Understanding Number Set Relations

This question asks us to identify a set of numbers from the given options that shares the same mathematical relationship as the two provided sets: (25, 30, 65) and (50, 50, 100). We need to find a consistent rule that applies to all three numbers within each of these sets, using operations on the whole numbers themselves, not their individual digits.

Analyzing the Given Number Sets

Let's look closely at the two example sets to find a common pattern:

• Set 1: (25, 30, 65)
• Set 2: (50, 50, 100)

We need a rule that connects the first two numbers to the third number in both sets. Let the three numbers in a set be denoted as A, B, and C.

Exploring Potential Rules

Let's test some common mathematical relationships:

• Simple Addition: A + B = C?
  • Set 1: $25 + 30 = 55 \neq 65$
  • Set 2: $50 + 50 = 100$. This works for Set 2, but not Set 1. So, simple addition is not the general rule.
• Addition and Multiplication: A + B + K = C or A × k₁ + B × k₂ = C?
  • For Set 1: $25 + 30 = 55$. To get 65, we add 10 ($55 + 10 = 65$).
  • For Set 2: $50 + 50 = 100$. To get 100, we add 0 ($100 + 0 = 100$).

  The constant added is different (10 vs 0), suggesting this isn't a simple A+B+Constant rule.

• Relationship involving Difference: Let's consider the difference between the first two numbers, $|B-A|$.
  • Set 1: A=25, B=30. $|30-25| = 5$. Sum = $25+30=55$. How to get 65 from 55 and 5? $55 + 2 \times 5 = 55 + 10 = 65$.
  • Set 2: A=50, B=50. $|50-50| = 0$. Sum = $50+50=100$. How to get 100 from 100 and 0? $100 + 2 \times 0 = 100 + 0 = 100$.

  This relationship, $C = (A+B) + 2 \times |B-A|$, works for both given sets.

Testing the Discovered Rule on Options

Let's apply the rule $C = A + B + 2 \times |B-A|$ to the given options:

• Option 1: (44, 157, 14)
  A=44, B=157, C=14
  $A+B = 44 + 157 = 201$
  $|B-A| = |157 - 44| = 113$
  $C_{expected} = 201 + 2 \times 113 = 201 + 226 = 427$
  Actual C = 14. Since $427 \neq 14$, this option does not follow the rule.
• Option 2: (25, 132, 18)
  A=25, B=132, C=18
  $A+B = 25 + 132 = 157$
  $|B-A| = |132 - 25| = 107$
  $C_{expected} = 157 + 2 \times 107 = 157 + 214 = 371$
  Actual C = 18. Since $371 \neq 18$, this option does not follow the rule.
• Option 3: (45, 40, 75)
  A=45, B=40, C=75
  $A+B = 45 + 40 = 85$
  $|B-A| = |40 - 45| = |-5| = 5$
  $C_{expected} = 85 + 2 \times 5 = 85 + 10 = 95$
  Actual C = 75. Since $95 \neq 75$, this option does not follow the rule.
• Option 4: (35, 178, 15)
  A=35, B=178, C=15
  $A+B = 35 + 178 = 213$
  $|B-A| = |178 - 35| = 143$
  $C_{expected} = 213 + 2 \times 143 = 213 + 286 = 499$
  Actual C = 15. Since $499 \neq 15$, this option does not follow the rule.

My initially discovered rule $C = (A+B) + 2 \times |B-A|$ seems incorrect as it doesn't match any of the options, including the provided correct answer. Let's reconsider the given examples and the provided correct answer option to find the true pattern.

Let's re-examine the sets: (25, 30, 65), (50, 50, 100), and the presumed correct one (45, 40, 75).

• Set 1: (25, 30, 65)
• Set 2: (50, 50, 100)
• Option 3: (45, 40, 75)

Let's look at the relationship between A, B, and C again. Consider simple linear relationships like $C = xA + yB$.

• Set 1: $xA + yB = C \implies 25x + 30y = 65$
• Set 2: $xA + yB = C \implies 50x + 50y = 100$

From the second equation, dividing by 50 gives $x + y = 2$. So, $y = 2-x$. Substitute into the first equation: $25x + 30(2-x) = 65$ $25x + 60 - 30x = 65$ $-5x = 65 - 60$ $-5x = 5$ $x = -1$ Now find y: $y = 2 - x = 2 - (-1) = 2 + 1 = 3$. So the rule is $C = -1 \times A + 3 \times B$, or $C = 3B - A$.

Verifying the Rule $C = 3B - A$ on Given Sets

• Set 1 (25, 30, 65): A=25, B=30. $C = 3(30) - 25 = 90 - 25 = 65$. This matches!
• Set 2 (50, 50, 100): A=50, B=50. $C = 3(50) - 50 = 150 - 50 = 100$. This matches!

This rule $C = 3B - A$ is consistent for both given sets.

Applying the Rule $C = 3B - A$ to the Options

Let's use the rule $C = 3B - A$ to check the options:

• Option 1: (44, 157, 14)
  A=44, B=157
  $C_{expected} = 3 \times 157 - 44 = 471 - 44 = 427$
  Actual C = 14. Since $427 \neq 14$, this is incorrect.
• Option 2: (25, 132, 18)
  A=25, B=132
  $C_{expected} = 3 \times 132 - 25 = 396 - 25 = 371$
  Actual C = 18. Since $371 \neq 18$, this is incorrect.
• Option 3: (45, 40, 75)
  A=45, B=40
  $C_{expected} = 3 \times 40 - 45 = 120 - 45 = 75$
  Actual C = 75. Since $75 = 75$, this option follows the rule!
• Option 4: (35, 178, 15)
  A=35, B=178
  $C_{expected} = 3 \times 178 - 35 = 534 - 35 = 499$
  Actual C = 15. Since $499 \neq 15$, this is incorrect.

Only Option 3 follows the established rule $C = 3B - A$.

Conclusion

The relationship between the numbers (A, B, C) in the given sets is that the third number C is obtained by multiplying the second number B by 3 and then subtracting the first number A. This can be expressed as $C = 3B - A$. Applying this rule to the options, we found that only the set (45, 40, 75) satisfies this condition.

Let's summarize the rule and its application:

Therefore, the set (45, 40, 75) is related in the same way as the given sets.

Revision Table: Key Learnings

Additional Information: Solving Number Pattern Problems

Number pattern and analogy problems are common in reasoning and quantitative aptitude tests. Here are some strategies to tackle them:

• Look for simple arithmetic operations: addition, subtraction, multiplication, division between numbers.
• Consider operations involving squares, cubes, square roots, or cube roots.
• Check for patterns related to prime numbers, composite numbers, or specific number sequences.
• Investigate relationships involving the sum or difference of numbers, possibly multiplied by a constant.
• Sometimes, the pattern might involve the difference between consecutive terms, or the ratio between them.
• For sets of three numbers (A, B, C), common patterns relate C to A and B using combinations of operations. Examples include $C = A+B$, $C = A \times B$, $C = A^2 + B^2$, $C = k_1 A + k_2 B$, or relationships like the one found here ($C = 3B - A$).
• Always verify a potential rule with all given examples before applying it to the options.
• If a simple rule doesn't work, look for more complex relationships or relationships involving constants derived from the numbers themselves (like their difference or sum).
• Pay attention to any specific constraints mentioned in the question, such as operating only on whole numbers.