Solving Number Analogy Questions

Number analogy questions test your ability to identify the relationship between a pair of numbers and apply that same relationship to find a missing number in another pair. In this problem, we are given three pairs, with one number missing in the second pair:

14 : 203 :: 6 : ? :: 18 : 333

The task is to find the number that replaces the question mark (?) by discovering the rule connecting the first number to the second number in the given pairs and applying that rule to the third pair.

Analyzing the Relationship in the Pairs

Let's examine the relationship between the numbers in the complete pairs:

• Pair 1: 14 and 203
• Pair 3: 18 and 333

We need to find a pattern or rule ($n \rightarrow m$) that applies to both (14, 203) and (18, 333).

Exploring Potential Patterns

Often, number analogy patterns involve operations like addition, subtraction, multiplication, division, squaring, cubing, or a combination of these. Let's consider some possibilities for the first pair (14, 203):

• Is it addition/subtraction? $203 - 14 = 189$.
• Is it simple multiplication? $14 \times \text{something} = 203$. $203 \div 14 = 14.5$. Not a simple integer multiplication.
• Is it related to squaring the number? $14^2 = 196$. This is close to 203. The difference is $203 - 196 = 7$. So, $14^2 + 7 = 203$.

Verifying the Pattern with the Second Known Pair

Let's test the pattern $n^2 + k = m$ with the third pair (18, 333), where $n=18$ and $m=333$.

• $18^2 = 324$.
• The difference is $333 - 324 = 9$. So, $18^2 + 9 = 333$.

We see that the added value $k$ is different for $n=14$ (it's 7) and for $n=18$ (it's 9). We need to find a relationship for $k$ based on $n$. Notice that $7$ is half of $14$ ($14/2$), and $9$ is half of $18$ ($18/2$).

This suggests the pattern might be $m = n^2 + n/2$. Let's confirm this:

• For $n=14$: $14^2 + 14/2 = 196 + 7 = 203$. This matches the first pair.
• For $n=18$: $18^2 + 18/2 = 324 + 9 = 333$. This matches the third pair.

The pattern is consistently $n^2 + n/2$.

Applying the Pattern to Find the Missing Number

Now we apply the discovered pattern to the second pair: 6 : ?

Here, $n=6$. According to the pattern, the missing number (?) should be $6^2 + 6/2$.

Calculation:

Missing number = $6^2 + 6/2$

Missing number = $36 + 3$

Missing number = $39$

Conclusion

The missing number is 39. Let's check the options provided.

• Option 1: 39
• Option 2: 37
• Option 3: 20
• Option 4: 25

Our calculated number, 39, matches Option 1.

Revision Table: Key Learnings

Additional Information: Number Series and Reasoning

Number analogy is a common type of question in logical reasoning and quantitative aptitude sections of competitive exams. They require careful observation and logical thinking to identify underlying mathematical relationships. These patterns can be simple or complex, involving arithmetic operations, powers, roots, prime numbers, or even digit manipulation.

Tips for solving number analogies:

• Look for basic arithmetic relations (addition, subtraction, multiplication, division).
• Consider squares or cubes of the numbers.
• Check for patterns involving the sum or product of digits.
• Sometimes the pattern relates the number to its position in the series or involves sequences like prime numbers or Fibonacci numbers.
• Practice is key to recognizing common patterns quickly.

Understanding number relationships is crucial for various types of reasoning problems, including number series completion and coding-decoding based on numerical values.