Solving Number Analogy Questions

This question asks us to identify the relationship between the numbers in the given sets and then find an option set that follows the same relationship. The sets provided are (144, 9, 3) and (225, 7, 8).

We are instructed to perform operations only on the whole numbers provided, not on their individual digits.

Analyzing the Given Number Sets

Let's examine the first set: (144, 9, 3).

• The first number is 144.
• The second number is 9.
• The third number is 3.

We need to find a connection between 144, 9, and 3. Let's try combining the second and third numbers using basic arithmetic operations (addition, subtraction, multiplication, division) and see if we can relate the result to the first number, 144.

• Sum of the second and third numbers: $9 + 3 = 12$. Notice that $12^2 = 144$.
• Difference of the second and third numbers: $9 - 3 = 6$. $6^2 = 36$, $6 \times 12 = 72$, etc., none directly relate to 144.
• Product of the second and third numbers: $9 \times 3 = 27$. $27 \times \text{something} = 144$? No simple integer multiplication.
• Division of the second and third numbers: $9 \div 3 = 3$. $3^2 = 9$, $3 \times \text{something} = 144$? No simple integer multiplication.

The sum relationship seems promising: The first number is the square of the sum of the second and third numbers.

Let's verify this pattern with the second set: (225, 7, 8).

• The first number is 225.
• The second number is 7.
• The third number is 8.

Using the identified pattern: Sum of the second and third numbers is $7 + 8 = 15$. The square of this sum is $15^2 = 225$. This matches the first number in the set.

So, the established relationship is: First Number = (Second Number + Third Number)$^2$

Testing the Options for the Number Relation

Now we will apply this relation to the given options to find the set that follows the same pattern.

Option 1: (168, 5, 8)

• Second number = 5
• Third number = 8
• Sum = $5 + 8 = 13$
• Square of sum = $13^2 = 169$

The first number in the option is 168. Since $169 \neq 168$, this option does not follow the pattern.

Option 2: (64, 5, 3)

• Second number = 5
• Third number = 3
• Sum = $5 + 3 = 8$
• Square of sum = $8^2 = 64$

The first number in the option is 64. Since $64 = 64$, this option follows the pattern.

Option 3: (81, 6, 2)

• Second number = 6
• Third number = 2
• Sum = $6 + 2 = 8$
• Square of sum = $8^2 = 64$

The first number in the option is 81. Since $64 \neq 81$, this option does not follow the pattern.

Option 4: (120, 7, 4)

• Second number = 7
• Third number = 4
• Sum = $7 + 4 = 11$
• Square of sum = $11^2 = 121$

The first number in the option is 120. Since $121 \neq 120$, this option does not follow the pattern.

Only Option 2 matches the number relation found in the original sets (144, 9, 3) and (225, 7, 8).

Summary of Number Relation Analysis

Based on the analysis, the set (64, 5, 3) is related in the same way as the numbers of the given sets.

Revision Table: Number Analogy Basics

Additional Information: Solving Number Relations

Number relation and analogy questions are common in logical reasoning and quantitative aptitude tests. To solve these effectively, consider the following strategies:

• Look for basic operations: Start with simple arithmetic like addition, subtraction, multiplication, and division between the numbers.
• Consider squares and cubes: Often, numbers in these sets are perfect squares or cubes, or related to squares/cubes of combinations of other numbers in the set.
• Check for ratios or proportions: Sometimes numbers might be related by a constant ratio or a simple proportion.
• Look for sequences: In sets with more numbers, there might be an arithmetic or geometric progression, or another type of sequence.
• Combine numbers: Try operations on two numbers to see if they produce the third, or if a combination of two relates to the third. For example, (A, B, C), check relations like A = B+C, A = B*C, A = B$+$C$^2$, etc.
• Factorization: For larger numbers, prime factorization might reveal a hidden relationship.
• Be mindful of constraints: Pay close attention to rules given, such as the constraint in this problem about not breaking down digits.
• Test all options: Once a potential pattern is found, rigorously test it against all the given options to ensure only one fits the relationship.

Practicing different types of number relation problems helps in quickly recognizing common patterns.