Fractions are really cool!

Five out of four people have trouble with fractions, Steven Wright.

A fraction is a fancy way of representing the division of a whole into parts, for example, half of a pizza (¹⁄₂), a quarter of an apple (¹⁄₄), etc.

How can you learn about them?

You can use JFractionLab. It’s a free (it is licensed by the GNU GPL license), multi-platform (it runs on Windows, macOS, and Linux), and easy to use program for practicing fractions at school or at home.

It gives you graphical feedback, so you can always see and understand what you are doing.

It runs on Windows, GNU/Linux, and macOS, but it requires the installation of the Java Runtime Environment. If you use Ubuntu or other Debian-based distributions, type this command in your terminal: sudo apt install jfractionlab.

Launch it (JFractionLab), and select one category: Clicking the numerator, Defining or comparing fractions, or Extending fractions.

Observe that ³⁄₆ is the fraction that is represented in the screenshot. The pie is divided into six parts and there are only three pieces which are colored yellow.

⁴⁄₈ is lower than ⁸⁄₈ ( = 1), in other words, (almost) everyone prefers eating eight pieces of a yummy pizza (the whole pizza) rather than only four (half of it).

Equivalent fractions and reducing fractions

As you can see ⁶⁄₂₀ and ³⁄₁₀ look quite different, but they represent the same value, the same area colored in yellow.

These two fractions are equivalents. You can make equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

In our example, if you multiply the numerator (3), and the denominator (10) of our second fraction (³⁄₁₀) by the same number (2), you will get the first fraction (⁶⁄₂₀). The idea is simple. It is the same to have 3 slices of a pizza that has been cut into 10 equal slices as having 6 slices of a pizza that has been cut into 20 slices.

Every fraction can be written in its lowest terms; simply put, it cannot be further reduced. This is called an irreducible fraction (or fraction in lowest terms or reduced fraction). A fraction is irreducible if and only if the numerator and denominator are coprimes: the only number that divides both of them evenly is 1.

It can be obtained by dividing both the numerator and denominator by the greatest common divisor.

Let’s help you reduce any fractions to its lowest terms. You may use WolframAlpha, Maxima (just type your fraction) or Python.

wxMaxima is a document based interface for Maxima. Maxima is a free (it is released under the terms of the GNU General Public License) computer algebra system. It is written in Common Lisp and runs on macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. Installing it in windows is fairly simple. If you are in Linux (Ubuntu, Debian, etc.), open a terminal and type: sudo add-apt-repository ppa:blahota/wxmaxima. sudo apt-get update. sudo apt-get install wxmaxima.

To start using wxMaxima right away, start typing your command (8/20;) and press Shift + Enter to evaluate it.

Hi@Welcome:~ python
Python 3.8.6 (default, May 27 2021, 13:28:02) 
>>> from fractions import Fraction # Fraction module in Python
>>> print (Fraction(8, 20)) # It calculates the irreducible fraction
2/5
>>> print (Fraction(6, 14))
3/7
>>> import math
>>> print (math.gcd(8, 20)) # The Highest Common Factor (gcd) can be computed in python using a single function offered by the math module. 
4
>>> print (math.gcd(6, 14))
2

Fraction Arithmetic.

How do you add fractions together? Do they have the same denominator? If so, add the two numerators and write the sum over the same denominator: ²⁄₃ + ⁵⁄₃ = ⁷⁄₃.

If the fractions do not have the same denominator, we must make sure that they do. For instance, ²⁄₃ + ³⁄₅. This step is done by finding a common denominator. A way of doing this is by multiplying both denominators (3 x 5 = 15) and rewriting both fractions using this product.

Take into account that you need to multiply both numerator and denominator by the same number to get equivalent fractions. Our fractions, ²⁄₃ + ³⁄₅, are rewritten to, = ¹⁰⁄₁₅ + ⁹⁄₁₅, which in turn is equal to = ¹⁹⁄₁₅. Finally, rewrite, if necessary, the result as a simplified or reduced fraction.

How do you subtract fractions? If they have the same denominator, subtract both numerators while leaving the denominator unchanged: ⁷⁄₃ - ⁵⁄₃ = ⁴⁄₃.

As it was explained before, if they do not share the same denominator, it will be necessary to make sure that they do. For example, ³⁄₄ - ²⁄₃ = ⁹⁄₁₂ - ⁸⁄₁₂ = ¹⁄₁₂. Rewrite, if necessary, the result as a simplified or reduced fraction.

Microsoft Math Solver provides free step by step solutions to a variety of Math problems.

Let’s learn to multiply fractions. It will be easy peasy, lemon squeezy. You only need to multiply both numerators and denominators. For instance, ³⁄₄ * ²⁄₃ = ⁶⁄₁₂. Rewrite, if necessary, the result as a simplified or reduced fraction, ⁶⁄₁₂ = ¹⁄₂.

Finally, let’s divide some fractions. Multiply the first fraction by the reciprocal of the second fraction. You just need to turn the second fraction upside down or, in other words, flip it over, so that the numerator becomes the denominator and the denominator becomes the numerator. For instance, ³⁄₄ / ²⁄₃ = ⁹⁄₈. Simplify, if necessary the result.

In our article Fractions in Python, you can learn how to work with fractions in Python.