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Fractions – basics, visualisation and simplification

Imagine every number represents a unit of measurement but in a concrete form, for example: pizzas, chocolate, carrots… The number $1$ will represent $1$ pizza, $2$ obviously $2$ pizzas, and so on.

What happens if you have only one pizza, and you would like to share it with a friend?

Instinctively you know that, if you want to be fair, you’ll divide that pizza into two equal parts. If our pizza represents number $1$, how would we write down one half of the pizza? This is where fractions come into the story.

What exactly is a fraction?
A fraction is a number which represents the number of parts of a whole.

All fractions come in the form of $\displaystyle{\frac{a}{b}}$. The top number in a fraction is called the numerator, and the bottom number is called denominator. The denominator shows how many parts is a whole divided in, and the numerator shows the number of those parts in this particular fraction. The line that separates them is called the fraction line.

So, when we write down one half in the form of a fraction, it looks like this: $\displaystyle{\frac{1}{2}}$.

half of pizzaIn the context of pizza, this fraction means that you divided a pizza into two equal parts and took only one part.

What if another friend joins you and he also wants an equal part? Then you’ll divide pizza into three parts and take only one part. This means you’ll get or ‘one third’ of a pizza.

Visualizing equivalent fractions and simplification

Let’s say it’s time for dessert, and you want to share a cake with friend. Of course, the easiest way is to simply cut that cake in half and have each of you take one part. But you can’t eat that much at once, so you have to divide it into more parts to make it more manageable. If you divided it into $6$ equal parts and each of you took $3$ parts, both of you still got exactly one half of the cake.

two parts from one

This means that the value of one half is equal to the value of three sixths, and we can write that down as:

$\displaystyle{\frac{1}{2} = \frac{3}{6}}$.
We can divide this cake in as many parts as we want.

$\displaystyle{\frac{1}{2} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = …}$

These fractions are called equivalent fractions because they share the same value, and this leads us to a very important property of fractions:

If we multiply both the numerator and denominator by the same positive number, the value of the fraction will not change.

This property also gives us a way to simplify a fraction whose numerator and denominator are large numbers. We can accomplish this by dividing the numerator and the denominator with their greatest common divisor.

Example 1. Simplify the fraction.

$\displaystyle{\frac{6}{12} = ?}$
 

The greatest common divisor of 6 and 12 is 6. This means that we’ll divide both the numerator and the denominator by number 6, as a result we get:  $\displaystyle{\frac{6}{12} = \frac{1}{2}}$.

When solving various problems, you may also encounter fractions that contain unknowns. The recipe for treating them is the same as is for any other fraction.

Example 2. Simplify the fraction.

$\displaystyle{\frac{12ab}{2bc} = ?}$

The greatest common divisor of $12ab$ and $2bc$ is $2b$.

When we divide both the numerator and the denominator, we get: $\displaystyle{\frac{12ab}{2bc} = \frac{6ab}{c}}$.