Finding the Relationship Between Numbers in Sets

The question asks us to identify a set of fractions that follows the same numerical relationship observed in the two provided sets. We are given the rule that operations must be performed on the whole numbers (numerator and denominator) without breaking them down into individual digits.

The given sets are:

• Set 1: \(\left[\left(\frac{7}{9}\right),\left(\frac{31}{39}\right)\right]\)
• Set 2: \(\left[\left(\frac{3}{5}\right),\left(\frac{15}{23}\right)\right]\)

Let's analyze the relationship between the first fraction \(\frac{a}{b}\) and the second fraction \(\frac{c}{d}\) in each set.

Analyzing Set 1 Relationship

For Set 1, we have \(\frac{a}{b} = \frac{7}{9}\) and \(\frac{c}{d} = \frac{31}{39}\).

Let's look at how the numbers change from the first fraction to the second:

• Numerator: 7 becomes 31
• Denominator: 9 becomes 39

We need to find an operation or a set of operations involving the whole numbers 7 and 9 that results in 31 and 39, respectively. A common pattern in such problems is a linear transformation like multiplying by a constant and adding/subtracting another constant. Let's assume the relationship for the numerator is \(c = k \times a + m\) and for the denominator is \(d = k \times b + m\), where \(k\) and \(m\) are constants.

Using Set 1 values:

• For the numerator: \(31 = k \times 7 + m\)
• For the denominator: \(39 = k \times 9 + m\)

We have a system of two linear equations:

\(7k + m = 31 \quad (1)\)

\(9k + m = 39 \quad (2)\)

Subtract equation (1) from equation (2):

\((9k + m) - (7k + m) = 39 - 31\)

\(2k = 8\)

\(k = \frac{8}{2} = 4\)

Now substitute \(k=4\) into equation (1):

\(7 \times 4 + m = 31\)

\(28 + m = 31\)

\(m = 31 - 28 = 3\)

So, the potential relationship is: Numerator becomes \(4 \times a + 3\), and Denominator becomes \(4 \times b + 3\). In terms of fractions, the relationship is \(\frac{c}{d} = \frac{4a+3}{4b+3}\).

Verifying the Relationship with Set 2

Let's check if this relationship holds for Set 2: \(\left[\left(\frac{3}{5}\right),\left(\frac{15}{23}\right)\right]\). Here, \(a=3\) and \(b=5\).

Using the relationship \(\frac{4a+3}{4b+3}\):

• New Numerator: \(4 \times 3 + 3 = 12 + 3 = 15\)
• New Denominator: \(4 \times 5 + 3 = 20 + 3 = 23\)

The resulting fraction is \(\frac{15}{23}\), which matches the second fraction in Set 2. This confirms that the relationship is indeed \(\frac{c}{d} = \frac{4a+3}{4b+3}\).

Applying the Relationship to the Options

Now we need to examine each option to see which set follows this established relationship.

For each option \(\left[\left(\frac{a}{b}\right),\left(\frac{c}{d}\right)\right]\), we will calculate \(\frac{4a+3}{4b+3}\) and see if it equals \(\frac{c}{d}\).

Option 1: \(\left[\left(\frac{11}{13}\right),\left(\frac{47}{55}\right)\right]\)

Here, \(a=11\) and \(b=13\).

• Calculated Numerator: \(4 \times 11 + 3 = 44 + 3 = 47\)
• Calculated Denominator: \(4 \times 13 + 3 = 52 + 3 = 55\)

The calculated fraction is \(\frac{47}{55}\). This matches the second fraction in Option 1.

Therefore, Option 1 follows the same relationship.

Option 2: \(\left[\left(\frac{9}{13}\right),\left(\frac{37}{55}\right)\right]\)

Here, \(a=9\) and \(b=13\).

• Calculated Numerator: \(4 \times 9 + 3 = 36 + 3 = 39\)
• Calculated Denominator: \(4 \times 13 + 3 = 52 + 3 = 55\)

The calculated fraction is \(\frac{39}{55}\). This does not match \(\frac{37}{55}\).

Option 3: \(\left[\left(\frac{9}{11}\right),\left(\frac{32}{37}\right)\right]\)

Here, \(a=9\) and \(b=11\).

• Calculated Numerator: \(4 \times 9 + 3 = 36 + 3 = 39\)
• Calculated Denominator: \(4 \times 11 + 3 = 44 + 3 = 47\)

The calculated fraction is \(\frac{39}{47}\). This does not match \(\frac{32}{37}\).

Option 4: \(\left[\left(\frac{17}{19}\right),\left(\frac{36}{77}\right)\right]\)

Here, \(a=17\) and \(b=19\).

• Calculated Numerator: \(4 \times 17 + 3 = 68 + 3 = 71\)
• Calculated Denominator: \(4 \times 19 + 3 = 76 + 3 = 79\)

The calculated fraction is \(\frac{71}{79}\). This does not match \(\frac{36}{77}\).

Only Option 1 satisfies the numerical relationship found from the given sets.

Conclusion

Based on the analysis, the set of numbers that shares the same relationship as the given sets is Option 1: \(\left[\left(\frac{11}{13}\right),\left(\frac{47}{55}\right)\right]\), where the relationship between the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) is \(\frac{c}{d} = \frac{4a+3}{4b+3}\).

Revision Table: Understanding Number Set Relationships

Additional Information: Solving Quantitative Aptitude Problems

Quantitative Aptitude problems often involve identifying patterns and relationships between numbers. These can appear in various forms, including number series, analogies, coding-decoding, and set-based questions like this one. To solve such problems effectively, consider the following:

• Look for Common Operations: Think about basic arithmetic operations (addition, subtraction, multiplication, division) and how they might connect the numbers.
• Consider Transformations: Sometimes, the relationship involves multiplying by a constant, adding/subtracting a constant, squaring, cubing, or combinations of these.
• Check Both Numerator and Denominator: In fraction-based problems, the relationship might apply separately but similarly to both parts of the fraction.
• Test Hypotheses: Once you find a potential rule using one example (like Set 1 here), always test it with other examples provided (like Set 2) before applying it to the options.
• Systematic Checking: Go through each option methodically, applying the rule to see which one fits.
• Practice: Familiarity with common patterns and problem types comes with practice.

This question specifically tests your ability to find a consistent linear relationship applied symmetrically to both the numerator and the denominator of fractions within a set.