Understanding Numerical Analogies and Patterns

The question asks us to find the missing number in a numerical analogy. A numerical analogy shows a relationship between pairs of numbers. We need to figure out the specific pattern or rule that connects the first number to the second number in the given pairs and then apply that same rule to find the missing number.

The given analogy is: 8 : 128 :: 6 : ? :: 11 : 242

This can be read as "8 is to 128 as 6 is to ? as 11 is to 242".

Analyzing the Given Number Pairs to Find the Pattern

Let's look at the first pair of numbers: 8 and 128.

We need to find a mathematical operation or a series of operations that transforms 8 into 128.

• Is it multiplication? $8 \times N = 128$. $N = 128 / 8 = 16$. So, $8 \times 16 = 128$.
• Let's consider squares or cubes. $8^2 = 64$. $8^3 = 512$.

How can we get from 8 to 128 using squaring? We know $8^2 = 64$. If we multiply 64 by 2, we get $64 \times 2 = 128$. This looks like a possible pattern: square the first number and then multiply by 2.

Let's represent the first number as $A$ and the second number as $B$. The proposed pattern is $B = A^2 \times 2$.

Verifying the Pattern with Another Pair

Now, let's test this pattern with the second complete pair: 11 and 242.

Here, the first number is 11 and the second number is 242. According to our proposed pattern, the second number should be the square of the first number multiplied by 2.

Let $A = 11$. According to the pattern, $B = 11^2 \times 2$.

Calculate $11^2$: $11 \times 11 = 121$.

Now, multiply by 2: $121 \times 2 = 242$.

The calculated value 242 matches the second number in the pair (11 : 242). This confirms that the pattern $A^2 \times 2$ is the correct relationship for this numerical analogy.

The pattern is: The second number is equal to the square of the first number multiplied by two.

\( \text{Second Number} = (\text{First Number})^2 \times 2 \)

Using LaTeX for the pattern formula:

\( B = A^2 \times 2 \)

Applying the Pattern to Find the Missing Number

Now, we need to apply this established pattern to the pair with the missing number: 6 : ?

Here, the first number is 6. Let the missing number be $X$. According to the pattern, $X$ should be the square of 6 multiplied by 2.

\( X = 6^2 \times 2 \)

Calculate $6^2$: $6 \times 6 = 36$.

Now, multiply by 2: $36 \times 2 = 72$.

So, the missing number is 72.

Comparing the Result with Options

The missing number is 72. Let's check the given options:

• Option 1: 64
• Option 2: 72
• Option 3: 68
• Option 4: 70

Our calculated value, 72, matches Option 2.

Conclusion

The pattern in the numerical analogy 8 : 128 :: 6 : ? :: 11 : 242 is that the second number is twice the square of the first number. Applying this pattern to the number 6, we find that the missing number is 72.

\( 8^2 \times 2 = 64 \times 2 = 128 \)

\( 6^2 \times 2 = 36 \times 2 = 72 \)

\( 11^2 \times 2 = 121 \times 2 = 242 \)

Revision Table: Numerical Analogy Solution Steps

Additional Information on Numerical Reasoning and Analogies

Numerical analogies are a common type of question in reasoning tests. They assess your ability to identify patterns and relationships between numbers. The patterns can be simple arithmetic operations, or they can involve squares, cubes, roots, series, prime numbers, or combinations of operations.

Key strategies for solving numerical analogies:

• Look for simple relationships first: addition, subtraction, multiplication, division.
• Consider powers and roots: squares, cubes, square roots, cube roots.
• Think about combined operations: e.g., multiply and add, square and subtract.
• Look for patterns in the digits themselves or properties of the numbers (even/odd, prime/composite).
• If there are multiple pairs, always test your potential pattern with all given pairs to ensure it is consistent.
• Break down complex numbers to see if they are products of smaller numbers or squares/cubes.

Practicing different types of numerical reasoning problems helps in quickly identifying common patterns.