Understanding Number Pair Patterns

The question asks us to identify a mathematical relationship between the numbers in the given pairs and then find the option that follows the same rule. We are told that the operations are performed on the whole numbers, not on their individual digits. The given number pairs are (21, 66) and (32, 99).

Analyzing the Given Number Pairs

Let's look for a pattern in the pairs (21, 66) and (32, 99). We need to find an operation or a series of operations performed on the first number to get the second number.

• Consider the pair (21, 66). How can we get 66 from 21?
• If we multiply 21 by a small number, say 2, we get $21 \times 2 = 42$. This is too small.
• If we multiply 21 by 3, we get $21 \times 3 = 63$. This is close to 66. The difference is $66 - 63 = 3$.
• So, the pattern could be: Multiply the first number by 3 and then add 3. Let's write this as: Second number = (First number $\times$ 3) + 3.

Now, let's test this rule on the second given pair (32, 99).

• First number is 32. Apply the rule: $(32 \times 3) + 3$.
• $32 \times 3 = 96$.
• $96 + 3 = 99$.

The rule (First number $\times$ 3) + 3 successfully generates the second number in both given pairs (21, 66) and (32, 99).

The mathematical operation relating the numbers in the given sets is:
Second number = $\text{(First number} \times 3) + 3$

Testing the Options for the Same Pattern

We will now apply the identified rule to each of the given options to see which number pair follows the same pattern.

Based on our analysis, only the number pair (45, 138) in Option 3 follows the pattern where the second number is obtained by multiplying the first number by 3 and adding 3.

Conclusion on Number Pair Relationship

The pattern in the given sets (21, 66) and (32, 99) is that the second number is three times the first number plus three. Applying this relationship to the options reveals that the pair (45, 138) matches this pattern. This type of question tests your ability to identify mathematical operations linking numbers in a pair.

Revision Table: Number Pair Logic

Additional Information: Pattern Recognition in Reasoning

Pattern recognition is a common type of question in logical and quantitative reasoning tests. These questions require you to identify the underlying rule or sequence that connects a set of numbers, letters, or figures. Once the pattern is identified, you apply it to find the missing element or, as in this case, the set that follows the same rule.

• Types of Patterns: Patterns can involve arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, prime numbers, series based on positions, or combinations of these.
• Strategies:
  • Look for differences or ratios between consecutive numbers.
  • Check for common mathematical operations like multiplication, division, addition, or subtraction applied consistently.
  • Consider squares, cubes, or other powers of the numbers.
  • Sometimes, the pattern might involve two interleaved series.
• Important Note: Always test your hypothesized pattern against all the given examples before applying it to the options.

Mastering pattern recognition improves analytical skills and speed in solving reasoning problems for various exams.