Understanding Number Set Relationships

The question asks us to identify the option where the set of numbers shares the same mathematical relationship as the numbers in the two given sets. We need to find a pattern that applies to both (12, 24, 72) and (14, 28, 84).

Analyzing the Given Number Sets

Let's look closely at the first set: (12, 24, 72).

• The second number (24) is double the first number (12). Mathematically, $24 = 12 \times 2$.
• The third number (72) seems related to the first or second. Let's check the relationship with the first number: $72 = 12 \times 6$.

So, the relationship in the first set appears to be (x, $x \times 2$, $x \times 6$).

Now let's check this pattern with the second given set: (14, 28, 84).

• The first number is 14.
• Is the second number (28) equal to $14 \times 2$? Yes, $28 = 14 \times 2$.
• Is the third number (84) equal to $14 \times 6$? Yes, $84 = 14 \times 6$.

The pattern (x, $2x$, $6x$) holds true for both given sets. The operations are performed on the whole numbers as specified in the question.

Evaluating the Options for the Same Relationship

We will now check each option to see which one follows the established pattern (x, $2x$, $6x$).

Option 1: (20, 40, 160)

• First number (x) = 20.
• Second number: Check if it is $2x$. $20 \times 2 = 40$. This matches the second number in the option.
• Third number: Check if it is $6x$. $20 \times 6 = 120$. The third number in the option is 160.

This option does not follow the pattern (x, $2x$, $6x$).

Option 2: (24, 72, 216)

• First number (x) = 24.
• Second number: Check if it is $2x$. $24 \times 2 = 48$. The second number in the option is 72.

This option does not follow the pattern (x, $2x$, $6x$). We don't need to check the third number.

Option 3: (16, 32, 96)

• First number (x) = 16.
• Second number: Check if it is $2x$. $16 \times 2 = 32$. This matches the second number in the option.
• Third number: Check if it is $6x$. $16 \times 6 = 96$. This matches the third number in the option.

This option perfectly follows the pattern (x, $2x$, $6x$).

Option 4: (18, 36, 118)

• First number (x) = 18.
• Second number: Check if it is $2x$. $18 \times 2 = 36$. This matches the second number in the option.
• Third number: Check if it is $6x$. $18 \times 6 = 108$. The third number in the option is 118.

This option does not follow the pattern (x, $2x$, $6x$).

Conclusion

Based on the analysis, only Option 3: (16, 32, 96) shares the same relationship (x, $2x$, $6x$) as the numbers in the given sets (12, 24, 72) and (14, 28, 84).

Revision Table: Number Relationship Reasoning

• Identify the pattern in the given sets.
• Test the pattern with all given sets to confirm.
• Apply the identified pattern to each option.
• Select the option that matches the pattern.
• Remember the constraint about not breaking down numbers into digits.

Additional Information: Number Reasoning Problems

Number reasoning problems often involve finding mathematical relationships between numbers in a set or sequence. These relationships can be based on basic arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, or combinations of these. Identifying the pattern quickly is key to solving such problems efficiently. Sometimes, there might be multiple layers of relationships or patterns involving differences between consecutive numbers, ratios, or specific mathematical properties.