Fractions | Worksheets Questions Revision | MME

Fractions Worksheets, Questions and Revision

Level 1-3

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.

KS3 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.

KS3 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}}

KS3 Level 1-3
KS3 Level 1-3

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}}

KS3 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}}

KS3 Level 1-3
KS3 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}}

KS3 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}}

KS3 Level 1-3
KS3 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.

KS3 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}

KS3 Level 1-3

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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

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