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ArithmeticSimplifying fractions

Equivalent fractions Mixed number to improper fraction Reciprocal mathsThis topic is relevant for:

Here we will learn about multiplying and dividing fractions, including calculations involving whole numbers, proper and improper fractions, and mixed numbers.

There are also multiplying and dividing fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Multiplying and dividing fractions** is the skill of carrying out a calculation involving multiplication and division where one or more of the values is written as a fraction.

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

For example,

\frac{2}{5} \times \frac{3}{4}=\frac{2 \times 3}{5 \times 4}=\frac{6}{20}=\frac{3}{10}To **divide fractions**, we first calculate the **reciprocal** of the **dividing fraction** and then multiply the numerators together, and multiply the denominators together.

For example,

\frac{2}{5} \div \frac{3}{4}=\frac{2}{5} \times \frac{4}{3}=\frac{2 \times 4}{5 \times 3}=\frac{8}{15}Note that, unlike when we add and subtract fractions, we **do not need a common denominator to multiply and divide fractions.**

When we divide a quantity by a fraction, this is the same as multiplying by the reciprocal of this fraction. Let’s have a look at this diagrammatically.

For example, here is a circle, which has a value of 1.

Let’s divide this circle into halves (\div\frac{1}{2}).

Here we can see that the circle has been split into two halves.

This means that 1\div\frac{1}{2} is equal to 2.

Let’s look at another example, here are four squares.

If we wanted to divide these squares into two fifths (\div\frac{2}{5}), we would get,

If we total the number of two fifths in 4, we get 10.

This means that 4\div\frac{2}{5} is equal to 10.

How could we rearrange this division to become a multiplication instead?

Using 4\div\frac{2}{5}=10, as we are dividing 4 by two fifths, let’s multiply both sides by \frac{2}{5},

4=10\times\frac{2}{5}.When multiplying a quantity by a fraction, we multiply the numerator and divide by the denominator and so if we split the fraction into these two operations, we have

4=10\times{2}\div{5}.Rearranging this by using inverse operations, we need to multiply both sides by 5.

4\times{5}=10\times{2}And then divide both sides by 2.

4\times{5}\div{2}=10As 5\div{2} is the same as \frac{5}{2}, here we are now multiplying 4 by \frac{5}{2} or specifically,

4\times\frac{5}{2}=10.Notice what has happened to the fraction. The fraction is now “upside down” meaning that we have found the **reciprocal** of the fraction.

The division has also changed now to be a multiplication.

This means that 4\div\frac{2}{5} gives us exactly the same result as 4\times\frac{5}{2}.

In general, any quantity divided by a fraction is identical to the same quantity being multiplied by the reciprocal of that fraction,

\frac{a}{b}\div\frac{c}{d}\equiv\frac{a}{b}\times\frac{d}{c}In order to multiply with fractions:

**Convert any integer or mixed number to an improper fraction.****Multiply the numerators together and multiply the denominators together.****Simplify the fraction if possible.**

Get your free multiplying and dividing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free multiplying and dividing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEDetermine the value of \ 3\times\frac{5}{7}.

**Convert any integer or mixed number to an improper fraction.**

The integer 3 can be written as the fraction \frac{3}{1}.

We now have the question \frac{3}{1}\times\frac{5}{7}.

2**Multiply the numerators together and multiply the denominators together.**

3**Simplify the fraction if possible.**

This is an improper fraction which we should convert to a mixed number.

15\div 7 = 2 \ (remainder 1)

Therefore the mixed number has 2 as a whole number, and 1 remains to be divided by 7, giving 2\frac{1}{7}.

The solution is 2\frac{1}{7}.

Determine the value of \ \frac{2}{5}\times\frac{3}{5}.

**Convert any integer or mixed number to an improper fraction.**

Both of the numbers in the question are fractions and so we do not need to alter either fraction.

**Multiply the numerators together and multiply the denominators together.**

\frac{2}{5}\times\frac{3}{5}=\frac{2\times3}{5\times5}=\frac{6}{25}

**Simplify the fraction if possible.**

\frac{6}{25}

6 and 25 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is \frac{6}{25}.

Determine the value of \ \frac{5}{8}\times\frac{2}{3}\times\frac{1}{6}.

**Convert any integer or mixed number to an improper fraction.**

All of the numbers in the question are fractions and so we do not need to alter any of them.

**Multiply the numerators together and multiply the denominators together.**

\frac{5}{8}\times\frac{2}{3}\times\frac{1}{6}=\frac{5\times2\times1}{8\times3\times6}=\frac{10}{144}

**Simplify the fraction if possible.**

\frac{10}{144}

10 and 144 share a common factor of 2. Therefore we can divide the numerator and denominator by 2 in order to simplify the fraction.

\frac{10}{144}=\frac{10\div 2}{144\div 2}=\frac{5}{72}

The solution is \frac{5}{72}.

Determine the value of \frac{1}{6}\times2\frac{1}{4}.

**Convert any integer or mixed number to an improper fraction.**

The mixed number 2\frac{1}{4} needs to be converted to an improper fraction before we multiply the two numbers.

2\frac{1}{4}=\frac{(4\times{2})+1}{4}=\frac{8+1}{4}=\frac{9}{4}

We now have the question \frac{1}{6}\times\frac{9}{4}.

**Multiply the numerators together and multiply the denominators together.**

\frac{1}{6}\times\frac{9}{4}=\frac{1\times9}{6\times4}=\frac{9}{24}

**Simplify the fraction if possible.**

\frac{9}{24}

9 and 24 share a common factor of 3. Therefore we can divide the numerator and denominator by 3 in order to simplify the fraction.

\frac{9}{24}=\frac{9\div 3}{24\div 3}=\frac{3}{8}

The solution is \frac{3}{8}.

In order to divide with fractions:

**Convert any integer or mixed number to an improper fraction.****Find the reciprocal of the dividing fraction and rewrite the calculation with multiplication instead of division (remember ‘keep, change, flip’).****Multiply the numerators together and multiply the denominators together.****Simplify the fraction if possible.**

Determine the value of \ 2\div\frac{3}{4}.

**Convert any integer or mixed number to an improper fraction.**

The integer 2 can be written as the fraction \frac{2}{1}.

We now have the question \frac{2}{1}\div\frac{3}{4}.

**Find the reciprocal of the dividing fraction and rewrite the calculation with multiplication instead of division (remember ‘keep, change, flip’).**

The reciprocal of \frac{3}{4} is \frac{4}{3}.

Dividing by \frac{3}{4} is the same as multiplying by \frac{4}{3} and so we now have the question \frac{2}{1}\times\frac{4}{3}.

**Multiply the numerators together and multiply the denominators together.**

\frac{2}{1}\times\frac{4}{3}=\frac{2\times4}{1\times3}=\frac{8}{3}

**Simplify the fraction if possible.**

\frac{8}{3}

This is an improper fraction which we should convert to a mixed number.

8\div 3 = 2 \ ( remainder 2)

Therefore the mixed number has 2 as a whole number, and 2 remains to be divided by 3, giving 2\frac{2}{3}.

The solution is 2\frac{2}{3}.

Determine the value of \frac{1}{10}\div\frac{7}{10}.

**Convert any integer or mixed number to an improper fraction.**

Both of the numbers in the question are fractions and so we do not need to alter either fraction.

The reciprocal of \frac{7}{10} is \frac{10}{7}.

Dividing by \frac{7}{10} is the same as multiplying by \frac{10}{7} and so we now have the question \frac{1}{10}\times\frac{10}{7}.

**Multiply the numerators together and multiply the denominators together.**

\frac{1}{10}\times\frac{10}{7}=\frac{1\times10}{10\times7}=\frac{10}{70}

**Simplify the fraction if possible.**

\frac{10}{70}

10 and 70 share a common factor of 10. Therefore we can divide the numerator and denominator by 10 in order to simplify the fraction.

\frac{10}{70}=\frac{1}{7}

The solution is \frac{1}{7}.

**Note**: The fractions in the question both had the same denominator of 10. The final answer then simplified by a factor of 10. If you spot this earlier you can save time by simplifying during the calculation.

\frac{1}{10}\div\frac{7}{10}=\frac{1}{10}\times\frac{10}{7}=\frac{1\times 10}{10\times 7}=\frac{1\times 10}{10\times 7}=\frac{1}{7}

Determine the value of \ \frac{3}{4}\div\frac{9}{10}.

**Convert any integer or mixed number to an improper fraction.**

Both of the numbers in the question are fractions and so we do not need to alter either fraction.

The reciprocal of \frac{9}{10} is \frac{10}{9}.

Dividing by \frac{9}{10} is the same as multiplying by \frac{10}{9} and so we now have the question \frac{3}{4}\times\frac{10}{9}.

**Multiply the numerators together and multiply the denominators together.**

\frac{3}{4}\times\frac{10}{9}=\frac{3\times10}{4\times9}=\frac{30}{36}

**Simplify the fraction if possible.**

\frac{30}{36}

30 and 36 share a common factor of 6. Therefore we can divide the numerator and denominator by 6 in order to simplify the fraction.

\frac{30}{36}=\frac{30\div 6}{36\div 6}=\frac{5}{6}

The solution is \frac{5}{6}.

Determine the value of \ \frac{2}{3}\div{3}\frac{1}{5}.

**Convert any integer or mixed number to an improper fraction.**

The mixed number 3\frac{1}{5} needs to be converted to an improper fraction before we can carry out the calculation.

3\frac{1}{5}=\frac{(5\times{3})+1}{5}=\frac{15+1}{5}=\frac{16}{5}

We now have the question \frac{2}{3}\div\frac{16}{5}.

The reciprocal of \frac{16}{5} is \frac{5}{16}.

Dividing by \frac{16}{5} is the same as multiplying by \frac{5}{16} and so we now have the question \frac{2}{3}\times\frac{5}{16}.

**Multiply the numerators together and multiply the denominators together.**

\frac{2}{3}\times\frac{5}{16}=\frac{2\times5}{3\times16}=\frac{10}{48}

**Simplify the fraction if possible.**

\frac{10}{48}

10 and 48 share a common factor of 2. Therefore we can divide the numerator and denominator by 2 in order to simplify the fraction.

\frac{10}{48}=\frac{10\div 2}{48\div 2}=\frac{5}{24}

The solution is \frac{5}{24}.

Determine the value of \ \frac{3}{7}\div 5.

**Convert any integer or mixed number to an improper fraction.**

The integer 5 can be written as the fraction \frac{5}{1}.

We now have the question \frac{3}{7}\div\frac{5}{1}.

The reciprocal of \frac{5}{1} is \frac{1}{5}.

Dividing by \frac{5}{1} is the same as multiplying by \frac{1}{5} and so we now have the question \frac{3}{7}\times\frac{1}{5}.

**Multiply the numerators together and multiply the denominators together.**

\frac{3}{7}\times\frac{1}{5}=\frac{3\times1}{7\times5}=\frac{3}{35}

**Simplify the fraction if possible.**

\frac{3}{35}

3 and 35 do not share any common factors except 1 and so the fraction is already in its simplest form.

The solution is \frac{3}{35}.

**Believing that there needs to be a common denominator in order to multiply or divide with fractions**

When we add or subtract fractions there must be a common denominator. However, this is not a requirement when we multiply and divide with fractions.

When we multiply/divide with fractions they can have the same denominators or different denominators.

**Forgetting to write integers as fractions and then using a mathematically incorrect procedure**

Sometime students make the following sort of mistake,

3\times\frac{2}{5}=\frac{3\times2}{3\times5}=\frac{6}{15}.This is incorrect. If you remember to write the integer as an improper fraction first then this mistake can be avoided. The correct method is,

3\times\frac{2}{5}=\frac{3}{1}\times\frac{2}{5}=\frac{3\times2}{1\times5}=\frac{6}{5}=1\frac{1}{5}.**Forgetting to write mixed numbers as improper fractions first and then using a mathematically incorrect procedure**

Sometime students make the following sort of mistake,

2\frac{1}{3}\times\frac{4}{5}=2\frac{1\times4}{3\times5}=2\frac{4}{15}.This is incorrect. If you remember to write the mixed number as an improper fraction first then this mistake can be avoided. The correct method is,

2\frac{1}{3}\times\frac{4}{5}=\frac{7}{3}\times\frac{4}{5}=\frac{7\times4}{3\times5}=\frac{28}{15}=1\frac{13}{15}.**When dividing with fractions, finding the reciprocal of the wrong fraction**

Dividing by a fraction is the same as multiplying by the reciprocal of that fraction. For example,

\div\frac{3}{8} is the same as \times\frac{8}{3}.

However, sometimes students can get confused and find the reciprocal of the first fraction that is written instead of the reciprocal of the dividing fraction. They might write something like this,

\frac{2}{5}\div\frac{3}{8}=\frac{5}{2}\times\frac{3}{8}=\frac{5\times3}{2\times8}=\frac{15}{16}.This is incorrect. You must find the reciprocal of the dividing fraction. A simple way to remember this is by using the mnemonic **‘keep, change, flip’**.

**Keep** the first number/fraction the same. **Change** the division sign to a multiplication sign. **Flip** the second fraction upside down’. The correct method is,

Multiplying and dividing fractions is part of our series of lessons to support revision on fractions. You may find it helpful to start with the main fractions lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. Determine the value of 2\times\frac{8}{11}.

\frac{16}{22}

\frac{8}{22}

1\frac{5}{11}

2\frac{8}{11}

2\times\frac{8}{11}=\frac{2}{1}\times\frac{8}{11}=\frac{2\times{8}}{1\times{11}}=\frac{16}{11}=1\frac{5}{11}

2. Determine the value of \frac{3}{8}\times\frac{7}{8}.

\frac{21}{64}

\frac{24}{56}

3\frac{1}{21}

2\frac{5}{8}

\frac{3}{8}\times\frac{7}{8}=\frac{3\times{7}}{8\times{8}}=\frac{21}{64}

3. Determine the value of \frac{3}{13}\times\frac{7}{10}.

\frac{30}{91}

3\frac{1}{30}

\frac{121}{130}

\frac{21}{130}

\frac{3}{13}\times\frac{7}{10}=\frac{3\times{7}}{13\times{10}}=\frac{21}{130}

4. Determine the value of 4\frac{1}{3}\times\frac{2}{5}.

4\frac{2}{15}

5\frac{1}{5}

1\frac{11}{15}

\frac{2}{15}

\begin{aligned}
4\frac{1}{3}\times\frac{2}{5}&=\frac{(4\times{3})+1}{3}\times\frac{2}{5}\\\\
&=\frac{12+1}{3}\times\frac{2}{5}\\\\
&=\frac{13}{3}\times\frac{2}{5}\\\\
&=\frac{13\times{2}}{3\times{5}}\\\\
&=\frac{26}{15}\\\\
&=1\frac{11}{15}
\end{aligned}

5. Determine the value of 5\div\frac{3}{8}.

13\frac{1}{3}

1\frac{7}{8}

\frac{3}{40}

3\frac{1}{3}

5\div\frac{3}{8}=5\times\frac{8}{3}=\frac{5}{1}\times\frac{8}{3}=\frac{5\times{8}}{1\times{3}}=\frac{40}{3}=13\frac{1}{3}

6. Determine the value of \frac{5}{9}\div\frac{4}{9}.

\frac{4}{5}

1\frac{1}{4}

\frac{20}{81}

2\frac{2}{9}

\begin{aligned}
\frac{5}{9}\div\frac{4}{9}&=\frac{5}{9}\times\frac{9}{4}\\\\
&=\frac{5\times{9}}{9\times{4}}\\\\
&=\frac{45}{36}\\\\
&=\frac{5}{4}\\\\
&=1\frac{1}{4}
\end{aligned}

7. Determine the value of \frac{5}{7}\div\frac{6}{11}.

1\frac{13}{42}

\frac{42}{55}

\frac{30}{77}

2\frac{17}{30}

\frac{5}{7}\div\frac{6}{11}=\frac{5}{7}\times\frac{11}{6}=\frac{5\times{11}}{7\times{6}}=\frac{55}{42}=1\frac{13}{42}

8. Determine the value of \frac{7}{8}\div{2}\frac{1}{4}.

1\frac{31}{32}

2\frac{4}{7}

\frac{3}{4}

\frac{7}{18}

\begin{aligned}
\frac{7}{8}\div{2}\frac{1}{4}&=\frac{7}{8}\div\frac{(4\times{2})+1}{4}\\\\
&=\frac{7}{8}\div\frac{9}{4}\\\\
&=\frac{7}{8}\times\frac{4}{9}\\\\
&=\frac{7\times{4}}{8\times{9}}\\\\
&=\frac{28}{72}\\\\
&=\frac{7}{18}
\end{aligned}

1. A teacher draws a rectangle. The ratio of the width to the length of the rectangle is 1:2.

The width of the rectangle is {1}\frac{3}{4} centimetres.

What is the area of the rectangle?

**(2 marks)**

Show answer

2\times{1}\frac{3}{4}=2\times\frac{7}{4}=\frac{14}{4}\left(=3\frac{1}{2}\right)

**(1)**

**(1)**

2. Calculate \frac{1}{2}\times{1}\frac{3}{4}\div{1}\frac{2}{5}.

**(3 marks)**

Show answer

\frac{1}{2}\times{1}\frac{3}{4}=\frac{1}{2}\times\frac{7}{4}=\frac{7}{8}

**(1)**

**(1)**

**(1)**

3. Phil says,

“If I multiply 10 by a positive integer then the answer will always be bigger than 10.

If I multiply 10 by a positive fraction then the answer will always be smaller than 10. ”

Phil is incorrect.

By rewriting the rule or by using a counter example, explain why Phil is incorrect.

**(2 marks)**

Show answer

Rewritten rules such as,

-If I multiply 10 by a positive **proper** fraction then the answer will always be smaller than 10.

-If I multiply 10 by a positive **improper** fraction or **mixed number** then the answer will always be **bigger** than 10.

**(2)**

Counter example such as,

-Multiplying 10 by an improper fraction to show the answer is not smaller than 10.

For example, 10\times\frac{5}{4}=\frac{10}{1}\times\frac{5}{4}=\frac{50}{4}=12\frac{1}{2}.

-Multiplying 10 by a mixed number to show the answer is not smaller than 10.

For example, 10\times2\frac{1}{3}=\frac{10}{1}\times\frac{7}{3}=\frac{70}{3}=23\frac{1}{3}.

**(2)**

You have now learned how to:

- Carry out calculations involving multiplication and division with numbers that are written as fractions and/or mixed numbers
- Calculate exactly with fractions
- Use formal written methods to show your working out when calculating with fractions

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