Problem 2 - Fundamentals of Mathematics

Question

Find the reciprocal of each number. $$\frac{2}{3}$$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Step 1: Understand the Concept of a Reciprocal

The reciprocal of a number is simply 1 divided by that number. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). Reciprocals of a number and itself multiply to 1.

Step 2: Identify the Fraction

The fraction given in the problem is \( \frac{2}{3} \). We need to find the reciprocal of this fraction.

Step 3: Switch the Numerator and Denominator

To find the reciprocal of \( \frac{2}{3} \), switch the numerator and the denominator. The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Step 4: Verify the Reciprocal

Multiply the original fraction \( \frac{2}{3} \) by its reciprocal \( \frac{3}{2} \). The product should be 1: \( \frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1 \). This confirms that \( \frac{3}{2} \) is indeed the correct reciprocal.

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole. They are written in the form \( \frac{a}{b} \), where \( a \) is the numerator, and \( b \) is the denominator. The numerator signifies how many parts we are considering, whereas the denominator indicates the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{2}{3} \), 2 is the numerator and 3 is the denominator. This means we have 2 parts out of a total of 3 equal parts.
Fractions can be found in everyday life—for instance, in recipes that require half a teaspoon of salt or games that divide time into quarters. Understanding how to manipulate fractions, such as finding reciprocals, is crucial in math.

Numerator and Denominator

The terms numerator and denominator are essential components of any fraction.

• Numerator: It is the top part of a fraction and tells us how many parts of the whole or group are being considered.
• Denominator: It is the bottom part of a fraction and shows into how many equal parts the whole is divided.

To find the reciprocal of a fraction, you simply switch the positions of these two components. For instance, if you start with \( \frac{2}{3} \), you change the numerator to 3 and the denominator to 2, which results in the reciprocal \( \frac{3}{2} \).
This simple operation is a key concept in understanding and working with fractions, especially when doing division or solving equations involving fractions.

Multiplication Verification of Reciprocals

Multiplication verification is a method used to confirm that you have correctly found the reciprocal of a fraction.
To verify, multiply the original fraction by its reciprocal. The result should always be 1 because, by definition, multiplying a number by its reciprocal gives a product of 1. This is a foundational property of reciprocals.
Let's see how this works with our example of \( \frac{2}{3} \). When you multiply it by its reciprocal \( \frac{3}{2} \), the operation looks like this:

• Multiply the numerators: \( 2 \times 3 = 6 \)
• Multiply the denominators: \( 3 \times 2 = 6 \)

Thus, \( \frac{2}{3} \times \frac{3}{2} = \frac{6}{6} \). Simplifying the fraction \( \frac{6}{6} \) gives 1, confirming that \( \frac{3}{2} \) is indeed the correct reciprocal.
This method is an excellent way to ensure you're performing operations with fractions correctly.