Problem 63 - Algebra A Combined Function

Question

Find each multiplicative inverse or reciprocal. $$ \frac{5}{8} $$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \).

Step 1: Understanding the Concept

To find the multiplicative inverse (or reciprocal) of a number, you must find a number which, when multiplied by the original, results in 1. Mathematically, for a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

Step 2: Identifying the Numerator and Denominator

The given fraction is \( \frac{5}{8} \). Here, the numerator is 5, and the denominator is 8.

Step 3: Switching the Numerator and the Denominator

To find the reciprocal, switch the numbers in the numerator and the denominator of the fraction. Thus, the reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \).

Step 4: Verifying the Reciprocal

Multiply the original fraction \( \frac{5}{8} \) by its reciprocal \( \frac{8}{5} \). The product should equal 1: \( \frac{5}{8} \times \frac{8}{5} = \frac{5 \times 8}{8 \times 5} = \frac{40}{40} = 1 \). Since the product is 1, \( \frac{8}{5} \) is indeed the reciprocal of \( \frac{5}{8} \).

Key Concepts

Reciprocal

A reciprocal, also known as a multiplicative inverse, is what you multiply a number by to get 1. For example, if you have the fraction \( \frac{5}{8} \), its reciprocal is \( \frac{8}{5} \). The beauty of reciprocals is that when you multiply a number by its reciprocal, you are guaranteed to end up with 1. This is because reciprocals "undo" each other. Reciprocals play a critical role in operations involving fractions and can help simplify complex calculations. Understanding how to find the reciprocal of a number or fraction is a fundamental mathematical skill. For any fraction \( \frac{a}{b} \), simply swap the numerator and the denominator: \( \frac{b}{a} \). This new fraction is the reciprocal. It turns the division into multiplication in problem-solving scenarios, making math problems easier to handle.

Fraction

Fractions represent a part of a whole. They are written as \( \frac{a}{b} \), where \( a \) is called the numerator, and \( b \) is the denominator. The numerator indicates how many parts we have, while the denominator tells us how many equal parts the whole is divided into. Fractions can be added, subtracted, multiplied, and divided, making them versatile in a wide range of mathematical problems. They can also be converted to decimals or percentages, providing different ways to express the same value.
In the exercise example, \( \frac{5}{8} \) means that you have 5 out of 8 equal sections. Fractions can denote much more than parts of a pizza; they are critical in measurements, probabilities, and other real-world applications. Understanding the components of fractions and how they work is essential for mastering arithmetic and algebra.

Numerator and Denominator

The words "numerator" and "denominator" are key terms when dealing with fractions. The numerator, the number on top, tells us "how many" parts we have. Meanwhile, the denominator, the bottom number, tells us "into how many" equal parts the whole is divided. These parts make up fundamental math language and must be understood to handle any fraction effectively.
In the fraction \( \frac{5}{8} \), the numerator is 5, indicating the part of the whole we are considering, while the denominator is 8, showing the total parts the whole is split into. Visualizing the numerator as slices of a pie and the denominator as the number of slices that make up the whole pie can be a helpful analogy.

• Numerator: The part of the fraction that represents part of the entire set.
• Denominator: The part of the fraction that represents the total number of equal parts.

By understanding and remembering these roles, you'll be well-equipped to find reciprocals and perform other mathematical operations.

Multiplication

Multiplication is one of the basic arithmetic operations, alongside addition, subtraction, and division. In the context of fractions and reciprocals, multiplication is special because it demonstrates the reciprocal's power. When you multiply a fraction by its reciprocal, the result is always 1, such as \( \frac{5}{8} \times \frac{8}{5} = 1 \). Each term cancels the other because the product of the numerators equals the product of the denominators, converting to 1.
This property is vital in solving equations and converting fractions. Multiplying fractions involves multiplying the numerators together and the denominators together. For instance, \( \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} \) which simplifies to \( \frac{1}{2} \). Such operations highlight why understanding multiplication's role with fractions makes problem-solving efficient.