Solving Number Analogy Questions

This question asks us to find a relationship between the first pair of numbers and apply it to the third number to find the fourth. The given analogy is:

\(- \frac{9}{11} : \frac{11}{9} :: \frac{13}{2} : ?\)

Let's analyze the relationship between the first number, \(- \frac{9}{11}\), and the second number, \(\frac{11}{9}\).

We observe two things:

• The fraction has been inverted. The numerator and the denominator have swapped positions. This is also known as taking the reciprocal of the fraction.
• The sign has changed from negative to positive.

So, the operation applied to \(- \frac{9}{11}\) to get \(\frac{11}{9}\) is taking the reciprocal and changing the sign.

Mathematically, taking the reciprocal of \(- \frac{9}{11}\) gives \(- \frac{11}{9}\). Then, changing the sign of \(- \frac{11}{9}\) gives \(+ \frac{11}{9}\), or simply \(\frac{11}{9}\). However, let's consider the operation as a whole. If we take the negative of the reciprocal of \(- \frac{9}{11}\), we get \(- \left( \frac{1}{-\frac{9}{11}} \right) = - \left( -\frac{11}{9} \right) = \frac{11}{9}\). This operation is "take the negative reciprocal".

Alternatively, if we consider taking the reciprocal first, \(\frac{1}{-\frac{9}{11}} = -\frac{11}{9}\). Then, the sign changes from negative to positive. This implies multiplying by -1 after taking the reciprocal.

Let's test the operation "take the negative reciprocal".

Negative reciprocal of \(x\) is \(-\frac{1}{x}\).

For \(- \frac{9}{11}\), the negative reciprocal is \(- \frac{1}{-\frac{9}{11}} = - \left( -\frac{11}{9} \right) = \frac{11}{9}\). This matches the second number.

Now, we apply the same relationship to the third number, \(\frac{13}{2}\).

We need to find the negative reciprocal of \(\frac{13}{2}\).

The reciprocal of \(\frac{13}{2}\) is \(\frac{1}{\frac{13}{2}} = \frac{2}{13}\).

The negative reciprocal of \(\frac{13}{2}\) is \(- \frac{2}{13}\).

So, the missing number in the analogy is \(- \frac{2}{13}\).

Let's check the given options:

• Option 1: \(\frac{3}{7}\)
• Option 2: \(\frac{2}{13}\)
• Option 3: \(- \frac{2}{13}\)
• Option 4: \(- \frac{7}{3}\)

Our calculated number \(- \frac{2}{13}\) matches Option 3.

Thus, the relationship is taking the negative reciprocal.

Conclusion on Number Analogy

Based on the relationship observed in the first pair of numbers, applying the operation of taking the negative reciprocal to the third number \(\frac{13}{2}\) gives \(- \frac{2}{13}\).

The correct option is \(- \frac{2}{13}\).

Revision Table: Key Concepts for Number Analogies

Additional Information on Number Relationships and Analogies

Number analogies often test your understanding of basic mathematical operations and relationships between numbers. These can include:

• Addition or subtraction of a constant.
• Multiplication or division by a constant.
• Squaring, cubing, or taking roots.
• Operations involving reciprocals or negative reciprocals.
• Combining multiple operations.
• Looking at properties like prime numbers, composite numbers, odd/even numbers.

When solving number analogy questions, it's helpful to:

• Examine the first pair closely to find a consistent pattern or operation.
• Test the identified pattern with the third number.
• Check if the result matches any of the given options.
• If no simple pattern is found, consider more complex relationships or a combination of operations.

Understanding concepts like reciprocals and how signs change during operations is crucial for solving problems involving fractions like the one discussed here. The negative reciprocal relationship is common in topics like slopes of perpendicular lines in coordinate geometry, but it also appears in abstract number pattern questions.