Understanding Number Analogies and Relationships

This question asks us to find a set of numbers from the options that shares the same mathematical relationship between its elements as found in the given sets. We are provided with two example sets: (8, 44, 3) and (12, 60, 3). A key rule is that operations must be performed on the whole numbers themselves, not on their individual digits.

Analyzing the Given Sets to Find the Pattern

Let's examine the relationship between the numbers in the first set, (8, 44, 3). We need to figure out how 8, 44, and 3 are connected. Let's consider the first number (8), the middle number (44), and the third number (3).

• Could the middle number be a result of operations involving the first and third numbers?
• Let's try simple operations like addition, subtraction, multiplication, or a combination.

Looking at the numbers, we notice that the middle number is significantly larger than the first and third numbers. This might suggest multiplication or addition combined with multiplication.

Let's try combining the first and third numbers first:

• Sum: \(8 + 3 = 11\)
• Difference: \(8 - 3 = 5\) or \(3 - 8 = -5\)
• Product: \(8 \times 3 = 24\)

Now, let's see if any of these results can be related to the middle number, 44, perhaps by multiplying by a constant factor or adding/subtracting a constant.

• Sum is 11. If we multiply 11 by 4, we get \(11 \times 4 = 44\). This matches the middle number!

So, a potential pattern is: Middle Number = 4 × (First Number + Third Number).

Verifying the Pattern with the Second Set

Let's check if this pattern holds true for the second given set: (12, 60, 3).

• First Number = 12
• Third Number = 3
• Middle Number = 60

According to our potential pattern: Middle Number = 4 × (First Number + Third Number)

Let's calculate 4 × (First Number + Third Number):

\(4 \times (12 + 3) = 4 \times 15 = 60\)

The calculated value, 60, matches the middle number in the second set. This confirms that the relationship Middle Number = 4 × (First Number + Third Number) is the correct pattern.

Applying the Pattern to the Options

Now we will test each option to see which set follows the same pattern.

Option 1: (17, 56, 9)

• First Number = 17
• Third Number = 9
• Middle Number = 56

Let's apply the pattern: \(4 \times (17 + 9) = 4 \times 26 = 104\)

The calculated value (104) is not equal to the middle number (56). So, this option does not follow the pattern.

Option 2: (12, 68, 5)

• First Number = 12
• Third Number = 5
• Middle Number = 68

Let's apply the pattern: \(4 \times (12 + 5) = 4 \times 17 = 68\)

The calculated value (68) is equal to the middle number (68). This option follows the pattern.

Option 3: (8, 50, 2)

• First Number = 8
• Third Number = 2
• Middle Number = 50

Let's apply the pattern: \(4 \times (8 + 2) = 4 \times 10 = 40\)

The calculated value (40) is not equal to the middle number (50). So, this option does not follow the pattern.

Option 4: (23, 98, 5)

• First Number = 23
• Third Number = 5
• Middle Number = 98

Let's apply the pattern: \(4 \times (23 + 5) = 4 \times 28 = 112\)

The calculated value (112) is not equal to the middle number (98). So, this option does not follow the pattern.

Based on the analysis, only Option 2 follows the same relationship between its numbers as the given sets.

Conclusion

The set of numbers (12, 68, 5) follows the same relationship as the given sets, where the middle number is four times the sum of the first and third numbers.

Revision Table: Key Concepts

Additional Information: Solving Number Series and Analogies

Solving number series and analogy problems often involves looking for common mathematical operations, sequences, or relationships between numbers. Here are some common strategies:

• Arithmetic Operations: Look for patterns involving addition, subtraction, multiplication, or division between consecutive numbers or specific positions (like first and third, first and second, etc.).
• Powers and Roots: The pattern might involve squares, cubes, square roots, etc.
• Sequences: Sometimes the numbers follow known sequences like prime numbers, Fibonacci sequence, etc.
• Digit-Based Operations: Although explicitly excluded by the rule in this specific question, some problems involve operations on the digits of the numbers (e.g., sum of digits, product of digits). Always check the specific rules given.
• Position-Based Rules: The relationship might depend on the position of the number in the set (e.g., the third number is derived from the first two).

Testing potential patterns systematically, as done in this solution, is the key to finding the correct answer.