Fractions - The Basics

Fractions - The Basics

What is a fraction?

The best way to consider a fraction is as part of a pie…

…where the pie represents the number 1.

= 1

For example, half of the pie would be represented in maths as…

$\frac{1}{2}$

i.e.

$\small \frac{1}{2}$ literally means one divided by two.

The two numbers contained in the fraction each have a name: the number on the top is called the numerator and the number on the bottom is called the denominator.

It is important to note here that not all numbers can be represented as a fraction - a fact that is not immediately obvious and indeed this is an instance where the great Pythagoras got his maths wrong! Numbers that can be represented by fractions are called rational numbers and those that can't are irrational numbers. For example, pi ( $\small \pi$ or 3.1415927…) can't be represented as a fraction. Neither can $\small \sqrt{2}$ . Both of these numbers are irrational.

Whole Numbers As Fractions

Whole numbers can be represented as a fraction because, for example, two divided by one is just 2. So the number '2' as a fraction is written as

$\frac{2}{1}$

To write a whole number as a fraction simply write that whole number over 1.

$n \equiv \frac{n}{1}$

…putting it mathematically :-)

Equivalent Fractions

Below is a table showing how different fractions fit into the number 1. It is worthwhile seeing if you can understand this table before you read on…

$A fractions table$

See that 2 lots of quarters $\small \left(\frac{1}{4}+\frac{1}{4}\right)$ represents the same amount as one half $\small \left(\frac{1}{2}\right)$ .

In other words… $\frac{1}{2} = \frac{2}{4}$

One divided by two leaves me with a half of one. Two divided by four leaves me with a half of two. The amount represented by both fractions is one half. Technically speaking, the ratio of the numerator and demoninator in both fractions is the same. Although the numerators and denominators are different in both fractions both of these fractions represent the same amount. They are equivalent. See if you can spot other equivalent fractions in the table above.

This idea that we can convert a fraction into an equivalent fraction - same amount but different numbers top and bottom - becomes very important when we start working with them, as you will now see.

Adding Fractions

If we are adding fractions that have the same denominator then this is easy… $\frac{3}{14} + \frac{5}{14} = ?$

Because we have 3 lots of 14^ths and 5 lots of 14^ths then the sum of these two fractions will be 8 (5 + 3) lots of 14^ths, or $\small \frac{8}{14}$ . The sum is easy because the denominators are the same. Maths with fractions becomes complicated when the denominators are different, for example:-

$\frac{1}{4} + \frac{2}{5} = ?$

The trick to adding these two fractions together is to convert one or both of the fractions into equivalent fractions so that the denominators become the same number. In other words, if we find a common denominator then they will be easy to add together!

$\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} = \frac{5}{20} = \frac{6}{24}$ …

$\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20} = \frac{10}{25} = \frac{12}{30}$ …

Can you spot the two fractions from each list that have the same denominators? That's right: - $\small \frac{5}{20}$ and $\small \frac{8}{20}$ .

So our question can be rewritten as

$\frac{5}{20} + \frac{8}{20} = \frac{13}{20}$

Subtracting Fractions

The trick is the same when subtracting fractions: we need to find a common denominator. For example…

$\frac{7}{9} - \frac{1}{4} = ?$

Convert $\small \frac{7}{9}$ and $\small \frac{1}{4}$ into equivalent fractions where the denominators are both the same…

$\frac{7}{9} = \frac{14}{18} = \frac{21}{27} = \frac{28}{36} = \frac{35}{45} = \frac{42}{54}$ …

$\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} = \frac{5}{20} = \frac{6}{24} = \frac{7}{28} = \frac{8}{32} = \frac{9}{36} = \frac{10}{40}$ …