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Chapter 1: Problem 79

Find the reciprocal of each number, if it exists. $$ -\frac{4}{5} $$

Short Answer

Expert verified
The reciprocal of \(-\frac{4}{5}\) is \(-\frac{5}{4}\).

Step by step solution

01

Understand what a reciprocal is

A 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 \(\frac{b}{a}\).
02

Identify the given fraction

The given fraction is \(-\frac{4}{5}\).
03

Find the reciprocal of the fraction

To find the reciprocal of \(-\frac{4}{5}\), swap the numerator and the denominator. Thus, the reciprocal of \(-\frac{4}{5}\) is \(-\frac{5}{4}\).
04

Verify the result

Multiply \(-\frac{4}{5}\) by its reciprocal \(-\frac{5}{4}\) to ensure the product is 1:\[-\frac{4}{5} \times -\frac{5}{4} = \frac{4 \times 5}{5 \times 4} = \frac{20}{20} = 1.\]Since the result is 1, the reciprocal is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal of a Fraction
A reciprocal is an important concept in mathematics. It is the value that you multiply the original number by to get 1.
For example, the reciprocal of 2 is \(\frac{1}{2}\), because \(2 \times \frac{1}{2} = 1\). Similarly, for a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
Another example is \(\frac{3}{4}\). Here, the reciprocal would be \(\frac{4}{3}\), because \( \frac{3}{4} \times \frac{4}{3} = 1\).
When dealing with negative numbers, the reciprocal also changes its sign. For instance, for \(-\frac{4}{5}\), the reciprocal is \(-\frac{5}{4}\), and we can verify that \(-\frac{4}{5} \times -\frac{5}{4} = 1\).
Understanding Fractions
A fraction represents a part of a whole. It consists of two numbers - a numerator and a denominator.
The numerator (the top number) indicates how many parts we have. The denominator (the bottom number) indicates how many parts the whole is divided into.
For example, in the fraction \(\frac{3}{4} \), 3 is the numerator, and 4 is the denominator. This means we have 3 parts out of a total of 4 parts.
Negative fractions like \(-\frac{4}{5}\) indicate a direction or value less than zero. The negative sign applies to the whole fraction, meaning it could be written as \(-1 \times \frac{4}{5}\).
Multiplication and Verification
Multiplication of fractions involves multiplying the numerators together and the denominators together. For example, to multiply \( \frac{a}{b} \times \frac{c}{d}\), we calculate: \(\frac{a \times c}{b \times d}\).
To verify the reciprocal, multiply the original fraction by its reciprocal. If the result is 1, the reciprocal is correct.
Let's verify \(-\frac{4}{5}\) and \(-\frac{5}{4}\):
\( -\frac{4}{5} \times -\frac{5}{4} = \frac{20}{20} = 1 \).
We see the result is 1, confirming our reciprocal as correct.
This method ensures our understanding and accuracy in finding reciprocals for any given fraction.

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