Understanding Number Analogies and Finding Patterns

Number analogy questions test your ability to find a relationship between two numbers and apply that same relationship to a third number to find a missing fourth number. The given analogy is:

\(6 : 252 \;\;::\;\; 5 : ?\)

This means we need to discover the pattern or rule that connects 6 to 252, and then use that same rule to find the number that relates to 5.

Analyzing the Relationship between 6 and 252

Let's examine how 6 might be related to 252. We can consider basic mathematical operations like multiplication, squaring, cubing, addition, or subtraction, or a combination of these.

• If we multiply 6 by some number to get 252: \(252 \div 6 = 42\). The relationship could be multiplying by 42. If we apply this to 5: \(5 \times 42 = 210\). However, 210 is not one of the options.
• Let's consider powers of 6:
• \(6^2 = 36\)
• \(6^3 = 216\)

Notice that 216 is quite close to 252. The difference is \(252 - 216 = 36\). Interestingly, 36 is \(6^2\).

This suggests a possible relationship: The second number is the cube of the first number plus the square of the first number.

Let the first number be \(n\). The relationship appears to be \(n^3 + n^2\).

Let's verify this rule with the first pair (6 : 252):

For \(n=6\):

This matches the given relationship between 6 and 252.

Applying the Pattern to Find the Missing Number

Now, we apply the same rule \(n^3 + n^2\) to the third number, which is 5.

For \(n=5\):

So, the missing number related to 5 is 150.

Checking the Options

Let's compare our calculated missing number (150) with the given options:

• 125
• 150
• 225
• 176

Our calculated value, 150, matches option 2.

Therefore, the relationship is \(n^3 + n^2\), and applying this to 5 gives 150.

Number Analogy Revision Table

Additional Information on Number Analogy Patterns

Number analogies can involve various types of patterns. Understanding these common patterns can help solve such questions faster.

• Arithmetic Operations: Simple addition, subtraction, multiplication, or division.
• Squares and Cubes: The relationship might involve squaring (\(n^2\)), cubing (\(n^3\)), or combinations like \(n^2+1\), \(n^3-1\), \(n^2+n\), \(n^3+n^2\), etc.
• Prime Numbers: The numbers might be prime numbers or related to their sequence.
• Digit Operations: The pattern might involve operations on the digits of the number (e.g., sum of digits, product of digits).
• Sequence Based: Numbers following a specific sequence like Fibonacci.

Always start by checking simple relationships and then move towards more complex ones involving powers or combinations of operations when solving number analogy problems.