 Fractions | Worksheets Questions Revision | MME

## 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.

Level 1-3

## 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.

Level 1-3

## 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 $3\dfrac{3}{4}$ as an improper fraction.

$3 \dfrac{3}{4} = 3 + \dfrac{3}{4} = \dfrac{12}{4} + \dfrac{3}{4} = \textcolor{black}{\dfrac{15}{4}}$

Level 1-3

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

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

Level 1-3

## 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

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

Level 1-3

## Skill 5: Multiplying Fractions

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

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

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

Level 1-3

## 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

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

Level 1-3

## 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.

Level 1-3

## 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}

Level 1-3

### 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,

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

### Worksheets and Exam Questions

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

Level 1-4 New Official MME

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