Solving Number Analogy Questions by Identifying Patterns

Number analogy questions test your ability to find a specific relationship or pattern between numbers in a given set and apply that same pattern to other sets of numbers to find the matching one. The key is to carefully observe the numbers and try different mathematical operations (addition, subtraction, multiplication, division, squaring, cubing, etc.) or combinations of operations performed on the whole numbers.

The given sets are:

• (12, 9, 4)
• (18, 9, 6)

We need to find the relationship between the numbers in these sets. Let's denote the numbers in a set as A, B, and C, where the set is (A, B, C).

Analyzing the Relationship in the Given Sets

Let's look at the first set (12, 9, 4). What operations using 12 and 4 could result in 9? We can try:

• $12 + 4 = 16$
• $12 - 4 = 8$
• $12 \times 4 = 48$
• $12 \div 4 = 3$

The result of division, 3, seems promising. How can we get 9 from 3? We can multiply it by 3 ($3 \times 3 = 9$) or square it ($3^2 = 9$). Let's test the squaring idea.

Let's assume the pattern is $B = (A \div C)^2$.

Verifying the Pattern with Given Sets

Let's check if this pattern holds for both the given sets:

• For the set (12, 9, 4):
  Here, $A=12$, $B=9$, and $C=4$.
  According to the pattern, $B = (A \div C)^2$.
  $A \div C = 12 \div 4 = 3$.
  $(A \div C)^2 = 3^2 = 9$.
  This matches the second number $B=9$. So the pattern works for the first set.
• For the set (18, 9, 6):
  Here, $A=18$, $B=9$, and $C=6$.
  According to the pattern, $B = (A \div C)^2$.
  $A \div C = 18 \div 6 = 3$.
  $(A \div C)^2 = 3^2 = 9$.
  This matches the second number $B=9$. So the pattern also works for the second set.

The established pattern is $B = (A \div C)^2$, where A must be divisible by C to obtain a whole number for the division result, which is then squared.

Testing the Options

Now, we apply this pattern to the given options:

• Option 1: (20, 12, 10)
  Here, $A=20$, $B=12$, $C=10$.
  $A \div C = 20 \div 10 = 2$.
  $(A \div C)^2 = 2^2 = 4$.
  The calculated value (4) does not match the second number (12). So this option does not follow the pattern.
• Option 2: (14, 10, 7)
  Here, $A=14$, $B=10$, $C=7$.
  $A \div C = 14 \div 7 = 2$.
  $(A \div C)^2 = 2^2 = 4$.
  The calculated value (4) does not match the second number (10). So this option does not follow the pattern.
• Option 3: (15, 12, 7)
  Here, $A=15$, $B=12$, $C=7$.
  $A \div C = 15 \div 7$. This division does not result in a whole number. Based on the problem note about using whole numbers for operations, this set is unlikely to follow the pattern where $A \div C$ is squared to get B, as $A \div C$ is not a whole number.
• Option 4: (12, 16, 3)
  Here, $A=12$, $B=16$, $C=3$.
  $A \div C = 12 \div 3 = 4$.
  $(A \div C)^2 = 4^2 = 16$.
  The calculated value (16) matches the second number (16). So this option follows the pattern.

Conclusion

Only Option 4 follows the same relationship $B = (A \div C)^2$ as the given sets.

Revision Table: Key Analogy Concepts

Additional Information: Strategies for Number Analogy Problems

When tackling number analogy questions, consider these common strategies for identifying the pattern:

• Basic Arithmetic: Look for relationships involving addition, subtraction, multiplication, or division between the numbers in the set. For instance, is the third number the sum/difference/product/quotient of the first two?
• Powers and Roots: Check for squares, cubes, square roots, or cube roots. One number might be the square or cube of another number or a result of an operation on other numbers.
• Combinations of Operations: The pattern might involve multiple steps, like (A + B) $\times$ C = D, or (A $\div$ C) + K = B (where K is a constant).
• Ratios: Sometimes the ratio between numbers in the set is constant or follows a pattern.
• Sequence/Series: If there are multiple sets, the pattern might evolve across the sets, although this is less common for finding a single matching set.
• Position-Based Rules: The pattern often relates the first number (A), second number (B), and third number (C) in the set (A, B, C). Try relating A and C to get B, or A and B to get C, etc.

Always test your assumed pattern on all the given sets to ensure consistency before applying it to the options. Make sure your operations adhere to any specific notes provided, such as the one about using whole numbers in this question.