Fractions

Fractions can be illustrated by dividing a shape into equal parts, and shading a certain number of these parts.

The bottom number is the number of equal parts that diagram has been cut into, and the top number is the number of equal parts that has been shaded.

The name given to the number on the top of a fraction is the numerator, and the name given to the number on the bottom of a fraction is the denominator.

If you cut a cake into two equal pieces and eat one of them, you have eaten \(\frac{1}{2}\) (half) a cake.

If a cake is cut into five equal pieces and you eat three of them, you have eaten \(\frac{3}{5}\) (three fifths) of a cake.

\(\frac{1}{2}\) and \(\frac{3}{5}\) are examples of fractions - parts of a whole.

\(\frac{1}{3}\) is equivalent to \(\frac{2}{6}\) because the top and bottom numbers have been multiplied by \(2\).

\(\frac{4}{12}\) is equivalent to \(\frac{1}{3}\) because the top and bottom numbers have been divided by \(4\).

When writing equivalent fractions, do the same multiplication or division to the top and bottom numbers.

For example, if you multiply the top by 2, you must also multiply the bottom by 2.

You know that: \(\frac{4}{12} = \frac{2}{6} = \frac{1}{3}\)

\({4}\) and \({12}\) have a common factor, namely \({4}\), so \(\frac{4}{12}\) can be written as \(\frac{1}{3}\) (divide the top and the bottom by \(4\)).

\({2}\) and \({6}\) have a common factor, namely \({2}\), so \(\frac{2}{6}\) can be written as \(\frac{1}{3}\) (divide the top and the bottom by \({2}\)).

However, \({1}\) and \({3}\) have no common factors, so \(\frac{1}{3}\) cannot be simplified.

When a fraction cannot be simplified we say that it is its simplest form.

Whole numbers

A whole number can be written as \(\frac{2}{2}\), \(\frac{3}{3}\), \(\frac{4}{4}\), etc.

Mixed numbers

\(1\frac{2}{3}\) is known as a mixed number, because it is made up of a whole number and a fraction.

So \(1 \frac{2}{3}\) can be written as:

\(\frac{3}{3} + \frac{2}{3} = \frac{5}{3}\)

Improper fractions

\(\frac{5}{3}\) is called an improper fraction, because the top number is bigger than the bottom number.

Converting from a mixed number to an improper fraction

You can write the whole number part as a fraction, with the same denominator as the other fraction, and then add the fractions together.

Example

\(1 \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}\)

Here is another example:

\(2 \frac{1}{4} = 1 + 1 + \frac{1}{4} = \frac{4}{4} + \frac{4}{4} + \frac{1}{4} = \frac{9}{4}\)

Another way to work out how many quarters there are in \(2\) wholes is to multiply the whole number, \(2\), by the denominator, \(4\).

This gives \(8\) quarters.

Add one quarter to give the improper fraction \(\frac{9}{4}\).

Converting from improper fractions to mixed numbers

You can separate the fraction into as many whole numbers as possible, with a smaller remaining fraction.

Example

\(\frac{17}{5}= \frac{5}{5} + \frac{5}{5} + \frac{5}{5} + \frac{2}{5} = 3 \frac{2}{5}\)

Another way to convert an improper fraction is to find how many whole numbers you get, by using a division.

For example let's convert \(\frac{17}{5}\) to a mixed number again.

We start by dividing the top number by the bottom number.

\({17}\) divided by \({5}\) is \({3}\) remainder \({2}\).

So the whole number part is \({3}\), and the remainder \({2}\) means there are \(\frac{2}{5}\) left over.

So the answer is \(\frac{17}{5} = 3 \frac{2}{5}\)

Using a calculator

If your calculator has a fraction button you can use this to convert from improper fractions to mixed numbers.

Type in the improper fraction, press '\({=}\)', and the calculator will convert it to a mixed number.

Writing a number as a fraction of another

If you get \({7}\) out of \({10}\) in a test, you can write your score as \(\frac{7}{10}\).

\({7}\) expressed as a fraction of \({10}\) is \(\frac{7}{10}\).

Similarly, if there are \({20}\) socks in a drawer and \({4}\) of them are blue, \(\frac{4}{20}\) of the socks are blue.

\({4}\) expressed as a fraction of \({20}\) is \(\frac{4}{20}\).

We can put this into its simplest form by dividing the top and bottom numbers by \({4}\), so we get \(\frac{1}{5}\).

Which fraction is bigger: \(\frac{3}{4}\) or \(\frac{5}{7}\)?

It is hard to answer this question just by looking at the fractions.

However, if you write the fractions with the same bottom number, or denominator, the question will be easy.

\(\frac{3}{4}\) has a denominator of \(4\), and \(\frac{5}{7}\) has a denominator of \({7}\).

\({4}\) and \({7}\) both divide into \({28}\), so rewrite the fractions with a denominator of \({28}\).

\(\frac{3}{4}= \frac{21}{28}\)

\(\frac{5}{7}= \frac{20}{28}\)

It is easy to see that \(\frac{21}{28}\) is bigger than \(\frac{20}{28}\).

Therefore \(\frac{3}{4}\) is bigger than \(\frac{5}{7}\).

To compare fractions, first write them with the same bottom number, or denominator.