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Chapter 2: Problem 8

Find the reciprocal of the number. $$-\frac{3}{4}$$

Short Answer

Expert verified
The reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\).

Step by step solution

01

Understanding the premise

The number provided here is \(-\frac{3}{4}\). The task requires finding the reciprocal of this number.
02

Formulating the reciprocal

The reciprocal of a number is simply 1 divided by that number. Therefore, the reciprocal of \(-\frac{3}{4}\) would simply be \(\frac{1}{-\frac{3}{4}}\).
03

Calculating the reciprocal

To resolve the equation, one would need to simplify \(\frac{1}{-\frac{3}{4}}\) to \(-\frac{4}{3}\). This is achieved by multiplying the top and bottom of the fraction by -4.

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Key Concepts

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

Negative Fractions
When dealing with fractions, you might come across negative fractions like \(-\frac{3}{4}\). A negative fraction has either the numerator or the denominator as a negative number. In \(-\frac{3}{4}\), the negative sign is in front, indicating the entire fraction is negative.

Key points about negative fractions:
  • The negative sign can be attached to the numerator, the denominator, or in front of the fraction itself.
  • All three representations are equivalent, but typically, the negative sign is displayed at the front.
  • Converting between them does not change the value of the fraction.
For example, \(-\frac{3}{4}\), \(\frac{-3}{4}\), and \(\frac{3}{-4}\) are the same.

This understanding is crucial when finding reciprocals of negative fractions, as the negative sign remains unaffected.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers.

The denominator must not be zero. These numbers include positive fractions, negative fractions, whole numbers, and zero itself.
  • Whole numbers: These can be written as a fraction, e.g., \(5\) as \(\frac{5}{1}\).
  • Negative fractions: These fall into the category of rational numbers. \(-\frac{3}{4}\) is an example of a rational number.
  • Zero: Although it is a rational number, it cannot be in the denominator of a fraction.
Understanding rational numbers helps in identifying when a number is suitable for operations like finding reciprocals.

Always remember that any rational number, except zero, has a reciprocal.
Fraction Simplification
Simplifying fractions involves reducing them to their simplest form, ensuring the numerator and the denominator share no common factors other than 1.

While finding the reciprocal of a fraction such as \(-\frac{3}{4}\), simplification can be vital, though \(-\frac{3}{4}\) is already in its simplest form.

Steps to simplify a fraction:
  • Identify the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both by this GCD.
In our case, finding the reciprocal \(-\frac{4}{3}\) doesn鈥檛 require additional simplification as this result is already simplified. This rule holds true for any fraction derived from operations like reciprocals.

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Most popular questions from this chapter

Evaluate the expression. $$ \frac{10}{3}-\frac{2}{3} \cdot 4+5 $$

LEAVING A TIP In Exercises \(83-85\), use the following information. You and a friend decide to leave a \(15 \%\) tip for restaurant service. You compute the tip, \(T,\) as \(T=0.15 C,\) where \(C\) represents the cost of the meal. Your friend claims that an easier way to mentally compute the tip is to calculate \(10 \%\) of the cost of the meal plus one half of \(10 \%\) of the cost of the meal. Simplify the equation. What property did you use to simplify the equation?

LOGICAL REASONING Decide whether the statement is true or false If false, rewrite the right-hand side of the equation so the statement is true. $$ 7(15-6) \geq 7(15)-7(6) $$

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (2 x-4)(-3) $$

LOGICAL REASONING Decide whether the statement is true or false If false, rewrite the right-hand side of the equation so the statement is true. $$ -3.5(6.1+8.2) \stackrel{2}{=}-3.5(6.1)-3.5(8.2) $$
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