Analogy Question Analysis

The question asks us to identify the pair of numbers that shares the same relationship as the given pair, which is 8 : 448. This type of question falls under number analogy, where we need to find the pattern or rule connecting the two numbers in the first pair and then apply that same rule to the given options to find the matching pair.

Understanding the Given Pair: 8 : 448

Let's analyze the relationship between 8 and 448. We need to find a mathematical operation or a sequence of operations that transforms 8 into 448.

• First, let's consider basic operations like multiplication. Is 448 a direct multiple of 8? Yes, $448 \div 8 = 56$. So, $8 \times 56 = 448$. But how is 56 related to 8? $56 = 8 \times 7$. This gives us a pattern: $n : n \times (n \times 7)$. Let's test this later.
• Let's look at squares and cubes of 8. $8^2 = 64$ and $8^3 = 512$. The number 448 is close to 512. The difference is $512 - 448 = 64$. Notice that $64 = 8^2$.
• This suggests a possible pattern: the second number is the cube of the first number minus the square of the first number. Using the first number $n=8$:
  $n^3 - n^2 = 8^3 - 8^2 = 512 - 64 = 448$. This pattern fits the given pair 8 : 448 perfectly.

Alternatively, we can factor the expression $n^3 - n^2 = n^2(n-1)$. So the pattern can also be stated as the second number is the square of the first number multiplied by (the first number minus one).

Using the first number $n=8$:
$n^2 \times (n-1) = 8^2 \times (8-1) = 64 \times 7 = 448$. This also fits the given pair.

Both patterns ($n^3 - n^2$ and $n^2 \times (n-1)$) are mathematically equivalent.

Applying the Pattern to Options

Now, we will apply the pattern ($n^3 - n^2$ or $n^2(n-1)$) to the first number in each option pair and check if the result matches the second number in that pair.

• Option (a) 9 : 729
  Here, the first number is $n=9$.
  Applying the pattern: $9^3 - 9^2 = 729 - 81 = 648$.
  The second number in the pair is 729. Since $648 \neq 729$, this pair does not follow the same relationship. Note that 729 is $9^3$.
• Option (b) 7 : 1029
  Here, the first number is $n=7$.
  Applying the pattern: $7^3 - 7^2 = 343 - 49 = 294$.
  The second number in the pair is 1029. Since $294 \neq 1029$, this pair does not follow the same relationship.
• Option (c) 6 : 216
  Here, the first number is $n=6$.
  Applying the pattern: $6^3 - 6^2 = 216 - 36 = 180$.
  The second number in the pair is 216. Since $180 \neq 216$, this pair does not follow the same relationship. Note that 216 is $6^3$.
• Option (d) 11 : 1210
  Here, the first number is $n=11$.
  Applying the pattern: $11^3 - 11^2 = 1331 - 121 = 1210$.
  Alternatively: $11^2 \times (11-1) = 121 \times 10 = 1210$.
  The second number in the pair is 1210. Since $1210 = 1210$, this pair follows the same relationship.

Identifying the Correct Analogous Pair

Based on the analysis, only option (d) follows the same relationship as the given pair 8 : 448. The relationship is $n : n^3 - n^2$ or $n : n^2(n-1)$, where $n$ is the first number in the pair.

Revision Table: Number Analogy Patterns

Additional Information: Number Series and Analogies

Number analogy questions are common in various competitive exams. They test your ability to identify numerical patterns and relationships. The relationships can be based on fundamental arithmetic operations (addition, subtraction, multiplication, division), powers (squares, cubes), roots, prime numbers, or a combination of these. Sometimes, the pattern might involve the digits of the numbers themselves.

To solve number analogy problems effectively, consider these steps:

• Look at the given pair closely and try to find a relationship. Ask yourself how you can get the second number from the first number using arithmetic operations, squares, cubes, etc.
• Test simple relationships first. Is it multiplication by a constant? Is it adding or subtracting a constant?
• Consider squares, cubes, square roots, or cube roots. Is the second number related to the square or cube of the first number?
• Look for patterns involving the number itself, like $n \times (n+1)$, $n \times (n-1)$, $n^2+n$, $n^2-n$, $n^3+n$, $n^3-n$, etc.
• Once you find a plausible pattern, test it on the first number of each option pair. The option whose second number matches the result of applying the pattern is the correct answer.
• If multiple patterns seem possible for the given pair, test each one against the options until you find one that uniquely fits one option.

Practice with various types of number analogy problems helps in quickly recognizing common patterns and relationships.