Fractions | Worksheets Questions Revision | MME

# Fractions Worksheets, Questions and Revision

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## 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:

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.

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, changethe 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.

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

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

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

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,

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

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,

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

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

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

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,

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

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,

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

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