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Chapter 4: Problem 11

Find the reciprocal of the given number. $$-9 / 19$$

Short Answer

Expert verified
The reciprocal of \(-\frac{9}{19}\) is \(-\frac{19}{9}\).

Step by step solution

01

Understanding the Reciprocal

The reciprocal of a number is essentially what you multiply by the original number to get 1. For any fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). This means you essentially 'flip' the numerator and the denominator.
02

Identify Numerator and Denominator

In the fraction \(-\frac{9}{19}\), the numerator is \(-9\) and the denominator is \(19\). We need to flip these to find the reciprocal.
03

Calculate the Reciprocal

To find the reciprocal, switch the numerator and the denominator. Thus, the reciprocal of \(-\frac{9}{19}\) is \(-\frac{19}{9}\).

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Key Concepts

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

Fractions
Fractions are a way to represent numbers that are not whole. Think of them as parts of a "whole" pie. If you divide a pie into equal parts, each slice represents a fraction of the pie. The pie can be in any number of slices, for example,
  • If you divide the pie into 2 slices, each slice is represented as \(\frac{1}{2}\).
  • If the pie is split into 3 slices, each slice is \(\frac{1}{3}\).
Fractions consist of two numbers:
  • A numerator which tells you how many parts you have.
  • A denominator which states into how many equal parts the whole is divided.
Using fractions in math helps you handle scenarios where you need to work with parts of things, much like splitting slices of pizza between friends.
Numerator and Denominator
In any fraction, a numerator and denominator play an essential role. Understanding these two parts is key to working with fractions:
  • The numerator is the top number of a fraction. It denotes the number of parts we are considering or "the number being counted". For example, in \(\frac{3}{4}\), the "3" is the numerator, and it shows that we have 3 parts.
  • The denominator is the bottom number. It tells you how many total parts the whole is divided into. In the previous example \(\frac{3}{4}\), the "4" is the denominator, indicating the whole is divided into 4 parts.
These terms are fundamental in operations involving fractions, such as addition, subtraction, multiplication, and finding reciprocals. For example, to find the reciprocal of a fraction, you swap the numerator and the denominator.
Negative Fractions
Negative fractions bring an extra twist. They represent a value less than zero, essentially pointing in the opposite direction on a number line. Here's how they work:
  • A negative sign "\(-\)" can be placed either with the numerator or in front of the fraction. Placing it in either location means the same, for example, \(-\frac{4}{5}\) is equal to \(\frac{-4}{5}\).
  • Using negative fractions helps to denote quantities that are below a "neutral" or zero level.
Negative fractions are very common in math, particularly in contexts like accounting, temperature measurement, and more. When finding the reciprocal of a negative fraction, the negative sign stays with the numerator in the result as well, ensuring the fraction's value remains appropriate. For instance, if we find the reciprocal of \(-\frac{9}{19}\), it becomes \(-\frac{19}{9}\). This process keeps the fraction negative and accurately represents its value as the inverse.

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Most popular questions from this chapter

Simplify the complex rational expression. $$\frac{-\frac{3}{2}-\frac{5}{8}}{\frac{3}{4}-\frac{1}{2}}$$

Solve the equation and simplify your answer. $$-\frac{4}{9} x-\frac{8}{9}=\frac{1}{2} x-\frac{1}{2}$$

Is \(2 / 7\) a solution of the equation \(x-\frac{4}{9}=-\frac{8}{63}\) ?

Solve the equation and simplify your answer. $$x+\frac{8}{9}=\frac{2}{3}$$

Is \(1 / 3\) a solution of the equation \(x+\frac{3}{4}=\frac{13}{12}\) ?
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