## What you need to know

### Fractions

Fractions is one of the most fundamental topics in maths as it feeds into so many other areas so having a concrete understating of the following is key:

- Multiplying fractions
- Dividing fractions
- Adding and subtracting fractions
- Equivalent fractions
- Fraction, decimal and percentage conversions
- Recurring decimals and fractions
- Algebraic fractions

On this page we will look at fraction basics that link into all of the other fraction topics mentioned above. One of the keys to understanding this topic is simplifying fractions as that is a skill that is required in every related topic.

**Simplifying Fractions**

When simplifying fractions, the aim is to make the numbers on the numerator (the top of the fraction) and the denominator (the bottom of the fraction) smaller, without actually changing the value of the fraction. To do this, we have to be aware of the all-important rule: if you __multiply/divide the top and bottom of a fraction by the same number__, the value of the fraction is __unchanged__.

So, when simplifying fractions, we are going to be looking for common factors in the top and bottom to give us some number we can divide them both by. In other words, we will cancelling down the fraction.

### Adding & Subtracting Fractions

To add fractions we are going to use the rule mentioned above – we’re going to be multiplying the top and bottom of a fractions by the same thing in order to make the fractions have the same denominator – a common denominator.

Example: \dfrac{3}\textcolor{green}{{5}} + \dfrac{1}\textcolor{red}{{4}}.

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

\dfrac{1}{4}=\dfrac{1\times \textcolor{green}{5}}{4\times \textcolor{green}{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}=\dfrac{17}{20}

See the adding and subtracting fractions page for more examples.

### Multiplying Fractions

Multiplying fractions we simply multiply the numerators together and multiply the denominators.

Example: \dfrac{1}\textcolor{green}{{5}} \times \dfrac{2}\textcolor{red}{{3}}.

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

You can revise multiplying fractions here.

**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. Another way to remember how to divide fractions is KFC, the same meaning but the letters in a different order and more memorable to most, especially if you like fried chicken.

For more dividing fractions questions and revision visit the dedicated page.

### Example 1: Simplifying Fractions

Write \dfrac\textcolor{red}{{12}}\textcolor{green}{{30}} in its simplest form.

Okay, so immediately we can see that both 12 and 30 are even numbers, meaning they both have a factor of 2. Therefore, we get

\dfrac\textcolor{red}{{12}}\textcolor{green}{{30}}=\dfrac\textcolor{red}{{6}}\textcolor{green}{{15}}

Are there more common factors? Yes, both 6 and 15 are multiplies of 3. Therefore, we get

\dfrac\textcolor{red}{{6}}\textcolor{green}{{15}} = \dfrac\textcolor{red}{{2}}\textcolor{green}{{5}}

This time, there are no more common factors so we have fully simplified the fraction.

### Example 2: Adding and Subtracting Fractions

Work out \dfrac{1}{6} + \dfrac{3}{4}. Write your answer in its simplest form.

So, we want both denominators to be the same. The easiest way to choose what this denominator will be is to use the product of both denominators, here: 6 \times 4 = 24.

\dfrac{1}{6} = \dfrac{1 \times 4}{6 \times 4} = \dfrac{4}{24}

To make the bottom of the second fraction 24 it must be multiplied by 6, so we must also multiply the top by 6.

\dfrac{3}{4} = \dfrac{3 \times 6}{4 \times 6} = \dfrac{18}{24}

Now, just add the numerators.

\dfrac{4}{24} + \dfrac{18}{24} = \dfrac{22}{24}

Now all that remains is to simplify it. Cancelling out a factor of 2, it becomes \frac{11}{12}, which cannot be simplified further.

### Example 3: Multiplying Fractions

Work out \dfrac{4}{5} \times \dfrac{7}{12}. Write your answer in its simplest form.

Multiply across the top and across the bottom. Thus, our calculation looks like

\dfrac{4}{5} \times \dfrac{7}{12} = \dfrac{4 \times 7}{5 \times 12} = \dfrac{28}{60}

Now, we must simplify our answer. Both top and bottom have a factor of 4, so cancelling the 4 we get:

\dfrac{7}{15}

### Example 4: Dividing Fractions

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

We’re going to keep the first fraction the same, change the symbol to a multiplication, and flip the second fraction. This leaves us with.

\dfrac{1}{2} \div \dfrac{5}{9} = \dfrac{1}{2} \times \dfrac{9}{5}

Now, doing the multiplication we get

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

### Example Questions

1) Work out \dfrac{6}{13} \times \dfrac{4}{3}

Give your answer in its simplest form.

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}

2) Work out \dfrac{7}{10} - \dfrac{8}{3}

Give your answer in its simplest form.

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}

3) Work out \dfrac{9}{11} \div \dfrac{6}{7}

Give your answer in its simplest form.

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}

4) Work out \dfrac{5}{4} \times \dfrac{2}{3}

Give your answer in its simplest form.

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}

5) Work out 12\dfrac{1}{2} \div \dfrac{5}{8}

Give your answer in its simplest form.

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}=10

### Worksheets and Exam Questions

#### Fractions (Basics) - Exam Style Questions - MME

Level 1-4#### Fractions - Drill Questions

Level 1-4#### Fractions (Adding, Subtracting, Multiplying ,Dividing) - Drill Questions

Level 1-4#### Fractions Decimals & Percentages - Drill Questions

Level 1-4#### Fractions of amounts and percentages - Drill Questions

Level 1-4#### Fractions Of Amounts 2 - Drill Questions

Level 1-4### Videos

#### Fractions Grades 1-4 Q1

GCSE MATHS#### Fractions Grades 1-4 Q2

GCSE MATHS#### Fractions Grades 1-4 Q3

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