Working with percentages

A percentage usually describes a relative part of a whole, where the whole itself is written as 100%. A percentage can always be rewritten as a fraction.

Percentages as part of a whole

We use percentages when we want to describe a part of some whole. We can replace phrases like “a quarter of the class got a 1, the lowest grade” or “every other person is a man” with percentages. There is always a whole, for example a whole class of students, and then we say that a quarter of them got a 1. If there are 32 students in the class and a quarter of them got a 1, the number of students who got a 1 is

$$\frac{1}{4}\cdot 32=8\text{which is the same as}\frac{32}{4}=8$$

For a second example, take a town, let’s call it Millbrook, with roughly 60,000 inhabitants. If every other inhabitant is a man, then half of them are men, which we write as:

$$\frac{1}{2}\cdot 60000=30000\text{which is the same as}\frac{\mathrm{60,000}}{2}=\mathrm{30,000}$$

We could also say that “one in every hundred people in Millbrook is a mathematician”. Then Millbrook would have $\frac{1}{100}\cdot \mathrm{60,000}=600$ mathematicians. “One in every hundred” is the same as “one hundredth of the inhabitants”.

Now we can introduce the idea of a percentage. One per cent, written 1%, means one hundredth of the whole. So when we say “one in every hundred people in Millbrook is a mathematician” and “one per cent of the people in Millbrook are mathematicians”, we are saying the same thing. To find one per cent of a whole, we multiply the whole by $\frac{1}{100}$ or divide it by 100 – it comes to the same thing.

We can easily convert percentages to fractions and carry on working with those. If we say that “x% of the people in Millbrook have dark hair”, that is the same as saying “$\frac{x}{100}\cdot \mathrm{60,000}$ people in Millbrook have dark hair”. Then 100% represents the whole, that is, all the inhabitants. The sentence “100% of the students moved up to the next year” means that every student moved up. The sentence “0% of the students failed” says that nobody failed. Examples:

1. 35% of Millbrook’s inhabitants travel by bus. How many people travel by bus? 35% is the same as $\frac{35}{100}$ of the whole (“thirty-five hundredths of the whole”). We work it out as $\frac{35}{100}\cdot \mathrm{60,000}$. The easy way is to divide 60,000 by a hundred first and then multiply the result by 35. In effect, we are doing this:

   $$\frac{35}{100}\cdot \mathrm{60,000}=\frac{35\cdot \mathrm{60,000}}{100}=\frac{35\cdot 600}{1}=35\cdot 600=\mathrm{21,000}$$

   We can picture it like this: first we work out how many inhabitants correspond to one per cent: $\frac{1}{100}\cdot \mathrm{60,000}=600$. One per cent corresponds to 600 inhabitants. We want 35% of the inhabitants, so we just multiply these 600 inhabitants by 35: 35 · 600 = 21,000.

2. Oliver’s average pay is £80 a day. Today he earned only 75% of his average pay. How much did Oliver earn? The method is still the same. The whole is 80, and we convert 75% to the fraction $\frac{75}{100}$. We can simplify this fraction to $\frac{3}{4}$. We multiply the whole by this fraction:

   $$\frac{3}{4}\cdot 80=\frac{3\cdot 80}{4}=3\cdot 20=60$$

3. The next day Oliver made up for it and earned 150% of his average pay. How much did he earn? Can we have more than 100% when we said that 100% represents the whole? Yes, we can. Just as someone can earn “three times as much as usual”, someone can earn 150% of their average.

   We proceed in exactly the same way. We convert 150% to the fraction $\frac{150}{100}$ and multiply Oliver’s average pay, 80, by this fraction:

   $$\frac{150}{100}\cdot 80=\frac{150\cdot 80}{100}=\frac{150\cdot 4}{5}=30\cdot 4=120$$

Percentages with the unitary method

We can easily turn a part of a whole given as a percentage into a proportion problem. Let’s stay with Oliver, who earns £80 a day on average and today earned 75% of his average pay. Using the unitary method, we can write it like this:

$$\begin{array}{rcl}100\mathrm{\%}& \dots & 80\text{ pounds}\\ 75\mathrm{\%}& \dots & x\text{ pounds}\end{array}$$

This is direct proportion, so we get:

$$\frac{75}{100}=\frac{x}{80}$$

We multiply the equation by 80 and get:

$$\frac{75\cdot 80}{100}=x$$

From here we can easily work out the answer:

$$\begin{array}{rcl}\frac{75\cdot 80}{100}& =& x\\ \frac{75\cdot 4}{5}& =& x\\ x& =& 15\cdot 4\\ x& =& 60\end{array}$$

We can see that we got the same number.

What if we know the part but not the whole?

In the previous examples we always knew the exact value of the whole and had to work out a part of it given as a percentage. But what if we know the part and not the whole? Example: Harry has 120 trading cards with a big green monster on them. Even though that is an impressive number of green monster cards, he still has only sixty per cent of the number that Amelia has. How many cards does Amelia have?

Here we know the part but not the whole – we have to work the whole out. We know that 120 cards correspond to 60% of the cards Amelia has. To find out how many cards correspond to one per cent, we just divide a hundred and twenty by sixty. We get two: 120/60 = 2. One per cent of the cards is actually two green-monster cards. Since we are finding the whole – and the whole is always 100% – we multiply this number of cards by a hundred. That gives 200 cards, which is the correct answer.

We can check this by working out 60% of 200. We simply multiply

$$\frac{60}{100}\cdot 200=\frac{3}{5}\cdot 200=3\cdot 40=120$$

We can see that 60% of 200 is 120, which is exactly the number of green-monster cards Harry has.

Adding percentages

When you have to work out a problem involving percentages, it is always best to calculate the exact values instead of the percentages and then add those. A question might go like this: Jack earns £15,000 a year, George £20,000 and Lily £25,000. Now work out how much Isla earns, if she earns 30% of what Jack earns plus 40% of what George earns plus 25% of what Lily earns.

Here we have three expressions with percentages, but each expression comes from a different whole. Jack, George and Lily each earn a different amount of money. So first we work out 30% of Jack’s pay. One per cent of it is 15,000/100 = 150. Then we multiply 150 · 30 and get 4,500. Next comes 40% of George’s pay. One per cent of it is 20,000/100 = 200. Again we multiply by 40 and get 200 · 40 = 8,000. The last part is 25% of Lily’s pay. She earns 25,000, and one per cent of that is 25,000/100 = 250. Multiplying by 25 gives 250 · 25 = 6,250. Now we just add up all the partial results: 4,500 + 8,000 + 6,250 = 18,750. Isla earns £18,750, which is quite a nice salary.

However, sometimes we really do add percentages. We can add percentages only when they have the same base, the same whole. So if, say, Noah earned 20% of what George earns plus 30% of what George earns, we can add the percentages, because they have the same base – both expressions use George’s basic pay. The result is 20% + 30% = 50%. Noah earns half of what George earns, ten thousand pounds.

The tricky trap of adding percentages in sequence

Now imagine the following situation: in his first year Oscar earned £10,000. In the second year he got a rise and his pay went up by 30%. In the third year he got another rise, though this time only 10%. The question is, what is Oscar’s pay now?

This type of problem is very tempting to treat by adding the percentages as in the previous case. After all, we are – apparently – still using the same base, Oscar’s pay. So we work out 30% + 10% and add that to his basic pay. Oscar would then earn £14,000. But that is not how it works. Let’s go through it.

In the first year he earned £10,000. In the second year he earned £10,000 + 30%, which is £13,000, as you can no doubt work out for yourself. Now in the next year his pay went up by ten per cent. But watch out! The base is no longer the ten thousand he earned in the first year. The base is now thirteen thousand! And as we know, we can add percentages only if they are calculated from the same base. In our case the base is different. The first time it was 10,000 and now it is 13,000. So in the last year Oscar earned £13,000 + 10%, which is £14,300.

The answer of £14,000 was wrong, and we also cheated poor Oscar out of £300. Don’t be mean to Oscar!

Buy a maths textbook in the sale!

A classic example is also the one with a discount. Imagine that a shop has a maths textbook that you would love to get from Father Christmas for your birthday. After all, what else would you want? :-)

Before Christmas, though, the shop puts the price of the book up by 20 %. You decide to wait and buy it after Christmas. You made a good choice, because after the holidays the shop reduced the price of the book by 20 %. The question is: does the book cost the same as before the increase? Or does it cost more or less?

Some people’s common sense again says that the book will cost the same, but that is not true. Let’s work it out. Say the book cost £10. A 20% increase means the book became dearer by $\frac{20}{100}\cdot 10=2$ pounds. So the book cost £12.

But the 20% discount is no longer worked out from the base of £10, but from the base of £12! That means the book goes down by $\frac{20}{100}\cdot 12=2.40$ pounds. After the sale the book costs £9.60.

(But honestly, who would buy a maths textbook when you already have this website? ;-))

Per mille

Per mille is most often used for measuring alcohol in a driver’s blood. Some people wrongly think that a per mille is a thousandth of a per cent, but that is a mistake, so watch out. A per mille is a thousandth of the whole. Otherwise you calculate with per mille in exactly the same way as with per cent. If you want to find x per mille of 5,000, you work it out as $\frac{x}{1000}\cdot 5000$. Instead of 100, the denominator of the fraction is 1000. Three per mille of 5,000 is therefore $\frac{3}{1000}\cdot 5000=3\cdot 5=15$.

Per mille is written with the symbol ‰. We have $x‰=\frac{x}{10}\mathrm{\%}$.

There is more in the separate article on per mille.

Writing percentages

In British writing the sign goes straight after the number with no space: 25%. In words, per cent is written as two words: “a twenty per cent discount”. Please bear this in mind when you write, and don’t get too worked up when a newspaper gets it wrong :-).

Percentage points

Besides ordinary percentages we can also have percentage points. What is the difference? Imagine that two years ago 20% of people used the Firefox browser, but this year 30% of people use it. What is the increase between these two years? It would be a mistake to say that the increase is ten per cent. If Firefox was first used by 20% of people and then by 30%, it means that half as many people again use it now as before. That is a fifty per cent increase.

But it is a bit awkward to say the increase is 50% when it would be clearer to everyone if we said that the number of per cent went up by ten. That is exactly what percentage points are for. In this case we can say that the use of the Firefox browser has gone up by ten percentage points. We can use percentage points when we want to express the simple difference between two percentages.