Problem 23 - Fundamentals of Mathematics

Question

For the following problems, find the reciprocal of each number. $$\frac{2}{9}$$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{2}{9} \) is \( \frac{9}{2} \).

Step 1: Understand the Concept of Reciprocal

The reciprocal of a number is 1 divided by that number. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). So, to find the reciprocal, you simply swap the numerator and the denominator.

Step 2: Identify the Numerator and Denominator

In the fraction \( \frac{2}{9} \), the numerator is 2 and the denominator is 9.

Step 3: Swap the Numerator and Denominator

To find the reciprocal, swap the numerator and the denominator of the fraction. Therefore, the reciprocal of \( \frac{2}{9} \) becomes \( \frac{9}{2} \).

Step 4: Verify the Solution

The product of a number and its reciprocal should equal 1. Calculate \( \frac{2}{9} \times \frac{9}{2} = \frac{18}{18} = 1 \), confirming that \( \frac{9}{2} \) is indeed the reciprocal.

Key Concepts

Fraction Inversion

When dealing with fractions, understanding how to find the reciprocal—also called fraction inversion—is a fundamental skill. The reciprocal of a fraction is simply the result of flipping the numerator and the denominator. This is sometimes described as inverting the fraction.

If you have a fraction like \( \frac{a}{b} \), the reciprocal would be \( \frac{b}{a} \). This means that whatever was originally on top in the fraction under consideration (the numerator), becomes the bottom (the denominator), and vice versa.
This inversion process provides a valuable tool in various mathematical operations, especially when dealing with fraction division.

Numerator and Denominator

Understanding what constitutes the numerator and the denominator in a fraction is key to manipulating fractions correctly. The numerator is the top number, indicating how many parts we have, while the denominator is the bottom number, showing into how many parts the whole is divided.

In the fraction \( \frac{2}{9} \), 2 is the numerator because it is situated above the fraction line, and 9 is the denominator since it is below the line.

• Numerator: 2
• Denominator: 9

Being clear about these positions is crucial when finding the reciprocal, as it includes swapping these two elements.

Verify Reciprocal

Once you've found the reciprocal of a fraction, it's important to verify your solution. Verification involves multiplying the original fraction by its reciprocal. A successful product will always be 1 if you did everything correctly.

For example, when you take the original fraction \( \frac{2}{9} \) and its reciprocal \( \frac{9}{2} \), their multiplication gives \( \frac{2}{9} \times \frac{9}{2} = \frac{18}{18} = 1 \).
This result is a crucial check since it confirms that your inversion process was correct, ensuring that the switch of the numerator and denominator was executed properly.

Mathematical Operations

The concept of a reciprocal is intertwined with various mathematical operations. By knowing how to find and use the reciprocal, you're more prepared to tackle problems that involve dividing fractions.

For dividing by a fraction, you simply multiply by its reciprocal. For instance, dividing by \( \frac{2}{9} \) is the same as multiplying by \( \frac{9}{2} \).

• This operation makes complex fraction division a much simpler process.
• It also broadens your understanding of how different mathematical principles are interconnected.

Being comfortable with inverting fractions and using those inversions is a powerful asset in higher-level math skills.