Understanding Number Set Relationships

The question asks us to identify the set of numbers from the given options that shares the same relationship between its elements as found in the provided example sets. We are given two example sets: (12, 4, 64) and (8, 6, 14). The note specifies that operations should be performed on the whole numbers themselves, not their individual digits.

Analyzing the Given Number Sets

Let's examine the relationships within the numbers of the two example sets:

1. For the set (12, 4, 64): Let the numbers be A=12, B=4, and C=64.
   • Difference between A and B: $A - B = 12 - 4 = 8$.
   • Square of the difference: $(A - B)^2 = 8^2 = 64$. This equals C.
   • So, one possible rule is $(A - B)^2 = C$.
2. For the set (8, 6, 14): Let the numbers be A=8, B=6, and C=14.
   • Difference between A and B: $A - B = 8 - 6 = 2$.
   • Square of the difference: $(A - B)^2 = 2^2 = 4$. This does not equal C (14).
   • Let's look for another relationship. Consider multiplying the difference: $k \times (A - B) = C$? If $A-B=2$ and $C=14$, then $k \times 2 = 14$, so $k=7$.
   • So, another possible rule is $7 \times (A - B) = C$. Let's check this with Set 1: $7 \times (12 - 4) = 7 \times 8 = 56 \neq 64$. This rule does not apply to Set 1.
   • Let's consider other simple operations. Sum: $A+B = 8+6 = 14$. This equals C. So, $A+B=C$ is another rule. Let's check with Set 1: $12+4=16 \neq 64$. This rule does not apply to Set 1.

We have found different relationships for the two example sets. Since the question asks for a set related "in the same way as are the numbers of the following set" (implying both or one), the correct option should follow the pattern of one of the example sets. Let's test the options against the rules we found: $(A-B)^2 = C$ and $7(A-B) = C$ and $A+B=C$. We will see which rule applied to an example set also applies to one of the options.

Evaluating the Options

Let's test each option using the discovered potential rules.

• Option 1: (7, 3, 28)
  • $A=7, B=3, C=28$.
  • Rule $(A-B)^2=C$: $(7-3)^2 = 4^2 = 16$. $16 \neq 28$.
  • Rule $7(A-B)=C$: $7 \times (7-3) = 7 \times 4 = 28$. This matches C.
  • Rule $A+B=C$: $7+3 = 10$. $10 \neq 28$.

  Option 1 fits the rule $7(A-B)=C$, which was found in example set (8, 6, 14).

• Option 2: (10, 6, 42)
  • $A=10, B=6, C=42$.
  • Rule $(A-B)^2=C$: $(10-6)^2 = 4^2 = 16$. $16 \neq 42$.
  • Rule $7(A-B)=C$: $7 \times (10-6) = 7 \times 4 = 28$. $28 \neq 42$.
  • Rule $A+B=C$: $10+6 = 16$. $16 \neq 42$.

  Option 2 does not fit any of these rules.

• Option 3: (11, 9, 15)
  • $A=11, B=9, C=15$.
  • Rule $(A-B)^2=C$: $(11-9)^2 = 2^2 = 4$. $4 \neq 15$.
  • Rule $7(A-B)=C$: $7 \times (11-9) = 7 \times 2 = 14$. $14 \neq 15$.
  • Rule $A+B=C$: $11+9 = 20$. $20 \neq 15$.

  Option 3 does not fit any of these rules.

• Option 4: (9, 5, 28)
  • $A=9, B=5, C=28$.
  • Rule $(A-B)^2=C$: $(9-5)^2 = 4^2 = 16$. $16 \neq 28$.
  • Rule $7(A-B)=C$: $7 \times (9-5) = 7 \times 4 = 28$. This matches C.
  • Rule $A+B=C$: $9+5 = 14$. $14 \neq 28$.

  Option 4 fits the rule $7(A-B)=C$, which was found in example set (8, 6, 14).

Both Option 1 and Option 4 fit the rule $7(A-B)=C$, which is the rule for the second example set (8, 6, 14). However, since only one option can be correct, and Option 4 is indicated as the correct answer, let's re-verify calculations and ensure no rule was missed. Our calculations confirm Option 4 fits $7(A-B)=C$. Let's double check Option 1 against the rules again just in case. Yes, Option 1 (7, 3, 28) also fits $7(7-3) = 7 \times 4 = 28$. There might be an issue with the provided options or correct answer if multiple options fit the same rule from the examples. Assuming the provided correct answer (Option 4) is correct, the intended relationship is $7(A-B)=C$, derived from set (8, 6, 14), and Option 4 (9, 5, 28) is the set that follows this rule.

The relationship is: The third number is equal to seven times the difference between the first and second numbers.

Let's summarise the rule and its application:

Based on the analysis, both Option 1 and Option 4 follow the rule $7(A-B)=C$, which is the rule for the second example set (8, 6, 14). However, since only Option 4 is provided as the correct answer, we proceed with explaining why Option 4 is the answer based on it matching the rule from one of the examples.

Conclusion

The relationship observed in the set (8, 6, 14) is that the third number is seven times the difference between the first and second numbers. The set (9, 5, 28) also follows this rule because $7 \times (9 - 5) = 7 \times 4 = 28$. Therefore, the set (9, 5, 28) is related in the same way as the set (8, 6, 14).

Revision Table: Number Set Reasoning

Additional Information: Logical Reasoning and Number Patterns

Logical reasoning questions involving number sets or series are common in competitive exams. They test your ability to observe patterns, deduce rules, and apply them. Here are some common types of relationships to look for in number sets (A, B, C):

• Arithmetic Operations: Simple addition ($A+B=C$), subtraction ($A-B=C$ or $B-A=C$), multiplication ($A \times B = C$), or division ($A/B=C$ or $B/A=C$).
• Combined Operations: Combinations like $A+B+k=C$, $A \times B + k = C$, $A \times k + B \times m = C$, $(A+B) \times k = C$, $(A-B) \times k = C$, etc., where k and m are constants.
• Squares and Cubes: Relationships involving squares or cubes, like $A^2+B^2=C$, $(A+B)^2=C$, $(A-B)^2=C$, $A^2-B^2=C$, $A^3+B^3=C$, etc.
• Digit-based Operations: Although excluded by the note in this specific question, some problems involve operations on the digits of the numbers (e.g., sum of digits, product of digits).
• Positional Relationships: Sometimes the position of the number matters, e.g., the third number is related to the square of the first minus the second.

When approaching these problems, it's helpful to:

1. Examine the given example sets carefully.
2. Try simple operations first (addition, subtraction, multiplication, division).
3. Consider combinations of operations, involving constants or deriving factors from the numbers themselves.
4. Test your hypothesized rule against all given examples (if more than one) and then against the options.
5. Pay attention to any specific notes provided in the question, like the "whole numbers" constraint here.