed.
• Sometimes the pattern involves a constant multiplier or adder after an initial operation.
• Test the discovered pattern on all provided example sets to ensure consistency.
• Apply the confirmed pattern rigorously to each option.

Practice with different types of number sets helps in quickly identifying potential relationships.