An (arithmetic) **fraction** is an expression of the form $\dfrac{p}{q}$ where $p$ and $q$ are integers or algebraic expressions (see Algebraic Fractions). One way of thinking about a fraction is as a division that hasn't been done yet. The fraction $\dfrac{p}{q}$ is another way of writing $p\div q$.

We use fractions to express parts of a whole. For example, the fraction $\frac{3}{4}$ represents $3$ parts out of $4$.

We call $p$ the **numerator** and $q$ the **denominator**.

Fractions of the form $\dfrac{a}{b}$, where $a$ and $b \neq 0$ make up the set of rational numbers $\mathbb{Q}$. Such fractions are also called **arithmetic fractions**.

Arithmetic fractions can be *evaluated* by dividing $a$ by $b$. For example, evaluating $\frac{1}{2}$ gives $0.5$.

**Examples**: $\dfrac{3}{4},\;\;\;\dfrac{7}{225},\;\;\;\dfrac{1}{100000},\;\;\;-\dfrac{500}{2300}$

**Note**: Arithmetic fractions are closely related to decimals, ratios and percentages.

A fraction remains the same if we multiply both the numerator and denominator by the same non-zero number.

**Examples**:

\begin{align} \dfrac{1}{4}&= \dfrac{1 \times 2}{4\times 2} = \dfrac{2}{8}\\ \dfrac{2}{3}&=\dfrac{2 \times (-10)}{3\times (-10)} = \dfrac{20}{30}\\ \end{align}

We can **simplify** a fraction by cancelling all common factors of the numerator and the denominator. A fraction which has been simplified (the numerator and denominator have no more common factors) is said to be written in its **simplest form** or **lowest form**.

Arithmetic with fractions is typically easier if the fractions are in their simplest form.

Simplify: $\dfrac{24}{36}$

$24$ and $36$ have a factor of $4$ in common so we can cancel $4$ from the top and bottom: \[\dfrac{24 \div 4}{36\div 4} = \dfrac{6}{9}\] Since $6$ and $9$ have a factor of $3$ in common, we can cancel $3$ from the top and bottom: \[\dfrac{6\div3}{9\div3}=\dfrac{2}{3}\] Since $2$ and $3$ do not have any common factors, this fraction is now in its simplest form.

**This is the only way that you can simplify fractions.**

If two or more fractions have the same denominator, they are said to have a **common denominator**.

Adding and subtracting fractions is straightforward if they all have a common denominator: we just add or subtract the numerators.

**Example**: $\dfrac{1}{5} + \dfrac{3}{5} = \dfrac{4}{5}$

If the denominators are different, just adding or subtracting the numerators together doesn't make sense - what should the denominator of the result be? The solution is to restate both fractions over a common denominator before doing the addition (or subtraction). This common denominator is equal to the product of the denominators of the fractions we are adding (or subtracting). For example, suppose we want to calculate: \[\frac{3}{5}+\frac{7}{10}\] The common denominator will be $5\times 10=50$. To restate the first fraction over this denominator, we must multiply both the numerator and denominator by $10$: \[\frac{3\times 10}{5 \times 10}=\frac{30}{50}\] To restate the second fraction over this denominator, we must multiply both the numerator and denominator by $5$: \[\frac{7\times 5}{10 \times 5}=\frac{35}{50}\] Since the denominators of the two fractions are now the same ($50$), we can just add the numerators: \[\frac{30}{50}+\frac{35}{50}=\frac{65}{50}\] The final step is to write this fraction in its simplest form by cancelling all common factors in the numerator and denominator (if there are any). The highest common factor of $65$ and $50$ is $5$, so: \[\frac{65}{50}=\frac{65 \div 5}{50\div 5}=\frac{13}{10}\] Thus \[\frac{3}{5}+\frac{7}{10}=\frac{13}{10}\]

In **summary**:

1) Multiply the denominators to get a common denominator.

2) Restate each fraction over the common denominator.

3) Add (or subtract) the numerators so that you now have one fraction.

4) Simplify the fraction if required.

Calculate $\dfrac{3}{5}+\dfrac{1}{4}$.

\begin{align} \frac{3}{5}+\frac{1}{4} &= \frac{3\times4}{4 \times 5} + \frac{1 \times 5}{5\times4}\\ &= \frac{12}{20} + \frac{5}{20} \\ &=\frac{12+5}{20}\\ &=\frac{17}{20} \end{align}

Calculate $\dfrac{5}{6}-\dfrac{2}{3}$.

\begin{align} \frac{5}{6}-\frac{2}{3}&= \frac{5\times 3}{6\times 3} - \frac{2\times 6}{3 \times 6} \\ &= \frac{15}{18} - \frac{12}{18} \\ &=\frac{15-12}{18}\\ &=\frac{3}{18}\\ &=\frac{1}{6} \end{align}

The rule for multiplying fractions is multiply the numerators together and multiply the denominators together:

\[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]

Calculate $\dfrac{3}{7} \times \dfrac{5}{3}$.

\begin{align} \frac{3}{7} \times \frac{5}{3} &= \frac{3\times5}{7\times3}\\ &= \frac{15}{21}\\ &= \frac{5}{7} \end{align}

The rule for dividing fractions is: \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.\]

This is easily remembered as “flip the second fraction, then multiply”.

Sometimes this may be written as a fraction on top of a fraction, but the rule is the same: \[\frac{a/b}{c/d}=\frac{a}{b}\times\frac{d}{c}\text{.}\]

Calculate $\dfrac{1}{3}\div\dfrac{1}{4} $.

\begin{align} \frac{1}{3}\div\frac{1}{4} &=\frac{1}{3}\times\frac{4}{1}\\\\ &=\frac{1\times4}{3\times1}\\\\ &=\frac{4}{3}. \end{align}

Calculate $\dfrac{3}{5} \div \dfrac{2}{3}$.

\begin{align} \frac{3}{5}\div\frac{2}{3} &=\frac{3}{5}\times\frac{3}{2}\\\\ &=\frac{3\times3}{5\times2}\\\\ &=\frac{9}{10}. \end{align}

Simplify the fraction $\dfrac{2/3}{2-5/7}$.

Start by changing the fraction to a division symbol to see both parts side-by-side. \[\frac{2}{3} \div \left(2-\frac{5}{7}\right)\] Subtract $\frac{5}{7}$ from $2$ by putting them over a common denominator. \begin{align} 2 - \frac{5}{7} &= 2 \times \frac{7}{7} - \frac{5}{7} \\ &= \frac{14}{7} - \frac{5}{7} \\ &= \frac{14 - 5}{7} \\ &= \frac{9}{7}. \end{align} Now, we have: \[\frac{2}{3} \div \frac{9}{7}\] Use the rule for dividing fractions. \begin{align} \frac{2}{3} \div \frac{9}{7} &= \frac{2}{3} \times \frac{7}{9} \\ \\ &= \frac{2 \times 7}{3 \times 9} \\ \\ &= \frac{14}{27}. \end{align}

Prof. Robin Johnson computes $\dfrac{2}{3} + \dfrac{4}{5}$ and $\dfrac{5}{6} + \dfrac{1}{4} - \dfrac{7}{30}$.

Prof. Robin Johnson calculates $2 \div \dfrac{1}{4}$ and $3 \div \dfrac{2}{5}$.

Prof. Robin Johnson simplifies the fraction $\dfrac{3/4}{2-4/7}$.

Test yourself: Numbas test on fractions

Test yourself: Another Numbas test on fractions

- Fractions videos, worksheets and quizzes at BBC Skillswise.
- Various resources on fractions by mathcentre