Understanding Number Analogy Reasoning

This question asks us to identify the relationship between pairs of numbers and apply that same relationship to find a missing number. This type of problem is known as a number analogy or number reasoning question. We are given two complete pairs and one incomplete pair:

• First pair: 8 is related to 71
• Third pair: 12 is related to 151
• Second pair: 11 is related to ?

Our task is to find the rule connecting the numbers in the first and third pairs and then use that rule to find the missing number in the second pair.

Finding the Pattern in Number Pairs

Let's examine the relationship between the numbers in the given pairs.

Analyzing the First Pair (8 : 71)

We need to find a mathematical operation or a sequence of operations that transforms 8 into 71.

• Simply adding or subtracting a fixed number doesn't seem likely ($71 - 8 = 63$).
• Let's consider multiplication. $8 \times 8 = 64$. The result 64 is close to 71. The difference is $71 - 64 = 7$. So, the relationship could be squaring the number and adding 7: $8^2 + 7 = 64 + 7 = 71$.

Analyzing the Third Pair (12 : 151)

Let's test if the pattern found in the first pair ($ \text{Number}^2 + 7 $) also applies to the third pair (12 : 151).

• Apply the potential rule to 12: $12^2 + 7$.
• Calculate $12^2 = 144$.
• Add 7: $144 + 7 = 151$.

The result, 151, matches the second number in the third pair. This confirms that the pattern is indeed squaring the first number and adding 7.

Applying the Pattern to Find the Missing Number

Now that we have established the pattern ( $ \text{Number}^2 + 7 $ ), we can apply it to the second pair (11 : ?) to find the missing number.

• The first number in the second pair is 11.
• Apply the pattern: $11^2 + 7$.
• Calculate $11^2 = 121$.
• Add 7: $121 + 7 = 128$.

So, the missing number related to 11 in the same way is 128.

Verification with Options

Let's check if 128 is among the given options:

1. 132
2. 128
3. 125
4. 141

The calculated number, 128, matches option 2.

Conclusion

The relationship followed in the number analogy is that the second number is obtained by squaring the first number and adding 7. Applying this pattern to 11, we get $11^2 + 7 = 121 + 7 = 128$. Therefore, the missing number is 128.

Revision Table: Number Analogy Patterns

Reviewing common patterns in number analogies is key to solving such problems efficiently. Here are a few types:

• Arithmetic Operations: Addition, subtraction, multiplication, division, sometimes combined.
• Squaring or Cubing: Using squares ($n^2$) or cubes ($n^3$), often with an additional constant or a related number.
• Prime Numbers: The numbers might be prime numbers or related to them.
• Digit Operations: Operations on the digits of the number (sum of digits, product of digits, etc.).
• Series based on position: The relation might depend on the position of the number in a known series (e.g., Fibonacci series).

Additional Information: Strategies for Solving Number Analogy Problems

Solving number analogy questions requires observation, pattern recognition, and basic mathematical skills. Here are some tips:

• Look for Simple Patterns First: Start with basic operations like addition, subtraction, multiplication, or division.
• Consider Squares and Cubes: Many number analogies involve squares or cubes, often adjusted by adding or subtracting a small number.
• Check Differences and Ratios: Calculate the difference or ratio between the numbers in a pair. See if this difference or ratio is constant or follows a pattern.
• Examine Digits: For larger numbers, look at the relationship between the digits.
• Test the Pattern Consistently: Once you find a potential pattern from one pair, always test it with the other given pairs to ensure it holds true for all.
• Practice Regularly: The more you practice, the faster you will become at recognizing common patterns.

Understanding these strategies will help you confidently approach number analogy questions in various competitive exams and reasoning tests.