## What you need to know

Before adding fractions, it’s important to understand that if you multiply both top and bottom of a fraction by the same value, the fraction’s value doesn’t change. For example,

$\dfrac{2}{3}=\dfrac{4}{6},\,\,\,\,\text{ and }\,\,\,\,\dfrac{2}{3}=\dfrac{10}{15}$

In order to add fractions, 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 $\dfrac{3}{5} + \dfrac{1}{4}$.

To find a common denominator here, we will take the product of the two denominators: $5\times 4=20$. As it happens, this is the LCM of 5 and 4 anyway.

Now, to make the denominator of the first fraction 20, we’ll have to multiply its current denominator by 4. So, to do this without changing the value of the fraction, we’ll also have to multiply the top by 4.

$\dfrac{3}{5}=\dfrac{3\times 4}{5\times 4}=\dfrac{12}{20}$

For the second fraction’s denominator to be 20, we’ll have to multiply it by 5. So, we will also have to multiply the top by 5.

$\dfrac{1}{4}=\dfrac{1\times 5}{4\times 5}=\dfrac{5}{20}$

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

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

So, we have completed the fraction addition.

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

Example: Evaluate $\dfrac{4}{7} + \dfrac{5}{2}$

Our choice of common denominator in this case will be $7\times 2=14$. Again, is also the LCM of 7 and 2.

To make the denominator of the first fraction be 14, we’ll have to multiply its top and bottom by 2. To make the denominator of the second fraction be 14, we’ll have to multiply its top and bottom by 7. This looks like

\begin{aligned}\dfrac{4}{7}+\dfrac{5}{2} &= \dfrac{4\times 2}{14}+\dfrac{5\times 7}{14} \\ &=\dfrac{8}{14} + \dfrac{35}{14} \\ &=\dfrac{43}{14}\end{aligned}

Remember, in the final step you add the numerators, and the denominator is unchanged.

Example: Evaluate $8+\dfrac{5}{6}$.

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

$8=\dfrac{8}{1}$

Then, the problem becomes

$\dfrac{8}{1}+\dfrac{5}{6}$

This is now a familiar situation. This time, since $1\times 6=6$, the common denominator will be 6, meaning we’ll only have to change the first fraction – we will multiply its top and bottom by 6. Doing so, we get

\begin{aligned}\dfrac{8}{1}+\dfrac{5}{6} &= \dfrac{8\times 6}{6}+\dfrac{5}{6} \\ &=\dfrac{48}{6} + \dfrac{5}{6} \\ &=\dfrac{53}{6}\end{aligned}

Example: Evaluate $2\frac{5}{9}+\dfrac{1}{3}$. Write your answer in its simplest form.

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

$2\frac{5}{9}=\dfrac{(2\times 9)+5}{9}=\dfrac{23}{9}$

Now the calculation looks like

$\dfrac{23}{9}+\dfrac{1}{3}$

Now, for the common denominator we could use 27 (since it’s the product of 9 and 3). However, the LCM of 9 and 3 is just 9, and if we use 9 as our common denominator we will only have to change the second fraction.

Doing this, we get

\begin{aligned}\dfrac{23}{9}+\dfrac{1}{3} &= \dfrac{23}{9}+\dfrac{1\times 3}{9} \\ &=\dfrac{23}{9} + \dfrac{3}{9} \\ &=\dfrac{26}{9}\end{aligned}

This fraction is already in its simplest form, so we’re done. However, if we had chosen 27 as our common denominator, we would’ve had to simplify the fraction at the end.

## Adding and Subtracting Fractions Questions

We’re going to choose a common denominator of 24 – it’s the LCM of 8 and 12 and will make our calculations much simpler than if we used $8\times 12=96$.

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}{8}+\dfrac{5}{12} &= \dfrac{1\times 3}{24}+\dfrac{5\times 2}{24} \\ &=\dfrac{3}{24} + \dfrac{10}{24} \\ &=\dfrac{13}{24}\end{aligned}

This fraction is already in its simplest form, so we’re done. However, if you used a different common denominator, make sure your fraction was fully simplified to get the result above.

Writing $5$ as $\frac{5}{1}$, the calculation becomes

$\dfrac{9}{10}+\dfrac{5}{1}$

Using $1\times 10=10$ as a common denominator, we get

\begin{aligned}\dfrac{9}{10}+\dfrac{5}{1} &= \dfrac{9}{10}+\dfrac{5\times 10}{10} \\ &=\dfrac{9}{10} + \dfrac{50}{10} \\ &=\dfrac{59}{10}\end{aligned}