How did you do with the problems presented on my last post? I hope you did well. I have no comments yet so if you have questions you can leave a comment on the Comments page where I will answer your question or you can directly contact me on the Contact page.

Today, I thought I’d start on the topic of fractions. Apart from word problems, this topic seems to cause the most anxiety in maths students. Hopefully, I can simplify fractions in the next few posts.

So a fraction usually represents a part of a whole, that is a number between 0 and 1. I say “usually” because a fraction can be greater than one but I’ll get into that later. The notation for a fraction is a number over another number:

\[\frac{1}{2}

\]

The number on top of the line is called the numerator and the number below is called the denominator. The denominator represents how many pieces that the something is being divided into and the numerator represents the number of those pieces. So using the example above, the 2 means that something is being divided into 2 pieces and the 1 means the number of those pieces. So it should make sense that if I add two of these together, I should get the whole something, that is “1”:

\[\frac{1}{2}\hspace{0.33em}{+}\hspace{0.33em}\frac{1}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{2}\hspace{0.33em}{=}\hspace{0.33em}{1}

\]

I’ve shown two important things here. The first is that when you add fractions together that have the same denominator (that is the pieces are the same size), then you can simply add the numerators together to get the result. I also showed that when the numerator and the denominator are the same, this can be simplified to “1”. This makes sense here as two halves make up a whole. This also makes sense when you remember that 2 over 2 also indicates the division 2 ÷ 2 which is also 1.

Let’s look at smaller pieces of a whole. Now \[

\frac{1}{4}

\] means 1 of the 4 pieces of the whole of something. If I have two of these pieces and add them together I get:

\frac{1}{4}\hspace{0.33em}{+}\hspace{0.33em}\frac{1}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{4}

\]

Now some of you may know that this is the same as \[

\frac{1}{2}

\]. I will explain why this is so in my next post. Also, what if I have 1 of three pieces of a whole and 1 of four pieces of a whole? How much of the whole do I have? That is, what is

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

\]

Since the pieces are different sizes, I just can’t add the numerators together. So what can I do? That will be the topic of my next post.