Fractions bad. Pizza good.

February 4, 2008

 This is one of an occasional series of articles for my tutoring site: FlyingColoursMaths.co.uk. Do visit if you’re in need of maths tuition, or know somebody who is.

“Better cut it into six slices – I don’t think I could eat eight” – Yogi Berra.

Nobody likes fractions. Once you’ve figured out how to beat them senseless, perhaps they become almost tolerable, but even then remembering the rules and carefully applying them is the kind of thing the devil probably has lined up for when he runs out of other ideas.

Unfortunately, they’re on the exam paper. Always. Taking a GCSE paper at random – Foundation paper 2, June 2004 – 16 of the 100 marks (about a sixth of the total) have something to do with fractions. 16 marks could be the difference between a D and a B.

One big trick to working with fractions is to turn the ugly numbers into something you can easily imagine. I tend to work with pizza, because slicing it into bits is a natural thing to do, and also it tastes good. Another is to use your understanding of pizza to learn how to do some basic examples – so you can apply the method to more complicated questions.

I like to start with two quarter-pizzas. Adding together a quarter and a quarter is easy, you get two quarters (or a half, which we’ll come to shortly). 1/4 + 1/4 = 2/4. The size of the slices – the quarters – is the same for both of the fractions so all we have to do is add the top parts. 1/4 + 2/4 = 3/4 – a quarter slice and a half make up three quarters of a pizza. So far so easy.

Now, you might lose marks for writing 2/4 as your answer, because it’s not in its simplest form. Whenever you write a fraction down, you should look for a number that divides into both the top and the bottom. Here, you can divide the top and the bottom by two, making 1/2 – which is the same two quarters.

Why does this always work? It’s best to come at it from the other side. If you start from a half and divide it into two parts, you’ve doubled the number of slices (from 1 to 2) BUT you’ve halved the size of the slices (from a half (1/2) to a quarter (1/4). So now you have two quarters. You can do this with any fraction and any multiplier – if you multiply (or divide) the top and the bottom by the same number, you get another version of the same fraction. A half is also eight sixteenths (multiply the top and bottom by 8); 6/15 is the same as 2/5, if you divide the top and bottom by three.

This trick is really useful when it comes to adding slices of a different size. Let’s say I’m feeling greedy and want a half pizza, but you’re not really hungry and only want a quarter. Between us we eat 1/2 + 1/4. We know we can’t just add the things on top and on bottom – we’d get 2/6, which is a third, which is less than a half. That doesn’t make any sense. Instead, we need to add slices of the same size together.

What’s a good size of slice? Anything you can divide by 2 and 4. Eight is good. Four is also good (even better, because you don’t have to touch the quarter). But let’s do eight for the practice. We want to turn both fractions into eighths. That means we need to multiply the bottom of the half (2) by four. If we do that, we need to multiply the top by four as well, to get the same thing. 1/2 = 4/8. That makes sense. For the quarter, we need to multiply the bottom by two. To keep everything the same, we have to multiply the top by two too. A quarter is 2/8. Now we have both fractions the same underneath, so we can simply add the tops together – four slices plus two slices is six slices, and the slices are all 1/8 of a pizza. 1/2 + 1/4 = 4/8 + 2/8 = 6/8. And naturally, we can make that simpler – 6 and 8 are both even, so you can divide both by two to leave 3/4 of a pizza. Which is just what we’d expect.

So, next time you’re faced with a fraction problem (say, 1/2 + 1/3) – think about what you’d do to the pizza to make it all add up. I’ll leave that one as an exercise – as long as I can have the left-over slice :o)

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