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Chapter 4: Problem 173

Explain how you find the reciprocal of a negative fraction.

Short Answer

Expert verified
The reciprocal of \-\(\frac{a}{b}\) is \-\(\frac{b}{a}\).

Step by step solution

01

Understand What a Reciprocal Is

A reciprocal of a number is what you multiply by that number to get 1. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
02

Identify the Negative Fraction

Locate the given fraction, for example, \(-\frac{3}{4}\). Notice that it is negative.
03

Take the Reciprocal

To find the reciprocal of \(\frac{a}{b}\), simply flip the numerator and the denominator. The reciprocal of \-\(\frac{3}{4}\) is \(\frac{4}{3}\).
04

Retain the Negative Sign

Make sure to keep the negative sign from the original fraction. So, \-\(\frac{3}{4}\) becomes \-\(\frac{4}{3}\).

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Key Concepts

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

Understanding Fractions
Fractions represent parts of a whole. They are made up of two parts: the numerator and the denominator. The numerator is the top number and shows how many parts we have. The denominator is the bottom number and shows how many equal parts the whole is divided into. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator. Sometimes fractions can be negative, like \(-\frac{3}{4}\), indicating a value less than zero.
  • Numerator: The top part of the fraction
  • Denominator: The bottom part of the fraction
  • Negative Fraction: Indicates a value less than zero
Understanding fractions is crucial before diving into concepts like reciprocals and negative numbers.
Reciprocals in Mathematics
A reciprocal of a number is what you multiply by that number to get 1. When dealing with fractions, the reciprocal is achieved by flipping the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This principle remains consistent even for negative fractions. So, the reciprocal of \(-\frac{3}{4}\) is \(\frac{4}{3}\).
  • Reciprocal: Flipping the numerator and the denominator
  • Result should multiply back to 1
  • Negative reciprocal retains the negative sign
By understanding this concept, solving problems that involve reciprocals becomes straightforward and stress-free.
Dealing with Negative Numbers
Negative numbers are numbers that are less than zero. They are represented with a minus sign (-). For instance, in the fraction \(-\frac{3}{4}\), the negative sign indicates that the fraction is less than zero. When finding the reciprocal of a negative fraction, it is important to retain the negative sign in the reciprocal. So, the reciprocal of \(-\frac{3}{4}\) becomes \(-\frac{4}{3}\).
  • Negative Numbers: Less than zero, indicated by a minus sign
  • Negative Fraction: Contains a negative numerator, denominator, or both
  • Keeping the Negative Sign: Ensure the negative sign is retained in reciprocal
Understanding how to work with negative numbers is essential for correctly performing operations with negative fractions and their reciprocals.

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Most popular questions from this chapter

In the following exercises, simplify. $$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$$

In the following exercises, simplify. $$ \frac{9^{2}-4^{2}}{9-4} $$

In the following exercises, simplify. $$ \frac{2^{3}-2^{2}}{\left(\frac{3}{4}\right)^{2}} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{u+v}{w}\) when \(u=-4, v=-8, w=2\)

In the following exercises, subtract. $$-\frac{31}{30}-\frac{7}{30}$$
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