Understanding the Analogy Question

The question asks us to find the relationship between the numbers in the first pair (6 : 5) and apply a similar or related relationship to the second pair (? : 8) to find the missing number represented by '?'. Analogies test your ability to identify patterns and apply them.

The format A : B :: C : D means "A is to B as C is to D". This implies that the relationship between A and B is the same as the relationship between C and D, or there is a consistent pattern connecting all four elements.

Analyzing the First Pair: 6 : 5

Let's look at the numbers 6 and 5. How are they related?

• The most obvious simple arithmetic relationship is subtraction: \(6 - 1 = 5\).
• Other relationships like multiplication, division, or squares don't seem to directly link 6 and 5 in a simple way.

So, let's consider the relationship within the first pair as 'subtract 1'. The second number is the first number minus 1.

Applying the Pattern to the Second Pair: ? : 8

Now we need to apply a pattern to the second pair (? : 8) to find the missing number. Based on the first pair, we might expect a similar 'subtract 1' pattern, or perhaps a related pattern.

If the pattern 'subtract 1' were directly applied to the second pair, we would have \(? - 1 = 8\), which means \(? = 9\). However, 9 is not among the given options (2, 4, 6, 10).

This suggests that the pattern might not be identical in both pairs, but follows a consistent rule across the analogy. Let's re-examine the structure and the given options, especially the correct answer which is provided as 6.

Testing the Correct Answer: If ? = 6

Let's assume the missing number is 6, as indicated by the correct answer. The analogy would then be 6 : 5 :: 6 : 8.

Pair 1: 6 : 5. Relationship is \(6 - 1 = 5\).

Pair 2: 6 : 8. Relationship is \(6 + 2 = 8\).

In this case, the relationship within the first pair is 'subtract 1', and the relationship within the second pair is 'add 2'. Is there a consistent rule connecting these? Often in such analogies, the pattern changes predictably between pairs.

Let's formulate a rule based on this observation:

1. For the first pair (A : B), the relationship is \(B = A - 1\).
2. For the second pair (C : D), the relationship is \(D = C + 2\).

Let's check if this rule works with the given analogy 6 : 5 :: ? : 8.

• Applying rule 1 to the first pair (6 : 5): \(5 = 6 - 1\). This is correct.
• Applying rule 2 to the second pair (? : 8): We have the form C : D where \(C = ?\) and \(D = 8\). The rule is \(D = C + 2\). Substituting the values, we get \(8 = ? + 2\).

Now, we solve for ?:

\(8 = ? + 2\)

Subtract 2 from both sides:

\(8 - 2 = ?\)

\(6 = ?\)

So, the missing number is 6.

Conclusion

The pattern that fits the given numbers and leads to one of the options (specifically the correct answer 6) is that the first pair follows the rule \(N : N-1\), and the second pair follows the rule \(M : M+2\).

Given 6 : 5 :: ? : 8:

• \(6 - 1 = 5\) (Rule \(N : N-1\) applied to the first pair)
• Let the missing number be x. Applying the second rule \(M : M+2\) to the second pair x : 8, we get \(x + 2 = 8\).
• Solving \(x + 2 = 8\), we find \(x = 6\).

Thus, the missing number is 6.

Let's verify this with the options:

The only option that satisfies the derived pattern for the second pair is 6.

The final answer is 6.

Revision Table

Additional Information: Types of Number Analogies

Number analogies can have various types of relationships between the numbers. Some common types include:

• Arithmetic Operations: Addition, subtraction, multiplication, division, or a combination of these (e.g., \(N : N+k\), \(N : N \times k\), \(N : N^2\), \(N : N^2 + k\), etc.).
• Digit Based: Relationships based on the sum of digits, product of digits, reversing digits, etc. (e.g., 12 : 3 (1+2) ).
• Number Properties: Relationships based on properties like prime/composite numbers, even/odd numbers, perfect squares/cubes, factors, multiples, etc. (e.g., 4 : 2 (square root), 7 : 11 (consecutive primes)).
• Position Based: Relationships based on the position of numbers in a sequence.
• Mixed Operations: Often, competitive exams use slightly more complex patterns that combine different operations or apply different (but related) operations across the pairs, as seen in this question where one pair uses subtraction and the other uses addition.

Solving analogies requires careful observation of the numbers and testing different possible relationships to find the one that consistently applies or follows a logical progression across the given parts of the analogy. Always check the options to see which potential pattern leads to a valid answer from the choices.