$\frac{1}{3}\hspace{0.33em}{+}\hspace{0.33em}\frac{1}{4}$

You cannot add these directly because the sizes of the pieces of the whole are different sizes. The $\frac{1}{3}$ represents 1 piece (the numerator) of something divided into 3 pieces (the denominator) while the $\frac{1}{4}$ represents 1 piece of the same something divided into 4 pieces. It would be like adding 1 meter to 1 centimetre and getting 2 meters or centimetres? Either unit would be a wrong answer. You have to first convert one or both of the units to the same unit so that you can then add the numbers together. The same thing has to happen with fractions with different denominators: you got to find equivalent fractions where the size of the pieces (the denominators) are the same size.

In my last post, you saw that I can multiply top and bottom of a fraction by the same number to get an equivalent fraction. This is true because you are effectively multiplying by 1. It will be this skill that will help us add the two fractions.

Looking at the denominators 3 and 4, I want to find numbers to multiply them by to get the same number. If I multiply the 3 by 4 and the 4 by 3, I will get 12. This will be the new denominator that I want to use to convert each of the fractions.

So first, let’s convert the $\frac{1}{3}$ to an equivalent fraction with 12 in the denominator. To get the 12 in the denominator, I need to multiply the 3 by 4. But I cannot just do this in the bottom, I must also multiply the numerator by 4 as well to get an equivalent fraction:

$\frac{{1}\hspace{0.33em}\times\hspace{0.33em}{4}}{{3}\hspace{0.33em}\times\hspace{0.33em}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{4}{12}$

So $\frac{4}{12}$ is the same fraction as $\frac{1}{3}$ only it now represents 4 pieces of something divided into 12 pieces. Let’s do the same with the $\frac{1}{4}$.

If I multiply the 4 by 3, I will get 12, but again, this must also be done in the numerator:

$\frac{{1}\hspace{0.33em}\times\hspace{0.33em}{3}}{{4}\hspace{0.33em}\times\hspace{0.33em}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{12}$

So now we have the original fractions converted to fractions with the same size pieces. We can now add them directly:

$\frac{1}{3}\hspace{0.33em}{+}\hspace{0.33em}\frac{1}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{4}{12}\hspace{0.33em}{+}\hspace{0.33em}\frac{3}{12}\hspace{0.33em}{=}\hspace{0.33em}\frac{7}{12}$

So that’s how fractions with different denominators are added together:

1. Find a common denominator
2. Convert each fraction by multiplying each denominator by the number needed to get the common denominator and multiply the corresponding numerator by the same number, then
3. Add the new equivalent fractions together by just adding the numerators.

Now for this problem, you found the common denominator by just multiplying the two denominators 3 and 4 together. This method will always work but you may end up with an equivalent denominator bigger than it has to be. In order to work with smaller numbers and reduce the work to simplify the resulting answer (that is get rid of common factors), we want to find what is called the least common denominator (LCD). This is also called the least common multiple (LCM). I will cover the way to find the LCD in my next post.