Representing fractions

13.2 Representing fractions

Instead of writing “four sevenths”, we may write \(\frac{4}{7}\). This is called common fraction notation.

The number \(7\) below the line is called the denominator and it shows that the whole was divided into \(7\) equal pieces. Therefore, each piece is \(1\) seventh of the whole. The denominator shows the unit in which the number is expressed.

The number \(4\) above the line is called the numerator and it indicates the number of pieces.

common fraction

a fraction in which both the numerator and the denominator are integers

numerator

the number above the line in a fraction that tells how many parts of the whole are used

denominator

the number below the line in a fraction that tells how many parts there are in the whole

We can show a fraction using a diagram. Using a circle, we show the denominator as the total number of slices the circle is divided into, and the numerator as the number of shaded or unshaded slices.

\[\frac{4}{7} \leftarrow \frac{\text{number of shaded slices}}{\text{total number of slices}}\] \[\frac{3}{7} \leftarrow \frac{\text{number of unshaded slices}}{\text{total number of slices}}\]

We can also group the number of parts in a diagram to show different fractions. For example, this rectangle is divided into \(6\) identical squares:

If we shade \(4\) out of \(6\) squares, we can say that the shaded part represents \(\frac{4}{6}\) and the unshaded part represents \(\frac{2}{6}\).

We can also say that \(2\) out of the \(3\) parts are shaded. We shade \(2\) in every \(3\) parts of the rectangle. So the shaded part represents \(\frac{2}{3}\) and the unshaded part represents \(\frac{1}{3}\).

We call these fractions equivalent, so \(\frac{4}{6} = \frac{2}{3}\) and \(\frac{2}{6} = \frac{1}{3}\).

The same fraction can be expressed in different forms. For example, the fraction \(\frac{3}{4}\) can be expressed in eighths as \(\frac{6}{8}\), in twentieths as \(\frac{15}{20}\), in sixtieths as \(\frac{45}{60}\), and in many other units.

These are all different ways of expressing the same fraction. They are called equivalent fractions.

Equivalent fractions allow us to write the same number in different ways.

\[\frac{3}{4} = \frac{6}{8} = \frac{15}{20} = \frac{45}{60}\]

equivalent fractions

two or more fractions that represent the same amount (or value)

Equivalent fractions are the same value written in two or more different ways. It is exactly the same when we write \(4\) and \(\frac{4}{1}\). The values are the same – they only look different.

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Simplifying fractions

Simplifying a fraction means reducing the numbers in the fraction to the smallest possible numbers. We do this by cancelling a common factor from the numerator and the denominator. For example, in the fraction \(\frac{4}{6}\), both the numerator \(4\) and the denominator \(6\) are divisible by \(2\).

\(2\) is the common factor of \(4\) and \(6\); it is also the highest common factor.

\[\frac{4}{6} = \frac{2\ \mathbf{\times\ 2}}{3\ \mathbf{\times\ 2}} = \frac{2}{3} \times \frac{\mathbf{2}}{\mathbf{2}} = \frac{2}{3}\]

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Different types of fractions

You will find three types of fractions in fraction problems. Here is a summary of the three types:

• Proper fractions have a value of less than one; the numerator is always less than the denominator.
• Improper fractions are greater than or equal to one; the numerator is larger than or equal to the denominator.
• Mixed numbers (also called mixed fractions) have an integer part and a fraction part; the fraction part must be a proper fraction (less than one). We can write a mixed number as a sum of its integer part and the fractional part: \(2\frac{3}{5} = 2 + \frac{3}{5}\).

proper fraction

a fraction that has a value of less than one; the numerator is less than the denominator

improper fraction

a fraction that has a numerator larger than the denominator

mixed number

a number that is made up of a whole number and a fraction

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Converting mixed numbers to common fractions for calculations

To convert a mixed number to an improper fraction, start with the denominator; then follow the solid arrow around and multiply by the integer in the front; then continue to follow the dashed arrow, and add the numerator.

In other words, do this: \(3 \times 4 + 2 = 14\). This calculation gives you the numerator of the improper fraction.

Once we have the numerator, we must write the improper fraction using the same denominator as the mixed number, which gives us the answer:

\[4\frac{2}{3} = \frac{14}{3}\]

It is important to know that these are really just two different ways of writing the same number!

To convert an improper fraction to a mixed number, divide the numerator by the denominator.

Remember that the fraction bar is also a division symbol, so this fraction is a division problem:

\[\frac{5}{2} = 5 \div 2\]

Start by finding out how many times \(2\) fits into \(5\): \(2 \times 2 = 4\), which is \(1\) less than \(5\). (In other words, when you divide \(5\) by \(2\), there is an “extra” \(1\) left over that did not come out of the \(5\).)

When we put it all together it comes out like this:

\[\frac{5}{2} = 2\frac{1}{2}\]

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Fractions on a number line

We can locate fractions on a number line.

• All proper fractions are placed between \(0\) and \(1\) (or \(0\) and \(−1\) in the case of negative fractions).

In the diagram below, the dot shows where we reach \(7\) parts out of \(10\) pieces, which is the fraction \(\frac{7}{10}\).

In the diagram below, the dot shows where we reach \(3\) parts out of \(5\) pieces, which is the fraction \(\frac{3}{5}\).

Placing fractions on a number line helps when we compare fractions.

For example, to decide whether \(\frac{3}{5}\) is less than \(\frac{1}{2}\), greater than \(\frac{1}{2}\) or equal to \(\frac{1}{2}\), you can place these two fractions on the number line.

The diagram below shows the position of \(\frac{1}{2}\) on the number line. You can see that \(\frac{3}{5}\) is more (greater) than \(\frac{1}{2}\) (therefore \(\frac{3}{5} > \frac{1}{2}\)).

• Improper and mixed numbers (mixed fractions) are placed between two whole numbers. The integer part of the mixed number shows the number after which the fraction is placed on the number line.

The position of \(\frac{25}{9} = 2\frac{7}{9}\) is shown on the number line below.

The position of \(4\frac{3}{10}\) is shown on the number line below.

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