Finding the Number Pattern in Sets

The question asks us to identify a set of numbers from the options that shares the same mathematical relationship between its elements as observed in the two given sets: (49, 63, 441) and (7, 14, 14).

We need to find a rule that applies to both (49, 63, 441) and (7, 14, 14). Let's represent a general set as (a, b, c).

Analyzing the Given Sets to Find the Pattern

Let's examine the numbers in the first set: (49, 63, 441).

• How is 63 related to 49? $63 = 49 + 14$ or $63 = 49 \times \frac{63}{49} = 49 \times \frac{9}{7}$.
• How is 441 related to 49? $441 = 49 \times 9$. Note that $49 = 7^2$ and $441 = 21^2 = (3 \times 7)^2 = 9 \times 49$.
• How is 441 related to 63? $441 = 63 \times 7$.

Now let's examine the numbers in the second set: (7, 14, 14).

• How is 14 related to 7? $14 = 7 + 7$ or $14 = 7 \times 2$.
• How is the second 14 related to 7? $14 = 7 \times 2$.
• How is the second 14 related to the first 14? $14 = 14 \times 1$.

We are looking for a common pattern that links (49, 63, 441) and (7, 14, 14).

Let's consider relationships between the three numbers (a, b, c) in each set. From the first set, we saw $441 = 63 \times 7$. From the second set, we saw $14 = 14 \times 1$. The multiplier (7 in the first case, 1 in the second) doesn't seem directly related to the numbers in the set in an obvious way.

Let's look at the relationship we found: $c = b \times (\text{some number})$. For the first set, $441 = 63 \times 7$. For the second set, $14 = 14 \times 1$. The multiplier is 7 for the first set and 1 for the second set.

Let's try another observation from Set 1: $441 = 49 \times 9$. For Set 2: $14 = 7 \times 2$. This pattern $c = a \times (\text{multiplier})$ gives multipliers 9 and 2, which are different.

Let's revisit the calculation $441 = 63 \times 7$ from Set 1 and $14 = 14 \times 1$ from Set 2. Is there a relation involving the first number 'a'? Notice in Set 1, the multiplier is 7, which is not directly 49, 63, or 441, but it *is* the first number of the *second* given set. This might be a coincidence, or hint at a cross-set rule, but the question implies a rule within a single set.

Let's look closer at the relationship $c = a \times \frac{b}{7}$ that we briefly considered. For Set 1 (49, 63, 441): $a=49, b=63, c=441$. Let's check if $c = a \times \frac{b}{7}$. $49 \times \frac{63}{7} = 49 \times 9 = 441$. This matches $c$. For Set 2 (7, 14, 14): $a=7, b=14, c=14$. Let's check if $c = a \times \frac{b}{7}$. $7 \times \frac{14}{7} = 7 \times 2 = 14$. This matches $c$.

The pattern that holds for both sets is $\mathbf{c = a \times \frac{b}{7}}$. This can be rewritten as $7c = ab$.

Testing the Options Against the Pattern $7c = ab$

We will now check each option to see which set (a, b, c) satisfies the relation $7c = ab$.

Option 1: (7, 28, 35)

• $a = 7, b = 28, c = 35$
• Calculate $7c$: $7 \times 35 = 245$
• Calculate $ab$: $7 \times 28 = 196$
• Is $7c = ab$? $245 \ne 196$. Option 1 does not follow the pattern.

Option 2: (14, 28, 56)

• $a = 14, b = 28, c = 56$
• Calculate $7c$: $7 \times 56 = 392$
• Calculate $ab$: $14 \times 28 = 392$
• Is $7c = ab$? $392 = 392$. Option 2 follows the pattern.

Option 3: (7, 14, 45)

• $a = 7, b = 14, c = 45$
• Calculate $7c$: $7 \times 45 = 315$
• Calculate $ab$: $7 \times 14 = 98$
• Is $7c = ab$? $315 \ne 98$. Option 3 does not follow the pattern.

Option 4: (12, 18, 324)

• $a = 12, b = 18, c = 324$
• Calculate $7c$: $7 \times 324 = 2268$
• Calculate $ab$: $12 \times 18 = 216$
• Is $7c = ab$? $2268 \ne 216$. Option 4 does not follow the pattern.

Only Option 2 satisfies the relationship $7c = ab$ (or $c = a \times \frac{b}{7}$) that we found in the given sets.

Conclusion

The set of numbers that are related in the same way as the numbers in the sets (49, 63, 441) and (7, 14, 14) is (14, 28, 56), because for this set (a, b, c), the relationship $7c = ab$ holds true.

Revision Table: Key Concepts in Number Analogies

Additional Information: Strategies for Solving Number Pattern Questions

Solving number pattern and analogy questions requires systematic analysis. Here are some strategies:

• Look for basic arithmetic relations: Addition, subtraction, multiplication, division between numbers in the set. Is there a common difference or ratio?
• Check for squares, cubes, or other powers: Are the numbers related to squares or cubes of other numbers in the set or a base number?
• Examine ratios: Calculate the ratios between the first and second, first and third, or second and third numbers. Are the ratios constant across the given sets? (e.g., $a:b:c$ ratio).
• Consider combinations of operations: The relationship might involve more than one operation, like $c = a \times b + k$ or $c = (a+b) \times k$.
• Test relationships between pairs: How is the second number related to the first? How is the third number related to the first or second?
• Look for sequences: Sometimes the numbers follow a specific sequence rule (e.g., arithmetic progression, geometric progression, Fibonacci sequence, though less common in set analogies unless explicitly stated).
• Pay attention to constraints: The note in this question specified operating on whole numbers, not constituent digits, which is a common constraint in such problems.
• Systematically test possibilities: If a pattern isn't immediately obvious, list potential simple relationships and test them on the given sets.

In this specific problem, the pattern $c = a \times (b/7)$ which is equivalent to $ab = 7c$ is a less common but valid type of relationship involving a constant (7 in this case) and the numbers in the set. The key is finding a rule that consistently applies to all provided examples before testing the options.