This question is a number analogy problem where we need to find the relationship between the numbers in the given pairs and apply the same logic to find the missing number.

Understanding the Number Analogy Problem

In this type of question, two pairs of numbers are given, showing a specific relationship between the first and second numbers in each pair. We need to identify this relationship and then apply it to the third pair to find the missing number.

The given analogy is:

\(9 \ratio 702 \ratio\ratio 3 \ratio 18 \ratio\ratio 6 \ratio ?\)

This means:

• The relationship between 9 and 702 is the same as the relationship between 3 and 18.
• We need to find this relationship and apply it to 6 to find the missing number.

Analyzing the Relationships in the Given Pairs

Let's examine the first two pairs to find the pattern:

1. Pair 1: 9 ∶ 702
2. Pair 2: 3 ∶ 18

Let's try to find a mathematical operation or pattern connecting the numbers in each pair. Let 'n' be the first number in a pair.

Analyzing Pair 2 (3 ∶ 18) First

Starting with the smaller numbers often makes it easier to spot the pattern.

If \(n = 3\), the second number is 18.

• Is it multiplication? \(3 \times 6 = 18\). The multiplier is 6.
• Is it related to squares or cubes? \(3^2 = 9\), \(3^3 = 27\).
• Can we combine operations? \(3^2 + 9 = 18\). This uses the number itself.
• What about \(n^3 - k \times n\)? \(3^3 - 3 \times 3 = 27 - 9 = 18\). This works for Pair 2! The pattern is \(n^3 - 3n\).

Verifying the Pattern with Pair 1 (9 ∶ 702)

Now, let's check if the pattern \(n^3 - 3n\) holds for the first pair.

If \(n = 9\), let's calculate \(n^3 - 3n\):

\(9^3 - 3 \times 9 = 729 - 27\)

\(729 - 27 = 702\)

The calculated value (702) matches the second number in Pair 1. So, the pattern \(n^3 - 3n\) is consistent for both the given pairs.

Applying the Pattern to Find the Missing Number

Now, we apply the same pattern to the third pair (6 ∶ ?).

Here, the first number is \(n = 6\). We need to find the second number using the pattern \(n^3 - 3n\).

Missing number = \(6^3 - 3 \times 6\)

Calculate the terms:

• \(6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216\)
• \(3 \times 6 = 18\)

Subtract the second term from the first:

Missing number = \(216 - 18\)

Missing number = \(198\)

Conclusion

The missing number in the analogy is 198.

The pattern followed is \(n \ratio n^3 - 3n\), where n is the first number in the pair.

Step-by-Step Solution Summary

1. Identify the relationship between the numbers in the first two pairs: \(9 \ratio 702\) and \(3 \ratio 18\).
2. Determine the pattern. The pattern found is \(n \ratio n^3 - 3n\).
3. Verify the pattern with both pairs.
   • For \(n=3\): \(3^3 - 3(3) = 27 - 9 = 18\) (Matches)
   • For \(n=9\): \(9^3 - 3(9) = 729 - 27 = 702\) (Matches)
4. Apply the pattern to the third pair \(6 \ratio ?\).
   • For \(n=6\): \(6^3 - 3(6) = 216 - 18 = 198\)
5. The missing number is 198.

Revision Table: Number Analogy Pattern

Additional Information: Logical Reasoning and Number Series

Number analogy questions are part of logical reasoning and quantitative aptitude. They test your ability to find patterns and relationships between numbers.

Common patterns in number series and analogies include:

• Arithmetic progression (adding or subtracting a constant)
• Geometric progression (multiplying or dividing by a constant)
• Powers and roots (squares, cubes, square roots, cube roots)
• Combinations of operations (e.g., \(n^2 + n\), \(n^3 - n\), \(2n + 5\), etc.)
• Difference patterns (finding the pattern in the differences between consecutive terms)
• Digit operations (sum of digits, product of digits, etc.)
• Prime numbers, composite numbers, odd/even numbers

Solving these problems requires careful observation, trial and error, and knowledge of basic mathematical operations and number properties. Practicing different types of patterns helps improve pattern recognition skills.