Problem 25 - Algebra A Combined Function

Question

Find each reciprocal. \(\frac{2}{3}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Reciprocal

The reciprocal of a number is a value that, when multiplied by the original number, results in 1. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). This involves swapping the numerator and the denominator.

Step 2: Identify the Numerator and Denominator

In the fraction \( \frac{2}{3} \), the numerator is 2 and the denominator is 3. These are the numbers that we will swap to find the reciprocal.

Step 3: Swap the Numerator and Denominator

To find the reciprocal, swap the numerator and denominator of the fraction \( \frac{2}{3} \). This gives us the new fraction \( \frac{3}{2} \).

Step 4: Verify the Reciprocal

Multiply \( \frac{2}{3} \) by its reciprocal \( \frac{3}{2} \) to verify that the product is 1. Calculate: \(\frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6} = 1 \). This confirms that \( \frac{3}{2} \) is indeed the reciprocal of \( \frac{2}{3} \).

Key Concepts

Fractions

Fractions represent parts of a whole and are fundamentally composed of two numbers: a numerator and a denominator. Understanding fractions is essential to grasp how numbers are divided into equal parts. When you see a fraction like \( \frac{2}{3} \), it simply tells you that you have 2 parts out of a total of 3 equal parts.
Fractions can come in different forms:

• **Proper Fractions**: where the numerator is less than the denominator, such as \( \frac{2}{3} \).
• **Improper Fractions**: where the numerator is greater than or equal to the denominator, for example, \( \frac{5}{4} \).
• **Mixed Numbers**: consist of a whole number and a proper fraction, like 1\( \frac{1}{2} \).

Fractions are versatile because they relate closely to division, ratios, and percentages, making them useful in various mathematical applications. Understanding their basic structure will allow you to handle operations like addition, subtraction, multiplication, and finding reciprocals.

Numerator and Denominator

The numerator and the denominator are key components of a fraction, each playing a critical role.

• **Numerator**: The top number in a fraction, representing how many parts you have. In \( \frac{2}{3} \), 2 is the numerator.
• **Denominator**: The bottom number, indicating the total number of equal parts the whole is divided into. In the same fraction, 3 is the denominator.

The denominator tells us into how many parts the whole is divided, while the numerator tells us how many of those parts we have. Swapping the numerator and denominator of a fraction gives us its reciprocal. This switch inherently changes the relationship of the parts to the whole. The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \), as you invert the components.

Multiplication of Fractions

The multiplication of fractions is a straightforward process that involves multiplying the numerators together and the denominators together.
To multiply two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), you simply compute:\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]Multiplying fractions is particularly useful when validating ideas like reciprocals. For instance, to verify that \( \frac{3}{2} \) is the reciprocal of \( \frac{2}{3} \):

• Multiply the fraction by its reciprocal: \( \frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6} = 1 \).

This result shows that the product of a fraction and its reciprocal is always 1. This principle forms the basis of why reciprocals are useful in simplifying equations and solving problems across mathematics.