# 1 Percentages

News reports often mention the results of surveys and studies in ‘per cent’. Literally, ‘per cent’ means ‘per one hundred’, so ‘25 per cent’ means ‘25 out of one hundred’. As mentioned in the introduction, a per cent is just a specific kind of fraction – one that always has 100 as its denominator.

This means that it is important to remember that the per cent symbol, ‘%’, is not a unit as some people think. Instead it is a symbol that actually affects the number itself. Twenty-eight per cent is not equal to 28, it is actually , or 0.28

So, , not 28.

If you would like to see this explained visually, have a look at this short video.

#### Transcript

**We're asked to shade 20% of the square below.**Before doing that, let's just even think about what percent means. Let me just rewrite it. 20% is equal to-- I'm just writing it out as a word—20 percent, which literally means 20 per cent. And if you're familiar with the word century, you might already know that cent comes from the Latin for the word hundred. This literally means you can take cent, and that literally means 100. So this is the same thing as 20 per 100. If you want to shade 20%, that means, if you break up the square into 100 pieces, we want to shade 20 of them. 20 per 100. So how many squares have they drawn here? So if we go horizontally right here, we have one, two, three, four, five, six, seven, eight, nine, ten squares. If we go vertically, we have one, two, three, four, five, six, seven, eight, nine, ten. So this is a 10 by 10 square. So it has 100 squares here. Another way to say it is that this larger square-- I guess that's the square that they're talking about. This larger square is a broken up into 100 smaller squares, so it's already broken up into the 100. So if we want to shade 20% of that, we need to shade 20 of every 100 squares that it is broken into. So with this, we'll just literally shade in 20 squares. So let me just do one. So if I just do one square, just like that, I have just shaded 1 per 100 of the squares. 100 out of 100 would be the whole. I've shaded one of them. That one square by itself would be 1% of the entire square. If I were to shade another one, if I were to shade that and that, then those two combined, that's 2% of the entire square. It's literally 2 per 100, where 100 would be the entire square. If we wanted to do 20, we do one, two, three, four-- if we shade this entire row, that will be 10%, right? One, two, three, four, five, six, seven, eight, nine, ten. And we want to do 20, so that'll be one more row. So I can shade in this whole other row right here. And then I would have shaded in 20 of the 100 squares. Or another way of thinking about it, if you take this larger square, divide it into 100 equal pieces, I've shaded in 20 per 100, or 20%, of the entire larger square. Hopefully, that makes sense.

One of the reasons why percentages are useful is that you can make comparisons between different sets of data more easily than with the actual numbers.

Suppose that you are told that, over the course of one week, 345 people opted for a meat dish in one restaurant, but only 217 did in another. Does this mean that the meat dish was less popular in the second restaurant? You couldn’t conclude that unless you knew how many people in total had actually eaten in the two restaurants. In fact, in the first restaurant, 35 per cent of the total numbers of diners ordered a meat dish; in the second, 39 per cent. It is now immediately clear that the meat option was more popular in the second restaurant.

Hopefully, you can see from this example how useful showing data as a percentage can be for understanding of that data. This use of percentages will occur many times, not only your everyday life but also in other areas of study.

For example, if you ran a small bed and breakfast business and read that tourist numbers were expected to increase by 34 per cent across the UK by 2017, you might like to be able to work out what that could mean for your business. Or, you may be faced with a set of numbers breaking down the UK population into different age categories and for the purpose of your studies need to work these out as percentages. So knowing what a percentage can and can’t tell you, and how to present the raw data (the actual numbers) as a percentage, can relieve a lot of headaches!

So a percentage is just a specific kind of fraction and you know from your previous study that fractions and decimals are related to each other. This means that you can also write percentages as fractions and decimals. You’ll look at how to do this in the next section.