Solving Number Analogy Problems

Number analogy questions test your ability to find the relationship or pattern between a given set of numbers and then apply that same pattern to find a matching set from the options. It's important to remember the rule: operations should be performed on the whole numbers, not on individual digits.

Analyzing the Given Number Sets

We are given two sets of numbers that are related in a specific way:

• Set 1: (16, 7, 81)
• Set 2: (24, 16, 64)

Let's try to find the rule connecting the three numbers in each set. Let the numbers in a set be represented as A, B, and C.

Finding the Pattern in Set 1 (16, 7, 81)

Here, A = 16, B = 7, C = 81.

Let's explore possible relationships between A, B, and C. Often, the third number is derived from operations on the first two.

• Sum of the first two numbers: $A + B = 16 + 7 = 23$. This doesn't directly lead to 81.
• Difference of the first two numbers: $A - B = 16 - 7 = 9$.

Now, let's see if this difference relates to the third number, 81. We know that $9 \times 9 = 81$, which is $9^2$. So, the relationship could be $(A - B)^2 = C$.

Let's verify this pattern with the second given set.

Verifying the Pattern with Set 2 (24, 16, 64)

Here, A = 24, B = 16, C = 64.

Applying the potential pattern $(A - B)^2 = C$:

• Calculate the difference between the first two numbers: $A - B = 24 - 16 = 8$.
• Square the difference: $8^2 = 8 \times 8 = 64$.

The result, 64, matches the third number in Set 2. This confirms that the pattern is (First number - Second number)$^2$ = Third number.

So, the rule is: $C = (A - B)^2$.

Applying the Pattern to Options

Now we will check each option to see which one follows the same rule: $C = (A - B)^2$.

Option 1: (13, 7, 36)

Here, A = 13, B = 7, C = 36.

• Calculate the difference: $A - B = 13 - 7 = 6$.
• Square the difference: $6^2 = 6 \times 6 = 36$.

The calculated value (36) matches the third number in the set (36). This option follows the pattern.

Option 2: (48, 39, 18)

Here, A = 48, B = 39, C = 18.

• Calculate the difference: $A - B = 48 - 39 = 9$.
• Square the difference: $9^2 = 9 \times 9 = 81$.

The calculated value (81) does not match the third number in the set (18). This option does not follow the pattern.

Option 3: (37, 25, 100)

Here, A = 37, B = 25, C = 100.

• Calculate the difference: $A - B = 37 - 25 = 12$.
• Square the difference: $12^2 = 12 \times 12 = 144$.

The calculated value (144) does not match the third number in the set (100). This option does not follow the pattern.

Option 4: (21, 19, 8)

Here, A = 21, B = 19, C = 8.

• Calculate the difference: $A - B = 21 - 19 = 2$.
• Square the difference: $2^2 = 2 \times 2 = 4$.

The calculated value (4) does not match the third number in the set (8). This option does not follow the pattern.

Conclusion

Based on the analysis, only Option 1 follows the identified pattern where the third number is the square of the difference between the first two numbers.

Revision Table: Number Analogy Pattern

Additional Information: Types of Number Patterns in Reasoning

Number analogy and series problems often involve various types of patterns. Understanding these can help you identify the rule faster:

• Arithmetic Progressions: Numbers increase or decrease by a constant difference.
• Geometric Progressions: Numbers are multiplied or divided by a constant ratio.
• Difference Patterns: The difference between consecutive numbers follows a pattern (e.g., increasing differences, prime number differences).
• Square and Cube Patterns: Numbers are squares or cubes, or based on operations involving squares or cubes (like in this problem).
• Mixed Operations: A combination of addition, subtraction, multiplication, or division might be used in a specific sequence.
• Digit-based Operations (if allowed): Sometimes operations are performed on the individual digits of the numbers (but this was explicitly disallowed in this question).
• Positional Patterns: The position of the number in the sequence or set might influence the pattern.

Solving these problems effectively requires careful observation, systematic testing of potential patterns, and practice.