Understanding Number Analogy Questions

Number analogy questions test your ability to identify the relationship between two numbers and apply that same relationship to another pair of numbers. In this question, we are given three pairs, but one number is missing in the second pair. The pattern established in the first pair (8 : 448) and the third pair (11 : 847) must hold true for the second pair (9 : ?).

Analyzing the Given Pairs to Find the Pattern

Let's look at the first pair: 8 and 448.

We need to find a mathematical relationship between 8 and 448. Let's try some common operations:

• Could it be multiplication? $448 \div 8 = 56$. So, $8 \times 56 = 448$.
• How is 56 related to 8? $56 = 8 \times 7$.
• This suggests a possible relationship: the second number is the first number multiplied by (the first number multiplied by 7). In general, if the first number is $n$, the second number is $n \times (n \times 7)$.

Let's test this pattern with the first pair where $n=8$:

$$8 \times (8 \times 7) = 8 \times 56 = 448$$

This pattern works for the first pair.

Now, let's test this pattern with the third pair: 11 and 847. Here, $n=11$.

$$11 \times (11 \times 7) = 11 \times 77$$

To calculate $11 \times 77$: $11 \times 77 = 847$.

This pattern also works for the third pair. Therefore, the relationship is indeed $n : n \times (n \times 7)$.

Solving for the Missing Term in the Analogy

The analogy is 8 : 448 :: 9 : ? :: 11 : 847.

We need to find the number that relates to 9 in the same way. Using the pattern $n : n \times (n \times 7)$, where $n=9$, the missing term will be:

$$9 \times (9 \times 7)$$

First, calculate the value inside the parentheses:

$$9 \times 7 = 63$$

Now, multiply this result by the first number, which is 9:

$$9 \times 63$$

To calculate $9 \times 63$:

$$9 \times 60 = 540$$ $$9 \times 3 = 27$$ $$540 + 27 = 567$$

So, the missing term is 567.

Comparing the Result with the Options

Our calculated missing term is 567. Let's check the given options:

• Option 1: 576
• Option 2: 577
• Option 3: 567
• Option 4: 562

Our result, 567, matches Option 3.

Step-by-Step Solution Breakdown

1. Analyze the given pairs (8 : 448) and (11 : 847) to find the relationship between the first and second terms in each pair.
2. Discover the pattern is $n : n \times (n \times 7)$, where $n$ is the first term.
3. Apply this pattern to the third given term, 9, to find the missing term.
4. Calculate $9 \times (9 \times 7) = 9 \times 63 = 567$.
5. Identify the option that matches the calculated result.

The final answer is 567.

Revision Table: Key Concepts in Number Analogy

Additional Information on Analogy Reasoning

Analogy reasoning is a common type of question in many tests. It requires logical thinking and pattern recognition. While number analogies often involve arithmetic or simple algebraic relationships, other types of analogies can involve word meanings, synonyms, antonyms, parts and wholes, cause and effect, etc.

For number analogies, common patterns include:

• Addition or subtraction of a constant.
• Multiplication or division by a constant.
• Operations involving squares, cubes, or other powers.
• Operations involving square roots or cube roots.
• Combined operations (e.g., $n \times k + m$, $n^2 + k$).
• Relationships based on digits of the number.

Practice with different types of number series and patterns helps in quickly identifying the rule in analogy questions.