Adding and subtracting fractions - a simple guide

This hub on addition and subtraction of fractions is part of a series of "how to" hubs covering everyday topics in maths.  I will gradually be adding more of these hubs to the ones I've already done, and suggestions for topics are very welcome!

If you were shown a picture like the main illustration and asked to add the two purple bits together, chances are you'd probably say, "Oh, that's easy.  It's two thirds."  Mathematically, you would write it as follows:

What you've done is to add together the two numerators (the top parts of the fractions) and leave the denominators (the bottom parts of the fractions) the same.  So, along the same lines:

Note that I've tidied things up at the end, by converting to a mixed number and then cancelling (reducing) the fraction down.

If only adding fractions was always this simple. What happens when you come up against something like this?

You know that you'd be OK if the two denominators were the same, but they're not. However, you can make them the same by doing a bit of multiplication. The first question to ask yourself is "What's the lowest number that both the 3 and the 5 will divide into?" (In other words, what's the lowest common denominator of 3 and 5?). Then you can multiply everything out and add the result together.

Not sure what I mean here? Don't worry - all will become clear in a moment. If you do a bit of mental maths, you shouldn't have too much trouble working out that 15 is the lowest number that both the 3 and the 5 will divide into. The 3 goes into 15 five times, and the 5 goes 3 times. So 1/3 can be converted to 5/15 (by multiplying both the numerator and denominator by 5), and 1/5 can be converted to 3/15 by multiplying both the numerator and denominator by 3. It's important to multiply BOTH the numerator and denominator in each case (in other words, it's the reverse of what you do when you cancel fractions down). You can draw this as a picture as follows:

Subtracting fractions works exactly the same.  In the example below, I've worked out that 30 is the lowest common denominator of 5 and 6, and multiplied everything out accordingly:

Note that the initial answer (14/30) can be cancelled down to 7/15.

Hope this hub has been useful. Coming soon: adding and subtracting mixed numbers!

© Empress Felicity January 2010