# Fractions

## Fractions Revision

**Fractions**

**Fractions** is one of the most fundamental topics in maths as it feeds into many other areas, so having a concrete understating of the basics is important. There **8** key skills that you need to learn for fractions.

Make sure you are happy with the following topics before continuing.

**Skill 1: Simplifying Fractions**

To **simplify** a fraction, we divide the numerator (the top of the fraction) and the denominator (the bottom of the fraction) by the same amount, until we can’t simplify anymore.

**Note: **Simplifying fractions doesn’t change the value of the fraction

**Example: **Write \dfrac{12}{30} in its simplest form.

\textcolor{red}{12} and \textcolor{red}{30} both contain 6 as a factor, meaning we can divide both by 6.

\dfrac{12}{30}= \dfrac{12\div6}{30\div6} = \textcolor{black}{\dfrac{2}{5}}

2 and 5 are both prime numbers so cannot simplified anymore.

**Skill 2: Converting between Mixed and Improper Fractions**

**Mixed fractions** are numbers with an integer part and a fraction part, such as 6 \dfrac{1}{4}.

**Improper fractions** are fractions where the **numerator (top)** is bigger than the **denominator (bottom)**, such as \dfrac{9}{2}. We also call these **top-heavy fractions** for this reason.

We need to be able to convert between these two types.

**Example:** Write \textcolor{limegreen}{3} \textcolor{blue}{\dfrac{3}{4}} as an improper fraction.

We can write

\textcolor{limegreen}{3} \textcolor{blue}{\dfrac{3}{4}} = \textcolor{limegreen}{3} + \textcolor{blue}{\dfrac{3}{4}}

Then

3 + \dfrac{3}{4} = \dfrac{12}{4} + \dfrac{3}{4} = \textcolor{orange}{\dfrac{15}{4}}

**Skill 3: Adding Fractions**

When adding fractions you need to make sure you have a **common denominator**. Then we add the two fractions together by adding the numerators together.

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

\dfrac{3}{5}=\dfrac{3\times 4}{5\times 4}=\dfrac{12}{20}

\dfrac{1}{4}=\dfrac{1\times5}{4\times 5}=\dfrac{5}{20}

Now, to add two fractions with the same denominator, simply add the numerators together. Doing so, we get

\dfrac{3}{5} + \dfrac{1}{4}=\dfrac{12}{20} + \dfrac{5}{20}= \textcolor{black}{\dfrac{17}{20}}

**Skill 4: Subtracting Fractions**

When subtracting fractions you need to make sure you have a **common denominator**. Then we subtract one fraction from the other by subtracting one numerator from the other.

**Example:** Solve \dfrac{4}{5} - \dfrac{1}{2}

\dfrac{4}{5}=\dfrac{4\times2}{5\times 2}=\dfrac{8}{10}

\dfrac{1}{2}=\dfrac{1\times 5}{2\times 5}=\dfrac{5}{10}

Now, to subtract two fractions with the same denominator, simply subtract the second numerator from the first. Doing so, we get

\dfrac{4}{5} - \dfrac{1}{2}=\dfrac{8}{10} - \dfrac{5}{10}= \textcolor{black}{\dfrac{3}{10}}

**Skill 5: Multiplying Fractions**

To multiply fractions, we simply multiply the numerators together and multiply the denominators together.

**Example:** \dfrac{1}{5} \times \dfrac{2}{3}

\dfrac{1\times2}{5\times 3} = \textcolor{black}{\dfrac{2}{15}}

**Skill 6: Dividing Fractions**

For dividing fractions, remember the rule: **Keep, Change, Flip**.

This means, you must keep the first fraction as it is, change the division sign into a multiplication, and flip the second fraction. You then just work out the multiplication as normal.

**Example: **Work out \dfrac{1}{2} \div \dfrac{5}{9}. Write your answer in its simplest form.

We **keep** the first fraction the same, **change** the symbol to a multiplication, and **flip** the second fraction.

\dfrac{1}{2} \div \dfrac{5}{9} = \textcolor{Orange}{\dfrac{1}{2}} \textcolor{red}{\times} \textcolor{blue}{\dfrac{9}{5}}

Now, doing the multiplication we get

\dfrac{1}{2} \times \dfrac{9}{5} = \dfrac{1 \times 9}{2 \times 5} = \textcolor{black}{\dfrac{9}{10}}

**Skill 7: Calculations involving Mixed Fractions**

When doing calculations involving mixed fractions, it is easier to convert them to improper fractions first. We then perform the calculation using the above skills.

**Example**: Calculate \textcolor{red}{2 \dfrac{1}{4}} \times \textcolor{blue}{3 \dfrac{1}{2}}.

\textcolor{red}{2 \dfrac{1}{4}} = \dfrac{9}{4} \, and \, \textcolor{blue}{3 \dfrac{1}{2}} = \dfrac{7}{2}

Then

2 \dfrac{1}{4} \times 3 \dfrac{1}{2} = \dfrac{9}{4} \times \dfrac{7}{2} = \dfrac{9 \times 7}{4 \times 2} = \textcolor{black}{\dfrac{63}{8}}

We cannot simplify it any further, so our answer is in its simplest form.

**Skill 8: Fractions of Amounts**

To work out a fraction of an amount, for example money, we divide the amount by the denominator (bottom) and then multiply it by the numerator (top) or vice versa

**Example:** Calculate \dfrac{\textcolor{limegreen}{3}}{\textcolor{red}{5}} of \textcolor{blue}{\$ 170}.

Divide \textcolor{blue}{\$ 170} by \textcolor{red}{5} first and then multiply by \textcolor{limegreen}{3}:

\begin{aligned} \dfrac{3}{5} \text{ of } \textcolor{blue}{\$ 170} &= (\textcolor{blue}{\$ 170} \div \textcolor{red}{5}) \times \textcolor{limegreen}{3} \\ &= \$ 34 \times 3 \\ & = \textcolor{black}{\$ 102} \end{aligned}

## Fractions Example Questions

**Question 1:** Work out \dfrac{6}{13} \times \dfrac{4}{3}

Give your answer in its simplest form.

**[2 marks]**

When multiplying fractions, we multiply across the numerator (top) and multiply across the denominator (bottom),

\dfrac{6}{13} \times \dfrac{4}{3} = \dfrac{6 \times 4}{3 \times 13} = \dfrac{24}{39}

Identifying a common factor of 3, the answer simplifies to,

\dfrac{24}{39} = \dfrac{8}{13}

**Question 2:** Work out \dfrac{7}{10} - \dfrac{8}{3}

Give your answer in its simplest form.

**[2 marks]**

To subtract fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the first fraction by 3, and then multiplying the top and bottom of the second fraction by 10. Thus,

\dfrac{7}{10} - \dfrac{8}{3} = \dfrac{21}{30} - \dfrac{80}{30} = -\dfrac{59}{30}

**Question 3:** Work out \dfrac{9}{11} \div \dfrac{6}{7}

Give your answer in its simplest form.

**[2 marks]**

To divide fractions, we need to **Keep, Change, and Flip**.

Changing the division sign to a multiplication, and flipping the second fraction we get,

\dfrac{9}{11} \div \dfrac{6}{7} = \dfrac{9}{11} \times \dfrac{7}{6}

Hence,

\dfrac{9}{11} \times \dfrac{7}{6} = \dfrac{9 \times 7}{11 \times 6} = \dfrac{63}{66}

Both top and bottom have a factor of 3 which we can cancel, leaving us with

\dfrac{63}{66} = \dfrac{21}{22}

**Question 4:** Work out \dfrac{5}{4} \times \dfrac{2}{3}

Give your answer in its simplest form.

**[2 marks]**

When multiplying fractions, we multiply across the numerator (top) and multiply across the denominator (bottom),

\dfrac{5}{4} \times \dfrac{2}{3} = \dfrac{5 \times 2}{4 \times 3} = \dfrac{10}{12}

Identifying a common factor of 2, the answer simplifies to,

\dfrac{10}{12} = \dfrac{5}{6}

**Question 5:** Work out 12\dfrac{1}{2} \div \dfrac{5}{8}

Give your answer in its simplest form.

**[2 marks]**

To divide fractions, we need to **Keep, Change, and Flip**.

First we have to convert the mixed fraction to an improper fraction,

12\dfrac{1}{2} =\dfrac{25}{2}

Then changing the division sign to a multiplication, and flipping the second fraction we get,

\dfrac{25}{2} \div \dfrac{5}{8} = \dfrac{25}{2} \times \dfrac{8}{5}= \dfrac{25 \times 8}{2 \times 5} = \dfrac{200}{10}= 20