If you have ever cut a cake or a pizza into slices, you have already dealt with fractions.
A fraction represents a part of a whole or a collection of objects. It is a way to divide something into equal parts or portions.
A fraction has two parts: the numerator and the denominator. The numerator is the number on top of the fraction, and it represents how many parts of the whole or collection are taken. The denominator is the number below the line, and it represents how many equal parts the whole is divided into.
\frac{numerator}{denominator}
For example, in the fraction \frac{3}{4} , 3 is the numerator , and 4 is the denominator . The line separating the numerator from the denominator is known as the vinculum.
You can read more about the vinculum here .
Fractions are used in many areas of mathematics and everyday life, such as measuring ingredients for a recipe.
What Is A Fraction?
A fraction is a mathematical representation of a part of a whole.
It is used to describe a portion or section of any quantity out of a whole. Fractions are commonly written in the form \frac{a}{b} , where a is the numerator and b is the denominator.
Definition
A fraction is a numerical value that represents the quotient of two numbers, where the numerator represents the number of equal parts of the whole or collection that are taken, and the denominator represents the total number of equal parts that make up the whole or collection.
Parts of a Fraction
A fraction has two parts: the numerator and the denominator. The numerator is the number on the top of the fraction line, and it represents the number of equal parts of the whole or collection that are taken. The denominator is the number below the line, and it represents the total number of equal parts that make up the whole or collection.
Equivalent Fractions
Equivalent fractions are fractions that have different numerators and denominators but represent the same value.
For example, \frac{1}{2} and \frac{2}{4} are equivalent fractions because they both represent half of a whole.
You can read more about equivalent fractions here .
Proper and Improper Fractions
A proper fraction is a fraction where the numerator is less than the denominator. In contrast, an improper fraction is a fraction where the numerator is greater than or equal to the denominator.
An example of a proper fraction is \frac{1}{2} , whereas an example of an improper fraction is \frac{3}{2} .
Common Denominator
A common denominator is a shared multiple of the denominators of two or more fractions. Adding or subtracting fractions with different denominators requires finding a common denominator.
You can learn how to add fractions here .
Unit Fraction
A unit fraction is a fraction where the numerator is 1 .
For example, \frac{1}{2} is a unit fraction because the numerator is 1 .
Types of Fractions
When it comes to fractions, there are three main types: proper, improper, and mixed fractions.
Proper Fractions
A proper fraction is a fraction where the numerator is less than the denominator. In other words, the fraction represents a part of a whole that is less than one.
For example, \frac{1}{4} is a proper fraction because the numerator 1 is less than the denominator 4 .
Improper Fractions
An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In other words, the fraction represents a part of a whole that is greater than or equal to one.
For example, \frac{5}{3} is an improper fraction because the numerator 5 is greater than the denominator 3 .
Mixed Fractions
A mixed fraction is a combination of a whole number and a proper fraction. It is sometimes called a mixed number.
For example, 2\frac{1}{3} is a mixed fraction because it represents two whole units and one part of a third unit.
When working with mixed fractions, it is often helpful to convert them to improper fractions. To do this, you multiply the whole number by the denominator and then add the numerator.
The result becomes the new numerator, and the denominator stays the same.
Mixed \ Number \ \frac{numerator}{denominator} = \\ \ \\ \frac{(Mixed \ Number \times denominator ) + numerator}{denominator}
For example,
2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}
Understanding the different types of fractions is important when performing operations with fractions, such as addition, subtraction, multiplication, and division. It is also important when comparing fractions and converting between mixed and improper fractions.
Equivalent Fractions
Equivalent fractions are fractions that represent the same part of a whole, but have different numerators and denominators.
For example, \frac{1}{2} and \frac{2}{4} are equivalent fractions because they represent the same amount, which is half of a whole.
To find equivalent fractions, you can multiply or divide both the numerator and denominator by the same number .
For instance, to find an equivalent fraction of \frac{2}{3} , you can multiply both the numerator and denominator by 2 to get \frac{4}{6} . This fraction is equivalent to \frac{2}{3} because they both represent the same value.
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
It’s important to note that equivalent fractions are not unique. There are infinite equivalent fractions for any given fraction.
You can learn more about equivalent fractions here .
In summary, equivalent fractions are fractions that represent the same part of a whole but have different numerators and denominators.
Operations with Fractions
When working with fractions, it is important to understand how to perform basic operations such as addition, subtraction, multiplication, and division. By mastering these operations, you can solve complex problems involving fractions with ease.
Addition and Subtraction
Adding and subtracting fractions can easily be performed once the denominator of both fractions is the same.
When adding or subtracting fractions with different denominators, you need to find a common denominator.
To do this, you can either find the least common multiple (LCM) of the denominators or multiply the denominators together. Then, you can convert each fraction to an equivalent fraction with the same denominator and perform the operation.
To learn more about adding fractions read this resource .
Multiplication and Division
To multiply fractions, simply multiply the numerators and denominators separately and simplify the result if possible.
To divide fractions, invert the second fraction (i.e., switch the numerator and denominator) and multiply it by the first fraction.
Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their greatest common factor (GCF).
By understanding these basic operations and applying them correctly, you can solve a wide range of problems involving fractions.
Fractions in Real Life
Fractions are everywhere in our daily lives, from dividing up the pizza to measuring ingredients for a recipe. Understanding how fractions work in real-life situations can help you make better decisions and solve problems more effectively.
Pizza
Pizza is a perfect example of how fractions work in the real world. Imagine you order a whole pizza for yourself and a friend. If you want to share the pizza equally, you will need to divide it into two equal parts. This means each person will get half of the pizza. Half is represented as \frac{1}{2} .
Now, let’s say you order a pizza with four slices. If you want to eat two slices, you will be eating \frac{2}{4} of the pizza. This is also the same as \frac{1}{2} . It’s important to understand that fractions can be simplified, so \frac{2}{4} is the same as \frac{1}{2} .
Other Food
Fractions are also used in cooking and baking. When you measure out ingredients, you are often using fractions. For example, if a recipe calls for \frac{3}{4} cup of flour, you need to measure out three-quarters of a cup.
Another example is when you are making a smoothie. If you want to make a smoothie with equal parts of fruit and yogurt, you will need to use \frac{1}{2} cup of each.
In summary, fractions are an essential part of our daily lives, from sharing pizza to measuring ingredients for a recipe. By understanding how fractions work in real-life situations, you can make better decisions and solve problems more effectively.
Fractions Summary
Understanding fractions is an essential part of mathematics. Fractions represent parts of a whole or collection of objects. They are used in everyday life, from cooking and baking to measuring and building. In this section, we will provide you with a brief summary of fractions.
A fraction has two parts: the numerator and the denominator. The numerator is the number on the top of the fraction, and it represents the number of equal parts of the whole or collection that are taken. The denominator is the number on the bottom of the fraction, and it represents the total number of equal parts that make up the whole or collection.
Fractions can be proper, improper, or mixed. A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. A mixed fraction is a combination of a whole number and a proper fraction.
You can perform various operations with fractions, such as addition, subtraction, multiplication, and division. To add or subtract fractions, you need to find a common denominator. To multiply fractions, you simply multiply the numerators together and the denominators together. To divide fractions, you need to multiply the first fraction by the reciprocal of the second fraction.
In conclusion, fractions are an essential part of mathematics and everyday life. They represent parts of a whole or collection of objects and can be proper, improper, or mixed. You can perform various operations with fractions, such as addition, subtraction, multiplication, and division.