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Chapter 7: Problem 31

Vocabulary Define the term complex fraction. Give an example.

Short Answer

Expert verified
A complex fraction is a fraction wherein the numerator, the denominator, or both contain fractions. Example: \(\frac{3/4}{5/6}\)

Step by step solution

01

Define Complex Fraction

A complex fraction is a type of fraction where either the numerator, the denominator, or both contain fractions themselves.
02

Provide an Example

For instance, the fraction \(\frac{3/4}{5/6}\) is a complex fraction because it has fractions in both the numerator and the denominator.

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Key Concepts

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

Understanding Fractions
Fractions are a fundamental concept in mathematics. A fraction represents a part of a whole. For example, if you have half a pizza, you can represent this as the fraction \( \frac{1}{2} \).
A fraction is made up of two parts: the numerator and the denominator. Understanding both of these components is crucial to working with fractions effectively.
In the simplest form, a fraction like \( \frac{a}{b} \) means you have \(a\) parts of something that is divided into \(b\) equal parts.
  • The top part \, \((a)\) is the numerator.
  • The bottom part \, \((b)\) is the denominator.
Fractions can be simple, improper, or even complex depending on their components.
Discovering the Numerator
The numerator is the top number in a fraction. It tells you how many parts of the whole you have.
For instance, in the fraction \( \frac{3}{4} \), the numerator is \(3\). This means that out of 4 equal parts in total, you have 3 parts.
Knowing the numerator helps you understand the quantity of the fraction you are dealing with.
In complex fractions, the numerator itself can be another fraction. For example, in \( \frac{\frac{2}{3}}{5}\), the numerator, \( \frac{2}{3} \), is a fraction, making it more complex to handle.
Explaining the Denominator
The denominator is the bottom number in a fraction. It indicates how many equal parts the whole is divided into.
For example, in \( \frac{3}{5} \), \(5\) is the denominator, meaning the whole is divided into 5 equal parts.
Understanding the denominator is key to knowing the size of each part of the fraction.
In complex fractions, like \( \frac{4}{\frac{7}{8}} \), the denominator could be a fraction too. This complexity implies that each part is divided further, making fractions like these a little trickier to solve.
A Mathematical Example in Action
Let's dive into a mathematical example with complex fractions. Consider the complex fraction \( \frac{\frac{3}{4}}{\frac{5}{6}} \).
Here鈥檚 how you can break it down:
  • The numerator is \( \frac{3}{4} \).
  • The denominator is \( \frac{5}{6} \).
You can simplify complex fractions by dividing the numerator by the denominator.
This is done by multiplying the numerator by the reciprocal of the denominator:
\[\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5} \].
When you multiply these fractions, you get \( \frac{18}{20} \), which can be simplified to \( \frac{9}{10} \).
Practicing with examples like this improves your fraction manipulation skills, making you more comfortable with complex fractions.

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Most popular questions from this chapter

The equation \(1-\frac{8}{x-5}=\frac{3}{x}\) has an extraneous solution of \(5 .\)

\(6 y^{3}+3 y^{2}-2 y-1\)

\(\frac{5 x}{7}-\frac{2 x}{3}=\frac{1}{2}\)

Monthly Payment The approximate annual percent rate \(r\) (in decimal form) of a monthly installment loan is $$ r=\frac{\left[\frac{24(M N-P)}{N}\right]}{\left(P+\frac{M N}{12}\right)} $$ where \(N\) is the total number of payments, \(M\) is the monthly payment, and \(P\) is the amount financed. Simplify the expression.

\(\frac{a+3}{4}-\frac{a-1}{6}=\frac{4}{3}\)
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