Finding the Different Number Pair in Reasoning

In this type of reasoning question, we are given a set of number pairs, and we need to find the one pair that does not follow the same rule or pattern as the others. The task is to carefully examine each pair and discover the underlying relationship between the two numbers in the pairs.

Analyzing the Given Number Pairs

We have the following four number pairs:

• 6 : 180
• 5 : 100
• 4 : 52
• 7 : 294

Let's consider the relationship between the first number (let's call it 'n') and the second number (let's call it 'V') in each pair. We will look for a consistent mathematical pattern or rule that connects 'n' to 'V' for most of the pairs.

Exploring Potential Patterns

We can try simple operations like multiplication, squaring, cubing, or combinations of these. Let's look at the relationship between 'n' and 'V' for each pair:

• For 6 : 180, \( n=6, V=180 \).
• For 5 : 100, \( n=5, V=100 \).
• For 4 : 52, \( n=4, V=52 \).
• For 7 : 294, \( n=7, V=294 \).

Let's test a common pattern involving squares or cubes related to 'n'. Consider the pattern where V is related to \( n^2 \) or \( n^3 \).

Let's try testing the relationship \( V = n^2 \times (n-1) \):

• For the pair 6 : 180:
  Here \( n=6 \). Let's calculate \( n^2 \times (n-1) \):
  \( 6^2 \times (6-1) = 36 \times 5 = 180 \).
  This matches the given second number, 180. So, the pair 6 : 180 follows this pattern.
• For the pair 5 : 100:
  Here \( n=5 \). Let's calculate \( n^2 \times (n-1) \):
  \( 5^2 \times (5-1) = 25 \times 4 = 100 \).
  This matches the given second number, 100. So, the pair 5 : 100 follows this pattern.
• For the pair 4 : 52:
  Here \( n=4 \). Let's calculate \( n^2 \times (n-1) \):
  \( 4^2 \times (4-1) = 16 \times 3 = 48 \).
  This calculation gives 48, which is not equal to the given second number, 52. This pair does NOT follow the pattern \( V = n^2 \times (n-1) \).
• For the pair 7 : 294:
  Here \( n=7 \). Let's calculate \( n^2 \times (n-1) \):
  \( 7^2 \times (7-1) = 49 \times 6 = 294 \).
  This matches the given second number, 294. So, the pair 7 : 294 follows this pattern.

Alternatively, the pattern can also be expressed as \( V = n^3 - n^2 \), since \( n^2(n-1) = n^3 - n^2 \). Let's verify this equivalent pattern:

• For 6 : 180: \( 6^3 - 6^2 = 216 - 36 = 180 \). Matches.
• For 5 : 100: \( 5^3 - 5^2 = 125 - 25 = 100 \). Matches.
• For 4 : 52: \( 4^3 - 4^2 = 64 - 16 = 48 \). Does not match 52.
• For 7 : 294: \( 7^3 - 7^2 = 343 - 49 = 294 \). Matches.

Both forms of the pattern, \( V = n^2(n-1) \) and \( V = n^3 - n^2 \), lead to the same conclusion.

Identifying the Different Pair

From the analysis above, we can see that three of the number pairs (6:180, 5:100, and 7:294) follow the pattern where the second number is obtained by multiplying the square of the first number by one less than the first number, or equivalently, by subtracting the square of the first number from its cube. The pair 4 : 52 does not follow this pattern.

The expected value for \( n=4 \) based on the pattern \( V = n^2(n-1) \) or \( V = n^3 - n^2 \) is 48, but the given pair is 4 : 52.

Therefore, the number-pair that is different from the rest is 4 : 52.

Revision Table: Pattern Analysis

Additional Information: Numerical Reasoning Patterns

Numerical reasoning questions often involve identifying patterns between numbers. Common patterns include:

• Arithmetic Sequences: Adding or subtracting a constant value.
• Geometric Sequences: Multiplying or dividing by a constant value.
• Square and Cube Patterns: Involving squares \( n^2 \) or cubes \( n^3 \) of the number, or related values like \( n^2+c \), \( n^2-c \), \( n^3+c \), \( n^3-c \), \( n^2 \pm n \), \( n^3 \pm n \), \( n^3 \pm n^2 \), etc.
• Digit Operations: Sum or product of digits.
• Combinations of Operations: Patterns involving multiple steps or operations like in this question \( n^2 \times (n-1) \).
• Prime or Composite Numbers: The numbers might follow a rule related to them being prime or composite.

Solving these questions requires careful observation, testing different potential relationships, and applying basic arithmetic and algebraic principles.