Add and subtract fractions: Different denominators

​​In a nutshell

The easiest way to add and subtract fractions is to first make their denominators the same. This is true also when adding mixed numbers: make their fraction parts have the same denominator.

​​The lowest common denominator

When adding or subtracting fractions whose denominators are different and are also such that one is not an integer multiple of the other, you look for the lowest common denominator of the two fractions. This will be the best denominator to make your fractions have. If in doubt, set the denominator of each fraction to be the product of the denominators. Once the denominators are the same, add or subtract the numerators together and keep the denominator as the new denominator found.

Example 1

Calculate 

$$\dfrac{2}{5}+\dfrac{1}6$$​​

Here the denominators are $$5$$ and $$6$$, so you seek the lowest common multiple of these numbers. This is $$30$$. Hence you need to make both fractions such that their denominator is $$30$$. Starting with $$\dfrac{2}{5}$$, consider what is done to the denominator to turn it into $$30$$. It has to be multiplied by $$6$$, so you must also multiply the numerator by this:

$$\dfrac{2}{5}=\dfrac{2\times6}{5\times6}=\dfrac{12}{30}$$​​

For $$\dfrac16$$, the denominator must be multiplied by $$5$$:

$$\dfrac16=\dfrac{1\times5}{6\times5}=\dfrac{5}{30}$$​​

Now you have that 

$$\dfrac{2}{5}+\dfrac{1}6=\dfrac{12}{30}+\dfrac{5}{30}=\underline{\dfrac{17}{30}}$$​​

This answer cannot be simplified. This will often be the case when the lowest common denominator is used in the calculation. Otherwise, some cancellation and simplification will be needed at the end.

​​Mixed numbers

When adding or subtracting mixed numbers to each other or to standard fractions, the method rules applies. If adding/subtracting a mixed number to/from another mixed number, add together/subtract the integer parts, then add the fraction parts in the same way as done above. If adding/subtracting a mixed number to/from a standard fraction, first convert the mixed number into an improper fraction before following the same method as outlined above.

Example 2

Calculate

$$3\frac14-2\frac16$$​​

You can start by subtracting the $$2$$ from the $$3$$ to get $$1$$. Now calculate 

$$\dfrac14-\dfrac16$$​​

The lowest common denominator here is $$12$$. Find that $$\dfrac14=\dfrac{1\times3}{4\times3}=\dfrac{3}{12}$$ and that $$\dfrac16=\dfrac{1\times2}{6\times2}=\dfrac{2}{12}$$. Hence 

$$\dfrac14-\dfrac16=\dfrac{3}{12}-\dfrac{2}{12}=\dfrac1{12}$$

So you conclude that

$$3\frac14-2\frac16=\underline{1\frac1{12}}$$​​

If the answer is required as an improper fraction, simply rewrite the integer part as a fraction with the same denominator as the fraction part:

$$1\frac1{12}=1+\frac1{12}=\frac{12}{12}+\frac1{12}=\underline{\frac{13}{12}}$$​​