# Five foolish fraction foul-ups that keep costing you points

If you want to frighten an A-level maths student, you don’t need to produce a high-budget horror movie; you just need to produce a fraction. Or a fractionstein, as I like to call it.

I want to try to cure you of your irrational phobia of fractions. I agree, they’re ugly and more involved than certain other bits of maths, but really? They’re not so bad.

Here are five things about fractions you can fix - today, if you practise.

### 1. Saying ‘fractions are hard’

You are not Sir Alex Ferguson. Fractions are not susceptible to mind games. Telling a fraction that it’s the favourite to beat you doesn’t really affect anyone except you - and make it harder.

Have you ever sat down with a GCSE book and tried working through the exercises on fractions? Have you ever put any more thought into it than looking at it blankly and saying ‘this is hard’? Nah, thought not. It’s like complaining that playing guitar is really hard, when you haven’t learned the basic chord shapes.

Like anything else, it’s easy *once you know how to do it*, but you need to study to get to that point. And the first thing to do is stop complaining that it’s so mind-bogglingly hard and practise a bit.

### 2. Cancelling at random

Now, repeat after me: $\frac{x+3}{x}$ does not equal 3. No way, nohow, I’m going to install a QI klaxon in my classroom and set it off every time you say that.

*If you’re going to cancel a fraction, you have to cancel every term.*

Very simple. You can cancel a bracket, if it’s multiplied by everything else on top or on the bottom - $\frac{(x+3)(x+2)}{x+3}$ is the same thing as $x+2$ - but you can’t cancel bits and pieces.

When in doubt, factorise!

### 3. Not combining or splitting

Two related problems: quite often (especially with C3 and C4 questions), you get to something like $\frac{x^2 + 4x + 3}{x}$. The response is typically ‘what do I do now?’

What you do now, my simple-minded friend, is split the fraction up. You’re always allowed to break up the top of a fraction, a bit like you’d multiply out a bracket. Just make sure you have the same bottom on all of the split bits: $\frac{x^2 + 4x + 3}{x} = \frac{x^2}{x} + \frac{4x}{x} + \frac{3}{x} = x + 4 + \frac{3}{x}$

Much nicer. Notice that once you’ve separated off the fractions, you *can* cancel them separately. Just not when they’re all in one big glomp.

You also sometimes need to do the same thing in reverse: you have something like $\frac{3}{x} + \frac{x}{3}$ and for some reason don’t think of putting it into one fraction using the rules you’ve been using since year 6. You’d get $\frac{9 + x}{3x}$. Easy.

### 4. Stacking them

You know, for students who claims to hate fractions, some people really go out of their way to make more of them. Given something like: $\frac{3}{4} x = \frac{3}{2}$, instead of multiplying both sides by 4 (of which more in a minute) or writing $ x = \frac{3}{2} \times \frac{4}{3} = 2$, people write something like $ x = \frac{\frac{3}{2}}{\frac{3}{4}}$ and wonder why they’re confused.

It’s just like with minus signs: they’re not hard if you follow the rules, but don’t go out of your way to make more of them, that’s just daft.

*Whenever you’re dividing by a fraction, flip it upside down and multiply*. Seriously. You’ll thank me later.

### 5. Not getting rid

Related to that: fractions can be really useful tools. But, like a plumber with poor personal hygiene, you don’t want them to hang around any longer than is absolutely necessary.

If you’re trying to solve an equation with a fraction in, it’s almost always a good idea to multiply everything by the bottom of the fraction: and hey presto! No more fraction. You might get it back at the very end, but dealing with the numbers in the meantime is that much more pleasant.

I could go on for quarter-hours about the knots people tie themselves up into when fractions rear their smelly heads. But nine times out of ten, they’re not half as bad as they’re made out to be.

* Edited 2014-09-19 to fix LaTeX