Understanding Number Analogy and Set Relations

Number analogy questions test your ability to identify the relationship or pattern between numbers in a given set and apply that same relationship to find a similar set among the options. The key is to carefully analyze the given set and hypothesize possible rules or operations connecting the numbers.

The given set is (8, 7, 407).

Let's denote the numbers in the set as A, B, and C, so A = 8, B = 7, and C = 407. We need to find a relationship between A, B, and C that holds true.

Analyzing the Pattern in the Given Set (8, 7, 407)

We look for a mathematical operation or combination of operations performed on the first two numbers (8 and 7) that results in the third number (407).

Let's try common patterns involving squares, cubes, multiplication, and addition/subtraction:

• Trying simple operations: \(8 + 7 = 15\), \(8 \times 7 = 56\). Neither is 407.
• Trying powers: \(8^2 = 64\), \(7^2 = 49\), \(8^3 = 512\), \(7^3 = 343\).
• Let's try combining powers of A and B:
• Is it \(A^2 + B^2\)? \(8^2 + 7^2 = 64 + 49 = 113\). No.
• Is it \(A^3 + B^3\)? \(8^3 + 7^3 = 512 + 343 = 855\). No.
• Is it \(A^3 - B^3\)? \(8^3 - 7^3 = 512 - 343 = 169\). No.
• Let's try a combination of powers of A and B, like A squared and B cubed:
• Consider \(A^2 + B^3\): \(8^2 + 7^3 = 64 + 343 = 407\).

This matches the third number in the given set! So, the relationship is \(C = A^2 + B^3\).

Applying the Pattern to Test the Options

Now we will apply the discovered relationship \(C = A^2 + B^3\) to each of the given options to see which set follows the same rule.

Option 1: (12, 8, 555)

Here, A = 12 and B = 8. Let's calculate A² + B³:

\(12^2 + 8^3 = 144 + 512 = 656\)

The calculated third number is 656, but the given third number in the option is 555. This option does not follow the pattern.

Option 2: (10, 8, 576)

Here, A = 10 and B = 8. Let's calculate A² + B³:

\(10^2 + 8^3 = 100 + 512 = 612\)

The calculated third number is 612, but the given third number in the option is 576. This option does not follow the pattern.

Option 3: (11, 12, 450)

Here, A = 11 and B = 12. Let's calculate A² + B³:

\(11^2 + 12^3 = 121 + 1728 = 1849\)

The calculated third number is 1849, but the given third number in the option is 450. This option does not follow the pattern.

Option 4: (9, 6, 297)

Here, A = 9 and B = 6. Let's calculate A² + B³:

\(9^2 + 6^3 = 81 + 216 = 297\)

The calculated third number is 297, which matches the given third number in the option. This option follows the pattern \(C = A^2 + B^3\).

Therefore, the set (9, 6, 297) is related in the same way as the set (8, 7, 407).

Revision Table: Checking the Pattern

Additional Information on Number Reasoning

Number reasoning questions often involve identifying patterns based on:

• Basic arithmetic operations (addition, subtraction, multiplication, division).
• Squares and cubes of numbers.
• Combinations of operations and powers.
• Prime numbers, composite numbers, or other number properties.
• Digit-based operations (sum of digits, product of digits, etc.).
• Sequential patterns (arithmetic progression, geometric progression, etc.).

Practicing various types of number series and analogy problems helps in quickly recognizing potential patterns.