Finding the Number Relationship Pattern in Sets

The question asks us to identify the relationship between the numbers in the given sets and then find an option set that shares the same relationship. We are given two sets: (18, 69, 279) and (12, 45, 183).

We need to find a pattern or rule that connects the first number (A) to the second number (B) and the second number (B) to the third number (C) in both given sets. The rule states that operations should be performed on the whole numbers, not their constituent digits.

Analyzing the Given Number Sets

Let's look at the first set (18, 69, 279) and try to find a relationship between 18 and 69, and then between 69 and 279.

• Relationship between 18 and 69:
  • Difference: $69 - 18 = 51$
  • Let's try multiplication and addition/subtraction. $18 \times 3 = 54$. $54 + 15 = 69$. This involves addition.
  • Let's try $18 \times 4 = 72$. $72 - 3 = 69$. This involves subtraction.
• Relationship between 69 and 279:
  • Difference: $279 - 69 = 210$
  • Let's try multiplication and addition/subtraction based on the previous findings. If the multiplier is 4, $69 \times 4 = 276$. $276 + 3 = 279$. This involves addition.

From this analysis, a potential pattern emerges:

• Second number (B) = (First number (A) $\times$ 4) - 3
• Third number (C) = (Second number (B) $\times$ 4) + 3

Verifying the Pattern with the Second Given Set

Let's check if this pattern holds for the second given set (12, 45, 183).

• First number (A) = 12, Second number (B) = 45, Third number (C) = 183
• Check the rule for B: $B = (A \times 4) - 3$
  $45 = (12 \times 4) - 3$
  $45 = 48 - 3$
  $45 = 45$. This matches the second number.
• Check the rule for C: $C = (B \times 4) + 3$
  $183 = (45 \times 4) + 3$
  $183 = 180 + 3$
  $183 = 183$. This matches the third number.

The pattern $B = (A \times 4) - 3$ and $C = (B \times 4) + 3$ is consistent for both given sets.

Applying the Pattern to the Options

Now we will apply this established pattern to each of the given options to find the set that follows the same rule.

Option 1: (14, 39, 159)

• A = 14, B = 39, C = 159
• Check for B: $B = (A \times 4) - 3$
  $39 = (14 \times 4) - 3$
  $39 = 56 - 3$
  $39 = 53$. This is not equal to 39. Option 1 does not follow the pattern.

Option 2: (16, 67, 266)

• A = 16, B = 67, C = 266
• Check for B: $B = (A \times 4) - 3$
  $67 = (16 \times 4) - 3$
  $67 = 64 - 3$
  $67 = 61$. This is not equal to 67. Option 2 does not follow the pattern.

Option 3: (32, 125, 503)

• A = 32, B = 125, C = 503
• Check for B: $B = (A \times 4) - 3$
  $125 = (32 \times 4) - 3$
  $125 = 128 - 3$
  $125 = 125$. This matches the second number.
• Check for C: $C = (B \times 4) + 3$
  $503 = (125 \times 4) + 3$
  $503 = 500 + 3$
  $503 = 503$. This matches the third number.

Option 3 follows the established pattern for both relationships between the numbers in the set.

Option 4: (22, 85, 345)

• A = 22, B = 85, C = 345
• Check for B: $B = (A \times 4) - 3$
  $85 = (22 \times 4) - 3$
  $85 = 88 - 3$
  $85 = 85$. This matches the second number.
• Check for C: $C = (B \times 4) + 3$
  $345 = (85 \times 4) + 3$
  $345 = 340 + 3$
  $345 = 343$. This is not equal to 345. Option 4 does not fully follow the pattern.

Therefore, the option that shares the same number relationship as the given sets is (32, 125, 503).

Revision Table: Key Learnings from Number Pattern Questions

Solving number pattern and set-based reasoning questions often involves looking for common mathematical operations between consecutive numbers. Key strategies include:

• Finding the difference between consecutive numbers.
• Checking for simple multiplication or division.
• Looking for patterns involving multiplication/division combined with addition/subtraction.
• Considering squares, cubes, or other powers of numbers.
• Testing potential patterns systematically on all given examples before applying them to options.

Additional Information: Types of Number Series and Patterns

Number series and pattern recognition are common types of questions in logical and quantitative reasoning. Understanding different types of patterns can help solve these problems faster:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 5, 8, 11... difference is 3).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24... ratio is 2).
• Mixed Series: Involves a combination of operations (like the one in this problem: multiplication and addition/subtraction).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5...).
• Prime Numbers: Series composed only of prime numbers (e.g., 2, 3, 5, 7, 11...).
• Square or Cube Series: Terms are squares or cubes of natural numbers, or related to them.

Careful observation and systematic testing are crucial for identifying the correct number relationship pattern.