Understanding Number Set Relationships

This question asks us to identify a set of three numbers that shares the same mathematical relationship as the two given sets: (31, 47, 35) and (51, 67, 55). We are instructed to perform operations only on the whole numbers themselves, not on their individual digits.

Analyzing the Given Number Sets to Find the Pattern

Let's examine the first given set: (31, 47, 35). Let's call the numbers A, B, and C respectively. So, A = 31, B = 47, and C = 35.

Let's look at the differences between these numbers:

• Difference between B and A: \(B - A = 47 - 31 = 16\)
• Difference between C and A: \(C - A = 35 - 31 = 4\)

We can see that \(16\) is \(4\) times \(4\). This suggests a possible relationship: \(B - A = 4 \times (C - A)\).

Now, let's check this potential relationship with the second given set: (51, 67, 55). Here, A = 51, B = 67, and C = 55.

• Difference between B and A: \(B - A = 67 - 51 = 16\)
• Difference between C and A: \(C - A = 55 - 51 = 4\)

Again, we find that \(16 = 4 \times 4\). So, the relationship \(B - A = 4 \times (C - A)\) holds true for both given sets.

This is the pattern or relationship we need to find in the given options.

Testing the Options for the Same Number Set Relationship

We will now test each option provided to see which one follows the relationship \(B - A = 4 \times (C - A)\).

Option 1: (33, 47, 35)

Here, A = 33, B = 47, C = 35.

• Calculate B - A: \(47 - 33 = 14\)
• Calculate C - A: \(35 - 33 = 2\)
• Check the relationship: \(14 = 4 \times 2\) > \(14 = 8\)

The relationship does not hold for Option 1. This set is not related in the same way.

Option 2: (13, 29, 17)

Here, A = 13, B = 29, C = 17.

• Calculate B - A: \(29 - 13 = 16\)
• Calculate C - A: \(17 - 13 = 4\)
• Check the relationship: \(16 = 4 \times 4\) > \(16 = 16\)

The relationship holds true for Option 2. This set is related in the same way as the given sets.

Option 3: (32, 39, 27)

Here, A = 32, B = 39, C = 27.

• Calculate B - A: \(39 - 32 = 7\)
• Calculate C - A: \(27 - 32 = -5\)
• Check the relationship: \(7 = 4 \times (-5)\) > \(7 = -20\)

The relationship does not hold for Option 3. This set is not related in the same way.

Option 4: (21, 36, 35)

Here, A = 21, B = 36, C = 35.

• Calculate B - A: \(36 - 21 = 15\)
• Calculate C - A: \(35 - 21 = 14\)
• Check the relationship: \(15 = 4 \times 14\) > \(15 = 56\)

The relationship does not hold for Option 4. This set is not related in the same way.

Identifying the Correct Number Set

Based on our analysis, only Option 2 follows the same mathematical relationship \(B - A = 4 \times (C - A)\) as the given sets.

Revision Table: Number Set Analysis

This table summarizes the calculations for each set and option, making it easy to compare results and verify the pattern \(B - A = 4 \times (C - A)\).

Additional Information on Logical Reasoning and Number Sets

Number set and series problems are common in logical reasoning and quantitative aptitude tests. They require you to identify the underlying pattern or rule that relates the numbers in a given set or sequence. The patterns can involve basic arithmetic operations (addition, subtraction, multiplication, division), powers, roots, or combinations of these. Sometimes, the pattern might involve the position of the numbers within the set (like the relationship between the first, second, and third numbers here).

Key strategies for solving these types of problems include:

• Looking at differences between consecutive numbers.
• Looking at ratios between consecutive numbers.
• Considering differences or sums of alternate numbers.
• Checking for relationships between the first and last number, or first and middle number, etc., as demonstrated in this problem.
• Trying combinations of operations.
• Testing your hypothesized rule against all given examples before applying it to the options.

The constraint about not breaking down numbers into constituent digits is important. It guides us to treat 31 as the number 'thirty-one' rather than looking at '3' and '1' separately. This is typical in many competitive exams to prevent ambiguous solutions.