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Equivalent Fractions

Expand and simplify to find equivalent fractions.

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Equivalent Fractions
License: CC BY-NC 3.0
 

Stephanie is making her famous chili for her family for dinner tonight. The secret ingredient is \begin{align*}\frac{1}{2}\end{align*} teaspoon of cayenne pepper for an extra spicy kick. She has the cayenne pepper, but she can't find the \begin{align*}\frac{1}{2}\end{align*} teaspoon measuring spoon anywhere! All she can find is the 1 teaspoon, \begin{align*}\frac{1}{4}\end{align*} teaspoon, and \begin{align*}\frac{1}{8}\end{align*} teaspoon measuring spoons. How can Stephanie create a fraction equivalent to \begin{align*}\frac{1}{2}\end{align*} that will allow her to use the measuring spoons she has?

In this concept, you will learn how to write equivalent proper fractions, mixed numbers, and improper fractions.

Writing Equivalent Fractions

Whole numbers are the set of numbers \begin{align*}\{0, 1, 2, 3, ...\}\end{align*}. Whole numbers are numbers that are non-negative and contain no fractional parts.

Not all numbers are whole. A fraction describes a part of a whole number. Here are some examples of fractions.

  • \begin{align*}\frac{1}{2}\end{align*}  
  • \begin{align*}\frac{3}{4}\end{align*}
  • \begin{align*}\frac{9}{15}\end{align*}

The number written below the bar in a fraction is the denominator. The denominator tells you how many parts the whole is divided into. The numerator is the number above the bar in a fraction. The numerator tells you how many parts of the whole you have.

A whole can be divided into any number of parts. Equivalent fractions are fractions that describe the same part of a whole. Equivalent means equal.

Here is an example. All of the fractions below are equivalent to one another.

  • \begin{align*}\frac{1}{2}\end{align*} 
  • \begin{align*}\frac{4}{8}\end{align*}
  • \begin{align*}\frac{3}{6}\end{align*}
  • \begin{align*}\frac{2}{4}\end{align*}

One way to create equivalent fractions is through simplifying. If both the numerator and the denominator of a fraction are divisible by the same whole number, you can simplify the fraction by dividing both the numerator and the denominator by that whole number. Some fractions cannot be simplified. If the only number that both the numerator and the denominator are divisible by is 1, then the fraction is said to be in its simplest form.

Here is an example.

Find a fraction that is equivalent to \begin{align*}\frac{2}{4}\end{align*} by simplifying.

First, notice that both the numerator and the denominator of the fraction are divisible by 2.

Next, divide both the numerator and the denominator by 2.

\begin{align*}\frac{2{\div 2}}{4{\div}2}=\frac{1}{2}\end{align*}

The answer is \begin{align*}\frac{1}{2}\end{align*} is the simplified version of \begin{align*}\frac{2}{4}\end{align*}. They are equivalent fractions. \begin{align*}\frac{1}{2}=\frac{2}{4}\end{align*}.

Here is another example.

Find fractions that are equivalent to \begin{align*}\frac{6}{18}\end{align*} by simplifying.

First, notice that the numerator and denominator of the fraction are divisible by 2, 3, and 6. You can divide the numerator and denominator by any of these numbers to create equivalent fractions. To find the simplest form version of the fraction, divide by the largest number that both the numerator and denominator are divisible by (their greatest common factor). In this case, that number is 6.

\begin{align*}\begin{array}{rcl} \frac{6{\div 2}}{18{\div}2} &=& \frac{3}{9}\\ \frac{6{\div 3}}{18{\div}3} &=& \frac{2}{6}\\ \frac{6{\div 6}}{18{\div}6} &=& \frac{1}{3} * \text{simplest form} * \end{array}\end{align*}

The fractions \begin{align*}\frac{6}{18},\frac{3}{9},\frac{2}{6},\end{align*} and \begin{align*}\frac{1}{3}\end{align*} are all equivalent. \begin{align*}\frac{1}{3}\end{align*} is the simplest form version of the fractions.

Another way to create an equivalent fraction is by multiplying both the numerator and the denominator of your fraction by the same number. This is the reverse of simplifying. It doesn't matter which number you choose to multiply by.

Here is an example.

Create an equivalent fraction for \begin{align*}\frac{7}{8}\end{align*}.

First, notice that this fraction is already in simplest form so you cannot create an equivalent fraction by simplifying. You can create an equivalent fraction by multiplying both the numerator and the denominator by the same number.

Next, choose a number to multiply both the numerator and the denominator by. Let's choose 2.

\begin{align*}\frac{7{\times 2}}{8{\times}2}=\frac{14}{16}\end{align*}

The answer is \begin{align*}\frac{14}{16}\end{align*} is an equivalent fraction for \begin{align*}\frac{7}{8}\end{align*}.

Sometimes when working with fractions, you use numbers that consist of a whole number and a fraction. This is called a mixed number. Here are some examples of mixed numbers.

  • \begin{align*}1 \frac{1}{2} \end{align*} 
  • \begin{align*}2 \frac{2}{3}\end{align*}

When the numerator of a fraction is greater than or equal to the denominator of a fraction, you have an improper fraction. Improper fractions are greater than or equal to 1. Here are some examples of improper fractions.

  • \begin{align*}\frac{5}{2}\end{align*}
  • \begin{align*}\frac{8}{3}\end{align*} 
  • \begin{align*}\frac{11}{4}\end{align*}

Mixed numbers can be equivalent to improper fractions. To rewrite an improper fraction as a mixed number, divide the numerator by the denominator and keep the remainder as the numerator.

Here is an example.

Write \begin{align*}\frac{9}{2}\end{align*} as a mixed number.

First, divide 9 by 2.

9 divided by 2 equals 4 with a remainder of 1.

This means the whole number part of the mixed number will be 4. The remainder of 1 goes in the numerator of the fraction part of the mixed number. The denominator of the fraction part of the mixed number is the same as the original denominator, 2.

\begin{align*}\frac{9}{2}= 4 \frac{1}{2}\end{align*}

The answer is \begin{align*}\frac{9}{2}= 4 \frac{1}{2}\end{align*}.

To rewrite a mixed number as an improper fraction, find the new numerator by multiplying the denominator by the whole number and adding the original numerator. The denominator stays the same.

Here is an example.

Write \begin{align*}3 \frac{2}{3}\end{align*} as an improper fraction.

First, find the numerator of the improper fraction. Multiply the denominator (3) by the whole number (3) and add the original numerator (2).

\begin{align*}3 \times 3 + 2 =9 +2 =11\end{align*}

The numerator of the improper fraction is 11.

Next, find the denominator of the improper fraction. The denominator is the same as the original denominator.

The denominator of the improper fraction is 3.

The answer is \begin{align*}3 \frac{2}{3}=\frac{11}{3}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Stephanie and her famous chili.

She needs \begin{align*}\frac{1}{2}\end{align*} teaspoon of cayenne pepper, but she can only find the 1 teaspoon, \begin{align*}\frac{1}{4}\end{align*} teaspoon, and \begin{align*}\frac{1}{8}\end{align*} teaspoon measuring spoons. She needs to find an equivalent fraction for \begin{align*}\frac{1}{2}\end{align*} so that she can use the measuring spoons she has.

First, Stephanie should notice that \begin{align*}\frac{1}{2}\end{align*} is already in simplest form so she cannot create an equivalent fraction by simplifying. She can create an equivalent fraction by multiplying both the numerator and the denominator by the same number.

Next, she should choose a number to multiply both the numerator and the denominator by. Since she has a \begin{align*}\frac{1}{4}\end{align*} teaspoon, it makes sense to try to make the denominator of the equivalent fraction equal to 4. 2 times 2 is equal to 4, so multiply by 2.

\begin{align*}\frac{1{\times 2}}{2{\times}2}=\frac{2}{4}\end{align*}

The answer is \begin{align*}\frac{2}{4}\end{align*} is equivalent to \begin{align*}\frac{1}{2}\end{align*}. Stephanie could use the \begin{align*}\frac{1}{4}\end{align*} teaspoon twice to get \begin{align*}\frac{1}{2}\end{align*} teaspoon of cayenne pepper.

Example 2

Write \begin{align*}\frac{7}{3}\end{align*} as a mixed number.

First, divide 7 by 3.

7 divided by 3 equals 2 with a remainder of 1.

This means the whole number part of the mixed number will be 2. The remainder of 1 goes in the numerator of the fraction part of the mixed number. The denominator of the fraction part of the mixed number is the same as the original denominator, 3.

\begin{align*}\frac{7}{3}=2 \frac{1}{3}\end{align*}

The answer is \begin{align*}\frac{7}{3}=2 \frac{1}{3}\end{align*}.

Example 3

Simplify \begin{align*}\frac{10}{12}\end{align*}.

First, notice that both the numerator and the denominator of the fraction are divisible by 2.

Next, divide both the numerator and the denominator by 2.

\begin{align*}\frac{10{\div 2}}{12{\div}2}=\frac{5}{6} \end{align*}

The answer is \begin{align*}\frac{5}{6}\end{align*} is the simplified version of \begin{align*}\frac{10}{12}\end{align*}. They are equivalent fractions.

Example 4

Create an equivalent fraction for \begin{align*}\frac{3}{4}\end{align*}.

First, notice that this fraction is already in simplest form so you cannot create an equivalent fraction by simplifying. You can create an equivalent fraction by multiplying both the numerator and the denominator by the same number.

Next, choose a number to multiply both the numerator and the denominator by. Let's choose 6.

\begin{align*} \frac{3{\times 6}}{4{\times}6}=\frac{18}{24}\end{align*}

The answer is \begin{align*}\frac{18}{24}\end{align*} is an equivalent fraction for \begin{align*}\frac{3}{4}\end{align*}.

Example 5

Write \begin{align*}\frac{15}{2}\end{align*} as a mixed number.

15 divided by 2 equals 7 with a remainder of 1.

This means the whole number part of the mixed number will be 7. The remainder of 1 goes in the numerator of the fraction part of the mixed number. The denominator of the fraction part of the mixed number is the same as the original denominator, 2.

\begin{align*}\frac{15}{2}=7 \frac{1}{2}\end{align*}

The answer is \begin{align*}\frac{15}{2}=7 \frac{1}{2}\end{align*}.

Review

  1. Write four equivalent fractions for \begin{align*}\frac{6}{8}\end{align*}.

Write the following mixed numbers as improper fractions.

  1. \begin{align*}2\frac{5}{8}\end{align*} 
  2. \begin{align*}3\frac{2}{5}\end{align*} 
  3. \begin{align*}1\frac{1}{7}\end{align*} 
  4. \begin{align*}5\frac{4}{9}\end{align*} 

Write the following improper fractions as mixed numbers.

  1. \begin{align*}\frac{29}{28}\end{align*} 
  2. \begin{align*}\frac{12}{5}\end{align*} 
  3. \begin{align*}\frac{9}{2}\end{align*} 
  4. \begin{align*}\frac{17}{8}\end{align*} 
  5. \begin{align*}\frac{22}{3}\end{align*} 

Write three equivalent fractions for each of the following fractions.

  1. \begin{align*}\frac{2}{3}\end{align*} 
  2. \begin{align*}\frac{12}{28}\end{align*} 
  3. \begin{align*}\frac{3}{4}\end{align*} 
  4. \begin{align*}\frac{9}{10}\end{align*} 
  5. \begin{align*}\frac{7}{8}\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

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Vocabulary

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.

Equivalent

Equivalent means equal in value or meaning.

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.

Numerator

The numerator is the number above the fraction bar in a fraction.

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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