Understanding the Number Analogy Problem

This question asks us to identify a set of numbers from the options that shares the same relationship as the numbers in the two given sets: (16, 22, 28) and (14, 19, 23). We are instructed to perform operations on the whole numbers themselves, not their individual digits.

To solve this number analogy problem, we first need to discover the pattern or relationship that connects the three numbers within each of the given sets. Once we understand this relationship, we will apply it to the numbers in the options to find the matching set.

Analyzing the Given Number Sets to Find the Pattern

Let's examine the relationship between the numbers in the first set (16, 22, 28) and the second set (14, 19, 23).

Analyzing Set 1: (16, 22, 28)

Let the numbers be $a=16$, $b=22$, and $c=28$.

Let's find the differences between consecutive numbers:

• Difference between the second and first number: $b-a = 22 - 16 = 6$
• Difference between the third and second number: $c-b = 28 - 22 = 6$

In this set, the differences are 6 and 6. This is an arithmetic progression.

Analyzing Set 2: (14, 19, 23)

Let the numbers be $a=14$, $b=19$, and $c=23$.

Let's find the differences between consecutive numbers:

• Difference between the second and first number: $b-a = 19 - 14 = 5$
• Difference between the third and second number: $c-b = 23 - 19 = 4$

In this set, the differences are 5 and 4. This is not an arithmetic progression, and the pattern of differences (5, 4) is different from the first set (6, 6).

Since the two given sets have different simple arithmetic difference patterns, the underlying relationship must be something else that applies to both sets. Let's look for a relationship involving the numbers and their differences. Let the set be $(a, b, c)$, the first difference be $d_1 = b-a$, and the second difference be $d_2 = c-b$.

Let's test a potential pattern: Is there a consistent relationship between the first number ($a$) and the second difference ($d_2$)?

• For Set 1 (16, 22, 28): $a=16$, $d_2=6$. Let's calculate $a - d_2 = 16 - 6 = 10$.
• For Set 2 (14, 19, 23): $a=14$, $d_2=4$. Let's calculate $a - d_2 = 14 - 4 = 10$.

The pattern appears to be: The first number minus the second difference is equal to 10. Let's formally state the pattern:

Pattern: For a set $(a, b, c)$, the relationship is $a - (c-b) = 10$.

Checking the Options Against the Pattern

Now, we will apply this pattern $a - (c-b) = 10$ to each of the given options to find the set that follows the same rule.

Analyzing Option 1: (12, 18, 25)

Here, $a=12$, $b=18$, $c=25$.

• First difference $b-a = 18 - 12 = 6$.
• Second difference $c-b = 25 - 18 = 7$.

Check the pattern: $a - (c-b) = 12 - 7 = 5$.

Since $5 \neq 10$, this set does not follow the pattern.

Analyzing Option 2: (13, 21, 27)

Here, $a=13$, $b=21$, $c=27$.

• First difference $b-a = 21 - 13 = 8$.
• Second difference $c-b = 27 - 21 = 6$.

Check the pattern: $a - (c-b) = 13 - 6 = 7$.

Since $7 \neq 10$, this set does not follow the pattern.

Analyzing Option 3: (15, 26, 31)

Here, $a=15$, $b=26$, $c=31$.

• First difference $b-a = 26 - 15 = 11$.
• Second difference $c-b = 31 - 26 = 5$.

Check the pattern: $a - (c-b) = 15 - 5 = 10$.

Since $10 = 10$, this set follows the pattern.

Analyzing Option 4: (17, 24, 41)

Here, $a=17$, $b=24$, $c=41$.

• First difference $b-a = 24 - 17 = 7$.
• Second difference $c-b = 41 - 24 = 17$.

Check the pattern: $a - (c-b) = 17 - 17 = 0$.

Since $0 \neq 10$, this set does not follow the pattern.

Conclusion

Based on the analysis, only the set (15, 26, 31) follows the same relationship as the given sets (16, 22, 28) and (14, 19, 23), which is $a - (c-b) = 10$.

Revision Table: Key Concepts in Number Analogy

Additional Information: Tips for Solving Number Pattern Problems

Solving number pattern and number analogy problems often requires exploring different types of relationships between the numbers. Here are some common approaches to consider:

• Look for differences: Calculate the difference between consecutive numbers. See if there's a constant difference (arithmetic progression) or a pattern in the differences themselves (e.g., differences form an AP).
• Look for ratios: Calculate the ratio between consecutive numbers. See if there's a constant ratio (geometric progression).
• Combine operations: The pattern might involve a combination of addition, subtraction, multiplication, or division. For instance, $b = 2a+k$ or $c = a+b-k$.
• Relationships involving the position: Sometimes the rule depends on the position of the number in the set (first, second, third).
• Relationships between the first and third number: Check if the third number is related to the first number directly, perhaps through the middle number. For example, $c = a + k \times b$ or $c = a + (b-a) \times m$.
• Relationships involving squares, cubes, or prime numbers: While less common in basic problems, complex patterns might involve these.
• Test simple hypotheses first: Start by checking for basic arithmetic or geometric progressions before moving to more complex relationships like the one found in this problem ($a - (c-b) = constant$).

Practicing with various types of number pattern questions helps in developing the intuition to quickly spot the correct relationship.