## What you need to know

Subtracting fractions follows precisely the same method as adding fractions. You should click here and make sure you’re confident on how to add fractions before proceeding.

To recap, we will have to find a common denominator – some value that can become the denominator of both fractions. Then, we will manipulate the fractions accordingly to make them have the desired denominator. There are two main methods for choosing a common denominator:

1. Use the lowest common multiple (LCM) of the two denominators,

2. Use the product of the two denominators.

The first method is generally the best, whilst the second method will always work if you’re not sure what the lowest common multiple is. Read through the examples here to see both in action (note: often both methods will yield the same outcome anyway).

Example: Evaluate \dfrac{7}{10} - \dfrac{1}{6}. Write your answer in its simplest form.

To find a common denominator here, we will use 30, since it is the LCM of 6 and 10.

Now, to make the denominator of the first fraction 30, we’ll have to multiply its current denominator by 3. So, to do this without changing the value of the fraction, we’ll also have to multiply the top by 3.

\dfrac{7}{10}=\dfrac{7\times 3}{10\times 3}=\dfrac{21}{30}

For the second fraction’s denominator to be 30, we’ll have to multiply it by 5. So, we will also have to multiply the top by 5.

\dfrac{1}{6}=\dfrac{1\times 5}{6\times 5}=\dfrac{5}{30}

Now, to subtract two fractions with the same denominator, simply subtract the numerators. Doing so, we get

\dfrac{7}{10} - \dfrac{1}{6}=\dfrac{21}{30} - \dfrac{5}{30}=\dfrac{16}{30}

Finally, cancelling a factor of 2 from top and bottom, we get the fully simplified fraction to be

\dfrac{16}{30}=\dfrac{8}{15}

In this next example, we’ll see what this process looks like when done a bit more quickly.

Example: Evaluate \dfrac{1}{4} - \dfrac{8}{15}.

Our choice of common denominator will be 4\times 15=60. In this case, it is also the LCM of 4 and 15.

To make the denominator of the first fraction 60, we’ll have to multiply its top and bottom by 15. To make the denominator of the second fraction 60, we’ll have to multiply its top and bottom by 4. This looks like

\begin{aligned}\dfrac{1}{4}-\dfrac{8}{15} &= \dfrac{1\times 15}{60}-\dfrac{8\times 4}{60} \\ &=\dfrac{15}{60} - \dfrac{32}{60}\end{aligned}

Now, note that the fraction we’re taking away is bigger than the other fraction, so we’ll get a negative answer. There are no new rules about this, you should just subtract the numerators as you would subtract normally and expect a negative answer if the one on the right is bigger than the one on the left. So, we get

\dfrac{15}{60}-\dfrac{32}{60}=-\dfrac{17}{60}

Example: Evaluate 3-\dfrac{6}{11}.

To subtract a whole number and a fraction, we must first recognise that any number divided by 1 is itself. So, we can write

3=\dfrac{3}{1}

Then, the problem becomes

\dfrac{3}{1}-\dfrac{6}{11}

This is now a familiar situation. Since 1\times 11=11, the common denominator will be 11, meaning we’ll only have to change the first fraction – we will multiply its top and bottom by 11. Doing so, we get

\begin{aligned}\dfrac{3}{1}-\dfrac{6}{11} &= \dfrac{3\times 11}{11}-\dfrac{6}{11} \\ &=\dfrac{33}{11} - \dfrac{6}{11} \\ &=\dfrac{27}{11}\end{aligned}

Example: Evaluate 7\frac{1}{2}-\dfrac{4}{5}.

To subtract a mixed number and a fraction, first convert the mixed number to an improper fraction.

7\frac{1}{2}=\dfrac{(7\times 2)+1}{2}=\dfrac{15}{2}

Now the calculation looks like

\dfrac{15}{2}-\dfrac{4}{5}

We will use the common denominator of 2\times 5=10. So, we get

\begin{aligned}\dfrac{15}{2}-\dfrac{4}{5} &= \dfrac{15\times 5}{10}-\dfrac{4\times 2}{10} \\ &=\dfrac{75}{10} - \dfrac{8}{10} \\ &=\dfrac{67}{10}\end{aligned}

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### Example Questions

2) Evaluate \dfrac{7}{4} - 9.

The LCM of 6 and 9 is 18, so we’ll use 18 as a common denominator.

So, we have to multiply top and bottom of the first fraction by 3, and multiply top and bottom of the second fraction by 2. Doing this, we get

\begin{aligned}\dfrac{1}{6}-\dfrac{4}{9} &= \dfrac{1\times 3}{18}+\dfrac{4\times 2}{18} \\ &=\dfrac{3}{18} - \dfrac{8}{18} \\ &=-\dfrac{5}{18}\end{aligned}

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2) Evaluate \dfrac{7}{4} - 9.

Writing 9 as \frac{9}{1}, the calculation becomes

\dfrac{7}{4}-\dfrac{9}{1}

Using 1\times 4=4 as a common denominator, we get

\begin{aligned}\dfrac{7}{4}-\dfrac{9}{1} &= \dfrac{7}{4}-\dfrac{9\times 4}{4} \\ &=\dfrac{7}{4} - \dfrac{36}{4} \\ &=-\dfrac{29}{4}\end{aligned}

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