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People deal with numbers in countless daily situations. Many of these involve numbers that are not whole numbers but are some part, or fraction, of them. This lesson will define fractions, present some facts about them, and show different real-life applications.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Splitting a Pizza

Jordan invited her friends over for a pizza party. They wonder how to split a pizza so that everyone gets the same number of slices. Try different ways to split the pizza, then click on the slices to decide how much pizza each person will eat. External credits: @macrovector
Note that Jordan can only split the pizza into a minimum of and a maximum of slices.

## Presenting Rational Numbers

In order to state what part of the pizza each person will eat, a certain type of numbers should be used.

## Rational Numbers

The set of rational numbers, represented by the symbol is formed by all numbers that can be expressed as the ratio between two integers, where
All four of these examples are rational numbers. Note that integers are also rational because they can always be written as fractions with a denominator of
In other words, integer numbers is a subset of rational numbers. Real numbers that are not rational are called irrational numbers.

## Definition of a Fraction

In the definition of rational numbers, the word fraction showed up but was not explained in detail. To clarify any doubts, its definition will now be presented.

## Fraction

Fractions are a specific type of ratio that compares a part to a whole. Fractions are rational numbers written in the form where the numerator is the part and the denominator is the whole.

There are many possible ways of reading fractions, but one universal method is saying over Fractions where is less than are called proper fractions. Fractions where is greater than or equal to are called improper fractions. Fractions are also another way to write a division of the numerator by the denominator.
A fraction like can be simplified to or just It is important to keep in mind that the denominator of a fraction can never be equal to because the quotient of division by is always undefined.

## Determining Fractions

Jordan's friends know how much she loves chocolate, so three people brought her chocolate bars as a gift when they came for her pizza party. Each chocolate bar was divided into a different number of pieces. External credits: @freepik, @freepik, @freepik
Jordan decides to share the chocolate with her friends. The shaded pieces in the diagrams above show how much of each bar the kids ate.
a What part of the dark chocolate bar A do the shaded pieces represent?
b What part of the milk chocolate bar B do the shaded pieces represent?
c What part of the white chocolate bar C do the shaded pieces represent?

### Hint

a Find the total number of pieces the chocolate bar is divided into. Then count the number of shaded pieces.
b Divide the number of shaded pieces by the total number of pieces.
c Form a fraction and check whether it can be simplified by dividing its numerator and denominator by their greatest common factor.

### Solution

a Start by analyzing the dark chocolate bar A. External credits: @freepik
First, count the total number of pieces and the number of shaded pieces.
To find what part of the chocolate bar the shaded pieces represent, divide the number of shaded pieces by the total number of pieces.
Therefore, the shaded pieces represent (five-tenths) of the chocolate bar. Notice that both the numerator and denominator are divisible by This means that the fraction can be simplified by dividing its numerator and denominator by
This means that the shaded pieces represent one-half of the chocolate bar.
b Now consider the milk chocolate bar B. External credits: @freepik
Like in Part A, start by counting the total number of pieces of chocolate and the number of shaded pieces.
The part of the chocolate bar represented by the shaded pieces can be written as a fraction. The numerator of the fraction is the number of shaded pieces and the denominator is the total number of pieces.
This fraction can also be simplified. Since and are even numbers, they both can be divided by
Therefore, the shaded pieces represent or of the chocolate bar.
c Finally, examine the last chocolate bar, the white chocolate bar C. External credits: @freepik
Again, count the total number of pieces of chocolate and the number of shaded pieces.
Next, divide the number of shaded pieces by the total number of pieces to find what part of the chocolate bar the kids ate.
Since both the numerator and denominator are divisible by the fraction can be simplified.
The shaded pieces make up or of the chocolate bar.

## Identifying Fractions Describing the Graph

Consider a bar that is split into different parts. Find the fraction that describes the relationship between the shaded parts and the bar as a whole. Any shaded parts on the right-hand side indicate that the fraction is an improper fraction. Do not simplify the fractions. ## Equivalent Fractions

Fractions that have different numerators and denominators but represent the same value are called equivalent fractions. For example, the following fractions are equivalent because they are all equal to the same value, even though they look different.
Imagine that there are two friends, Emily and Maya, who really love chocolate cake. They have two identical cakes. Emily divided her cake into four parts and ate one. At the same time, Maya divided her cake into eight slices and ate two. External credits: @macrovector
As can be seen, Emily and Maya ate the same amount of cake. Therefore, and are equivalent fractions.
Equivalent fractions can be formed by multiplying or dividing the numerator and denominator by the same number. ## How to Simplify a Fraction

When a fraction has a large numerator and denominator, it can be hard to estimate its value. Simplifying such a fraction and finding an equivalent fraction with a smaller numerator and denominator can be helpful.

## Simplifying a Fraction

Fractions of the same value can be written using different pairs of numerators and denominators. This is why some fractions can be simplified to equivalent fractions with a smaller numerator and denominator. Consider the following example.
In order to simplify this fraction, there are three steps to follow.
1
Determine Whether the Fraction Can Be Simplified
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To determine whether the given fraction can be simplified, split the numerator and denominator into prime factors and see if there are any common factors other than
The numerator and denominator share factors and Therefore, the fraction can be simplified. If the numerator and denominator did not have common factors other than the fraction would be said to be simplified or written in its simplest form.
2
Find the Greatest Common Factor
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The greatest common factor (GCF) of the numbers and can be found by multiplying all their common factors.
If the numerator and denominator share only one common factor, then that factor is their GCF.
4
Reduce the Fraction
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Finally, to reduce the fraction, divide its numerator and denominator by their GCF.
As a result, an equivalent fraction of was obtained. It is the simplest form of the given fraction

### Extra

Simplification of Different Fractions
The applet below illustrates how different fractions are simplified. ## Pairing Equivalent Fractions

After having a lot of fun with her friends on Saturday, Jordan woke up rested and energized the next day and decided to do her math homework. When she finished, she texted her answers to her friend Emily and asked if she got the same results. Jordan was worried that she made a mistake. Suddenly, she realized that even though their fractions looked different, they actually had the same values. This means that they both solved the exercise correctly! Pair the equivalent fractions from Jordan's and Emily's results.

### Hint

Find the greatest common factor (GCF) of the numerator and denominator of a fraction, then use it simplify the fraction.

### Solution

Notice that Emily's fractions are written in their simplest form, but Jordan's fractions are not simplified. Therefore, to find the equivalent fractions, each of Jordan's fractions will be analyzed and simplified, one at a time.

### First Fraction

Begin by considering the first fraction from Jordan's list.
To simplify this fraction, its numerator and denominator should be divided by their greatest common factor (GCF). To find it, split and into prime factors.
As shown, and share two common factors. Their product is the GCF of these numbers.
Finally, divide both the numerator and denominator by to simplify the fraction.
Therefore, is an equivalent fraction to

### Second Fraction

Now, consider the second fraction from Jordan's list.
Again, find the GCF of the numerator and the denominator To do so, write each number as the product of prime numbers.
There are three common factors between and Multiply them to calculate the GCF of and
Next, simplify the fraction by dividing its numerator and denominator by
Therefore, and are equivalent fractions.

### Third Fraction

Similarly, examine the third fraction that Jordan wrote.
Just as before, determine the GCF of and by factoring the numbers into prime factors.
The numbers share only one common factor, This means that the GCF of and is Next, simplify the fraction by dividing the numerator and denominator by
It can be concluded that is an equivalent fraction to

### Fourth Fraction

Finally, consider the last fraction from Jordan's list.
Start by rewriting the numerator and denominator as the product of prime factors.
The common factor between and is This is their GCF. Divide both the numerator and denominator by to simplify the fraction.
It can be concluded that is an equivalent fraction to

## Simplifying Fractions

Consider the given fraction. Can it be simplified? If yes, write the given fraction in its simplest form. If the fraction is already simplified, write it as it is. ## Real Life Applications of Fractions

On top of the topics mentioned in this lesson, there are many other real-life applications of fractions. Here are some of them.

• Recipes: Cooking is full of fractions. For example, a recipe for four servings might suggest using teaspoon of vanilla extract and tablespoon of sugar. If someone wants to cook for only two people, they would need to use fractions to adjust the ingredients accordingly. External credits: @brgfx, @pikisuperstar, @brgfx
• Sports: Fractions are frequently used to analyze the performance of a particular player and team or determine statistics like shooting percentages. Note that although percentages often appear in such statistics, they are calculated with the usage of fractions. External credits: Image by brgfx
• Shopping: When there is a sale, fractions can be used to calculate the reduced price of a product. Sales tax and coupons also use fractions. • Tests and exams: Scores of tests, exams, and homework assignments are generally expressed as fractions, like • Money: A dime is of a dollar. A quarter is a of a dollar. Understanding fractions makes adding money quick and easy. External credits: Wikipedia, Wikipedia, Wikipedia, Wikipedia
These are just some of the situations where fractions are really useful, but there are so many more! Look around and try to find where else fractions might appear in day-to-day life.

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