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# Equivalent Fractions

## Expand and simplify to find equivalent fractions.

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Equivalent Fractions

Have you ever had a bake sale at school? It can be a great fundraiser. Take a look at this dilemma.

The Seventh grade class is having a bake sale to raise money for class projects and trips. Madison has decided to bake her favorite kind of cookie for the sale. She loves linzer cookies with jam inside.

“What are you doing?” asks her brother Kyle coming into the kitchen.

“I’m making cookies for the bake sale,” Madison explains as she takes out the flour and the measuring cups.

“Sure, now we need \begin{align*}2 \frac{1}{2}\end{align*} cups of flour. Here is the \begin{align*}\frac{1}{2}\end{align*} measuring cup, but I can’t find the 1 cup measuring cup. That’s okay though because I can measure five \begin{align*}\frac{1}{2}\end{align*} cups full of flour and that will be \begin{align*}2 \frac{1}{2}\end{align*} cups,” She explains to Kyle.

“You could also use the \begin{align*}\frac{1}{3}\end{align*} measuring cup and fill it up 8 times,” Kyle says picking up the \begin{align*}\frac{1}{3}\end{align*} measuring cup.

“I don’t think so,” Madison says. “I think that is too much flour.”

“No it isn’t,” Kyle argues.

Who is correct? To figure out whether \begin{align*}2 \frac{1}{2}\end{align*} cups of flour is equal to eight \begin{align*}\frac{1}{3}\end{align*} cups of flour, you will need to understand how to compare and figure out equivalent fractions. Pay attention during this Concept and by the end of it, you will know who is correct and who needs to rethink their figuring.

### Guidance

This Concept is all about fractions. To understand fractions, you will need to think about whole numbers too. Without whole numbers, it is impossible to understand fractions because a fraction is a part of a whole.

Whole numbers are numbers like 1, 8, 56, and 278—numbers that don’t contain fractional parts. Not all numbers are whole.

A Fraction describes a part of a whole number. You are certainly familiar with fractions in your everyday dealings with cooking. Consider a recipe that calls for \begin{align*}\frac{1}{2}\end{align*} cup of chocolate chips. You know that \begin{align*}\frac{1}{2}\end{align*} cup represents one-half of a whole cup.

\begin{align*}\frac{3}{4} \ or \ \frac{3}{4}\end{align*}

A fraction has certain parts. What are those parts?

The number written below the bar in a fraction is the denominator, which tells how many parts the whole is divided into. The numerator is the number above the bar in a fraction, which tells how many parts of the whole you have. In the recipe that calls for \begin{align*}\frac{1}{2}\end{align*} cup, the denominator is 2, so we know that one whole cup is divided into 2 parts. The numerator is 1, so we know that we need 1 of the 2 parts of the whole cup. Notice that the fraction can be written up and down with one number on top of the other or using a slash. With a slash, the first number is the top number or numerator and the bottom number is the second number.

A whole can be divided into an infinite number of parts. You can divide 1 cup of flour into thirds, sixths, tenths, and so on. Fractions which describe the same part of a whole are called equivalent fractions. Remember that the word equivalent means equal. For instance, if you measure out \begin{align*}\frac{2}{4}\end{align*} cup of flour, \begin{align*}\frac{3}{6}\end{align*} cup of flour or \begin{align*}\frac{1}{2}\end{align*} cup of flour, you will have the same amount of flour.

Therefore \begin{align*}\frac{2}{4}, \frac{3}{6}\end{align*} and \begin{align*}\frac{1}{2}\end{align*} are all equivalent fractions.

When we have been given a fraction, we can create a fraction that is equivalent to that fraction. We call this making equal fractions or making equivalent fractions.

How do we do make equivalent fractions?

The first way is to work on simplifying the fraction to make a fraction smaller.

To simplify a fraction, we can reduce the number in the numerator and denominator by dividing them by the same number. For example, \begin{align*}\frac{4}{8}\end{align*} can be rewritten as \begin{align*}\frac{1}{2}\end{align*} by dividing both the numerator and the denominator by 4. Note that not all fractions can be rewritten by dividing. If the only number that both the numerator and denominator are divisible by is 1, then the fraction is said to be in its simplest form.

Simplify \begin{align*}\frac{6}{18}\end{align*}

To simplify this fraction, we look for a number that we can divide into both the numerator and the denominator. In this case, the number is 6. We call 6 the Greatest Common Factor (GCF) of the numerator and the denominator. To simplify, we divide the numerator and the denominator by 6.

\begin{align*}\frac{6 \div 6}{18 \div 6}= \frac{1}{3}\end{align*}

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

The second way is through multiplying. We can create an equivalent fraction by multiplying the numerator and denominator by the same number. It doesn’t matter which number you choose, as long as the numbers are the same numbers.

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

To do this, we need to multiply the numerator and the denominator by the same number. 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*}.

Write four equivalent fractions for \begin{align*}\frac{8}{12}\end{align*}.

First, let’s see if we can reduce the numbers in the numerator and denominator. Are there any numbers, which can be divided into both 8 and 12? 8 and 12 are both divisible by 2 and 4. So, the fraction \begin{align*}\frac{8}{12}\end{align*} is not in its simplest form.

When we divide both the numerator and the denominator by 2, we get \begin{align*}\frac{4}{6}\end{align*} as an equivalent fraction. When we divide the numerator and the denominator by 4, we get \begin{align*}\frac{2}{3}\end{align*} as an equivalent fraction.

To find more equivalent fractions, we can multiply the numerator and denominator of \begin{align*}\frac{8}{12}\end{align*} by any number. Let’s multiply by 3. We get \begin{align*}\frac{24}{36}\end{align*} as an equivalent fraction to \begin{align*}\frac{8}{12}\end{align*}. If we multiply the numerator and denominator by 5, we get \begin{align*}\frac{40}{60}\end{align*} as an equivalent fraction to \begin{align*}\frac{8}{12}\end{align*}.

The answer is \begin{align*}\frac{2}{3}, \frac{4}{6}, \frac{24}{36}, \frac{40}{60}\end{align*}.

Notice that creating equivalent fractions in this example involved both simplifying and multiplying!!

There are other types of fractions too.

Sometimes when working with fractions, you use numbers which consist of a whole number and a fraction. This is called a mixed number. For example, if a recipe calls for more than 1 cup of flour but less than 2 cups of flour, you need to use a mixed number to describe exactly how much flour you need. A mixed number is written as a whole number with a fraction to the right of it. Some common mixed numbers include: \begin{align*}1 \frac{1}{2}\end{align*} or \begin{align*}2 \frac{2}{3}\end{align*}.

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

\begin{align*}\frac{2}{2}, \frac{3}{3}\end{align*}, and \begin{align*}\frac{10}{10}\end{align*} are all improper fractions that equal 1.

Why is this? Well, to understand this, you have to think about what the numerator and the denominator mean. The denominator is how many parts the whole is divided into. The numerator is how many of those parts you have. If you have two out of two parts, then you have the whole thing.

\begin{align*}\frac{5}{2}, \frac{8}{3}\end{align*}, and \begin{align*}\frac{11}{4}\end{align*} are all fractions that are greater than 1. These are called improper fractions.

Mixed numbers and Improper Fractions can be equivalent or equal to each other.

Improper fractions can be written as mixed numbers by dividing the numerator by the denominator and keeping the remainder as the numerator. Mixed numbers can be rewritten as improper fractions by multiplying the whole number in the mixed number by the denominator and adding the product to the numerator.

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

These two quantities are equal. This improper fraction is equal to the mixed number.

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

Remember, to write a mixed number as an improper fraction, we first multiply the whole number (3) by the denominator in the fraction, \begin{align*}3 \times 3 = 9\end{align*}. Next, we add this number to the numerator of the fraction, \begin{align*}9 + 2 = 11\end{align*}. We put this new number over the original denominator and we have our improper fraction.

Our answer is that \begin{align*}3 \frac{2}{3}\end{align*} can be written as the improper fraction \begin{align*}\frac{11}{3}\end{align*}.

Practice working with equivalent fractions.

#### Example A

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

Solution: \begin{align*}\frac{5}{6}\end{align*}

#### Example B

Create an equivalent fraction for\begin{align*}\frac{5}{6}\end{align*}

Solution: \begin{align*}\frac{10}{12}\end{align*}

#### Example C

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

Solution: \begin{align*}7 \frac{1}{2}\end{align*}

Here is the original problem once again.

The Seventh grade class is having a bake sale to raise money for class projects and trips. Madison has decided to bake her favorite kind of cookie for the sale. She loves linzer cookies with jam inside.

“What are you doing?” asks her brother Kyle coming into the kitchen.

“I’m making cookies for the bake sale,” Madison explains taking out the flour and the measuring cups.

“Sure, now we need \begin{align*}2 \frac{1}{2}\end{align*} cups of flour.Here is the \begin{align*}\frac{1}{2}\end{align*} measuring cup, but I can’t find the 1 cup measuring cup. That’s okay though because I can measure five \begin{align*}\frac{1}{2}\end{align*} cups full of flour and that will be \begin{align*}2 \frac{1}{2}\end{align*} cups,” She explains to Kyle.

“You could also use the \begin{align*}\frac{1}{3}\end{align*} measuring cup and fill it up 8 times,” Kyle says picking up the \begin{align*}\frac{1}{3}\end{align*} measuring cup.

“I don’t think so,” Madison says. “I think that is too much flour.”

“No it isn’t,” Kyle argues.

To solve this problem, we need to compare Madison’s measurement with Kyle’s measurement.

Madison’s measurement is \begin{align*}2 \frac{1}{2}\end{align*}.

Kyle’s measurement is 8 one-third cups which is \begin{align*}\frac{8}{3}\end{align*}.

Next, we compare the two quantities. Kyle thinks that his measurement is equal to Madison’s. To see if he is correct, we convert the improper fraction to a mixed number. That will make our comparison much easier.

\begin{align*}\frac{8}{3}=2 \frac{2}{3}\end{align*}

Now we compare \begin{align*}2 \frac{1}{2} < 2 \frac{2}{3}\end{align*}.

Madison is correct. If she uses Kyle’s measurement technique, she will have too much flour! Kyle needs to remember that two-thirds is greater than one-half.

You can check this by taking the fraction part of each and rewriting them with a common denominator.

You can see that two-thirds is greater than one-half.

### Vocabulary

Whole Number
a number that is a counting number like 5, 7, 10, or 22.
Fraction
a part of a whole.
Numerator
the top number in a fraction.
Denominator
the bottom number in a fraction. It tells you how many parts the whole is divided into.
Equivalent Fractions
equal fractions
Equivalent
equal
Simplifying
making a fraction smaller
Mixed Number
a whole number with a fraction
Improper Fraction
when the numerator is greater than the denominator in a fraction

### Guided Practice

Here is one for you to try on your own.

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

To write an improper fraction as a mixed number, we divide the numerator by the denominator. \begin{align*}7 \div 3 = 2R1\end{align*}. To finish, we write the remainder above the original denominator and write the whole number part of the quotient to the left of this new fraction.

Our answer is that \begin{align*}\frac{7}{3}\end{align*} can be written as the mixed number \begin{align*}2 \frac{1}{3}\end{align*}.

### Practice

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

Directions: Write the following mixed numbers as improper fractions

2. \begin{align*}2 \frac{5}{8}\end{align*}

3. \begin{align*}3 \frac{2}{5}\end{align*}

4. \begin{align*}1 \frac{1}{7}\end{align*}

5. \begin{align*}5 \frac{4}{9}\end{align*}

Directions: Write the following improper fractions as mixed numbers.

6. \begin{align*}\frac{29}{28}\end{align*}

7. \begin{align*}\frac{12}{5}\end{align*}

8. \begin{align*}\frac{9}{2}\end{align*}

9. \begin{align*}\frac{17}{8}\end{align*}

10. \begin{align*}\frac{22}{3}\end{align*}

Directions: Write three equivalent fractions for each of the following fractions.

11. \begin{align*}\frac{2}{3}\end{align*}

12. \begin{align*}\frac{12}{28}\end{align*}

13. \begin{align*}\frac{3}{4}\end{align*}

14. \begin{align*}\frac{9}{10}\end{align*}

15. \begin{align*}\frac{7}{8}\end{align*}

### Vocabulary Language: English

Denominator

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

Equivalent means equal in value or meaning.
Equivalent Fractions

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

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

Mixed Number

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

Numerator

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

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

Whole Numbers

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