Fractions. Representation, mixed numbers and operations

05/05/2026

The online fraction simulations on this page will help you better understand the concept of fractions in mathematics. We will discover different ways of representing fractions, mathematical operations with fractions, and what mixed numbers are.

This Thematic Unit is part of our Mathematics collection

Denominator

The lower number that determines how many equal parts the unit is divided into; it can never be zero due to mathematical impossibility.

Equivalent Fractions

Expressions that, despite having different numbers, represent the same proportion or decimal value after division.

Fraction

An expression representing the division of a quantity into equal parts, indicating how many of those parts are taken from the total.

Improper Fraction

A fraction whose numerator is equal to or greater than the denominator, indicating the value is equal to or greater than one whole.

Least Common Multiple

The smallest common multiple among several numbers; used to equalize denominators for adding or subtracting different fractions.

Mixed Number

A notation that combines a whole number and a proper fraction to easily represent values greater than one unit.

Numerator

The upper figure of a fraction that indicates the number of units or parts selected from the complete unit.

Proper Fraction

A type of fraction where the numerator is smaller than the denominator, always representing a value less than one whole unit.

Reciprocal

The result of transposing the terms of a fraction; multiplying it by the original fraction always results in the number one.

Simplification

The technique of reducing a fraction to its lowest terms by dividing both the numerator and denominator by a common divisor.

What are fractions in mathematics

Fractions are a fundamental part of mathematics. They are used to represent numbers that are not whole numbers, allowing to express relationships of parts of a whole. A fraction consists of two parts: the numerator and the denominator. The numerator indicates the number of parts that are taken or considered, while the denominator indicates how many parts the whole is divided into. For example, in the fraction 3/4, the numerator is 3 and the denominator is 4.

Representation of fractions in mathematics

Fractions can be represented in different ways. In addition to the common fraction notation, they can be expressed in decimal or percent form. For example, the fraction 1/2 is equal to 0.5 in decimal form and 50% in percent form.

Mathematical operations with fractions

Fractions allow you to perform mathematical operations such as addition, subtraction, multiplication and division. To add or subtract fractions, it is necessary to have the same denominator. If the denominators are different, common denominators must be found using the least common multiple (LCM) technique. To multiply fractions, multiply the numerators together and the denominators together. To divide fractions, multiply the first fraction by the inverse of the second fraction.

Mixed numbers

Mixed numbers are a combination of whole numbers and fractions. They are composed of a whole part and a fraction. For example, the mixed number 3 1/2 is composed of the whole number 3 and the fraction 1/2.

Denominator

The lower number that determines how many equal parts the unit is divided into; it can never be zero due to mathematical impossibility.

Equivalent Fractions

Expressions that, despite having different numbers, represent the same proportion or decimal value after division.

Fraction

An expression representing the division of a quantity into equal parts, indicating how many of those parts are taken from the total.

Improper Fraction

A fraction whose numerator is equal to or greater than the denominator, indicating the value is equal to or greater than one whole.

Least Common Multiple

The smallest common multiple among several numbers; used to equalize denominators for adding or subtracting different fractions.

Mixed Number

A notation that combines a whole number and a proper fraction to easily represent values greater than one unit.

Numerator

The upper figure of a fraction that indicates the number of units or parts selected from the complete unit.

Proper Fraction

A type of fraction where the numerator is smaller than the denominator, always representing a value less than one whole unit.

Reciprocal

The result of transposing the terms of a fraction; multiplying it by the original fraction always results in the number one.

Simplification

The technique of reducing a fraction to its lowest terms by dividing both the numerator and denominator by a common divisor.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Introduction

This simulation allows you to select among several functions and represent their derivative. It is possible to check how the derivative is modified by varying the parameters of the functions.

Build a fraction

Build fractions from graphs and numbers to earn stars in this fraction game or explore in the Fractions Laboratory. Test yourself on any level you want – try to collect as many stars as you can!

Mixed numbers

Explore fractions while serving yourself 1 and 1/2 cups of chocolate cake and drinking it with 1/3 cup of water! Create your own fractions using fun interactive objects. Match shapes and numbers to earn stars in the mixed numbers game. Try your hand at any level you like – try to collect as many stars as possible!

Fraction Matcher

Match shapes and numbers to earn stars in this fraction game. Test yourself at any level you like – try to collect lots of stars!

Equality of fractions

Build equivalent fractions with different denominators. Match shapes and numbers to earn stars in the game. Test yourself on any level you like – try to collect lots of stars!

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Leonhard Euler

1707

–

1783

Euler developed much of modern mathematical notation and made key contributions to analysis, topology, mechanics, and graph theory

“Nothing takes place in the universe without mathematics expressing it”

Carl Friedrich Gauss

1777

–

1855

Carl Friedrich Gauss developed fundamental methods in algebra, number theory, and statistics, consolidating the foundation of modern mathematics.

“Mathematics is the queen of the sciences and number theory its queen”

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1646

–

1716

Gottfried Wilhelm Leibniz was one of the founders of infinitesimal calculus, created modern mathematical notation, and contributed to mechanics and optics with groundbreaking theories

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Johannes Kepler

1571

–

1630

Kepler formulated the three laws of planetary motion, establishing elliptical orbits and the relationship between planetary period and distance from the Sun

“The heavens teach the geometry that nature follows”

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Test your knowledge

What is a fraction and what is it used for in mathematics?

A fraction is a way to represent a part of a whole or the relationship between two quantities, expressed as a numerator and a denominator. Fractions are used in mathematics to work with non-whole numbers, compare proportions, and perform operations with parts of a set. Understanding them is crucial because they appear in everyday situations like measuring ingredients, calculating discounts, dividing resources, or analyzing percentages, and they also form the foundation for more advanced concepts such as decimals, ratios, and equations.

What is the difference between proper, improper, and mixed fractions?

Proper fractions have a numerator smaller than the denominator, indicating a quantity less than the whole. Improper fractions have a numerator greater than or equal to the denominator, showing that the quantity is equal to or greater than the whole. Mixed fractions combine a whole number with a proper fraction, offering a clearer way to express quantities larger than one. This classification not only helps interpret fractions visually, but also simplifies operations and makes it easier to understand proportions in real-world problems.

How is it that sometimes fractions look bigger than whole numbers, even though they’re supposed to be “just parts”?

This happens with improper fractions or mixed numbers. Even though we talk about “parts,” the numerator can be larger than the denominator, meaning the value actually exceeds one. It’s like having 7 slices of a pizza cut into 4 pieces—you clearly have more than a whole pizza, and that’s exactly what an improper fraction tells you.

Does it make sense that simplifying a fraction changes anything if the value stays the same?

Actually, simplifying doesn’t change the value, but it makes calculations easier and results clearer. A simplified fraction is easier to compare, add, or subtract, and also more intuitive to understand. For example, adding 24/36 + 18/36 all the time is more complicated than 2/3 + 1/2, right? Simplifying reduces errors and helps interpret the results correctly.

So, what happens when denominators are different? How do we add fractions then?

You can’t just add numerators when denominators are different. First, you need to find a common denominator, which is like making all the slices the same size so they match. Once the denominators are the same, you just add the numerators. This method works for subtraction too and ensures the relationship between the parts stays correct.