Counting with percentages

Percentages usually refer to some relative part of a whole, with the whole itself expressed as 100%. Percentages can always be rewritten into a fraction.

Percentages as part of a whole

We use percentages when we want to express a part of a whole. Percentages can be replaced by expressions such as 'a quarter of the class got an F' or 'every second person is male'. We always have a whole, for example a whole class of pupils, and we say that a quarter of them got an F. If there are 32 pupils in a class and a quarter of them got an F, then the total number of pupils who got an F will be equal:

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

In the second case, we can have a city, say Opava. It has approximately 60 000 inhabitants. If every second inhabitant is male, then that means that half of them are male, we write it as:

$$ \frac{1}{2}\cdot 60, 000 = 30, 000 \quad \mbox{ which is the same as }\quad \frac{60{,}000}{2}=30{,}000 $$

Next, we could say that "every 100th person in Opava is a mathematician". Then it would be true that we have $\frac{1}{100}\cdot60,000=600$ mathematicians in Opava. "Every hundredth" is the same as "one hundredth of the population".

Now we can introduce the concept of percentage. One percent, we denote 1%, means one hundredth of the whole. This means that when we say that "every hundredth person in Opava is a mathematician" and "one percent of the people in Opava are mathematicians", we are saying the same thing. We calculate one percent of the whole by multiplying the whole by $\frac{1}{100}$ or dividing by 100, which is the same thing.

We can easily convert percentages into fractions and work with them further. If we say that "in Opava, x % of people have dark hair", then it is the same as saying "in Opava, $\frac{x}{100}\cdot60,000$" people have dark hair. 100% Then represents the whole, i.e. all the inhabitants. The sentence "100% of pupils passed to the next grade" means that all pupils passed. The sentence "0% of pupils failed" means that no one failed. Examples:

1. 35 % of the population of Opava rides buses. How many people ride buses? 35 % is the same as $\frac{35}{100}$ of the whole (= "thirty-five hundredths of the whole"). Let's calculate: $\frac{35}{100}\cdot 60,000$. This can be easily calculated by first dividing 60,000 by 100 and multiplying this result by 35. Basically, we make this adjustment:

   $$ \frac{35}{100}\cdot 60{,}000 = \frac{35\cdot60{,}000}{100}=\frac{35\cdot600}{1}=35\cdot600=21{,}000 $$

   We can think of it as first calculating the population that corresponds to one percent: $\frac{1}{100}\cdot60,000=600$. One percent corresponds to 600 inhabitants. We want to know 35 inhabitants, so we just multiply these 600 inhabitants by 35 · 600 = 21,000.

2. Thomas earns on average 500 crowns a day. Today he earns only 75% of his average wage. How much did Thomas earn? The procedure is still the same. The whole is 500 and we convert 75% to a fraction $\frac{75}{100}$. We can further reduce this fraction to the fraction $\frac{3}{4}$. We multiply the whole by this fraction:

   $$ \frac{3}{4}\cdot500=\frac{3\cdot500}{4}=3\cdot125=375 $$

3. The next day, Thomas fixed his appetite and earned 150% of his average wage. How much did he earn? Can we even have more than 100% since we said that 100% represents the whole? Yes, it can. Just as someone can make "three times the average", someone can make 150% of their average.

   We'll do exactly the same thing. We'll convert 150% to a fraction $\frac{150}{100}$ and multiply Thomas's average wage by that fraction, i.e. 500:

   $$ \frac{150}{100}\cdot500=\frac{150\cdot500}{100}=\frac{150\cdot5}{1}=150\cdot5=750 $$

Percentages as a trinomial

We can easily convert the part of the whole given by percentages into a trinomial. Let's stay with Thomas, who earns an average of 500 crowns per day and today earned 75% of his average wage. In fact, we can write it like this using the three-particle formula:

\begin{eqnarray} 100% &\quad\ldots\quad&500 \mbox{ crowns }\75% &\quad\ldots\quad&x \mbox{ crowns } \end{eqnarray}

Since this is a direct proportion, we get the form:

$$ \frac{75}{100} = \frac{x}{500} $$

Multiply the equation by 500 and we have:

$$ \frac{75\cdot500}{100}=x $$

And from there we can easily calculate the result:

\begin{eqnarray} \frac{75\cdot500}{100}&=&x\ \frac{75\cdot5}{1}&=&x\ x&=&75\cdot5}&&=&375 \end{eqnarray}

We can see that we got the same number.

What if we know the part and we don't know the whole?

In the previous examples, we always assumed that we knew the exact value of the whole and we had to calculate a certain fraction, given as a percentage. But what if we know that part and we don't know the whole? Example: Thomas has 120 magic cards with a big green monster. However, even though this is a respectable number of green monster cards, he still only has sixty percent of the cards compared to Jana. How many cards does Jana have?

Here we know a portion, but we don't know the whole - we have yet to calculate that. At the moment, we know that 120 cards corresponds to 60% of the cards Jana has. To find out how many cards correspond to one percent, we just need to divide the hundred and twenty by 60. We get two 120/60 = 2. One percent of the cards are actually two green-faced cards. Since we're counting the whole - which is always 100% - we multiply that number of cards by one hundred. And we have 200 cards, which is the correct result.

We can check this by counting 60% of 200: simply multiply

$$\frac{60}{100}\cdot200=\frac{3}{5}\cdot200=3\cdot40=120$$

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

Adding up the percentages

If we have to calculate an example where percentages are involved, it's always best to calculate exact values instead of percentages and then add them up. So the assignment might read something like this: Martin earns 15 000, Stanislav 20 000 and Lucy 25 000 crowns. Now calculate how many crowns Simona earns, who earns 30% of what Martin earns plus 40% of what Stanislav earns plus 25% of what Lucka earns.

Here we have three expressions with percentages, but each expression comes from a different whole. Martin, Stanislav and Lucka each earn a different amount of money. So first we calculate 30% of Martin's salary. That's 15,000/100 = 150. Then we multiply by 150 · 30 to get 4,500. Next up is 40% of Stanislav's salary. That's 20,000/100 = 200. Again, multiply by 40 and you get 200 · 40 = 8 000. The last part is 25% of Lucy's. She earns 25,000, of which one percent is 25,000/100 = 250. By multiplying by 25, we endow 250 · 25 = 6 250. Now we just add up all the partial results: 4,500 + 8,000 + 6,250 = 18,750. Simona earns 18,750 crowns, which is quite a nice income.

However, it may happen that we actually add up the percentages. We can only add percentages if they have the same base, the same whole. So, for example, if Kuba earned 20% of what Stanislav earned plus 30% of what Stanislav earned, we can add the percentages together because they are the same base - both expressions count Stanislav's basic salary. So the result will be 20% + 30% = 50%. Kuba earns half of what Stanislav earns, ten thousand.

The tricky cascading addition of percentages

Now imagine the following situation: Milan earned 10,000 the first year. But the second year he got a raise and his salary was increased by 30%. The next year, the third, he got a raise again, and again they raised his salary, though this time only by 10%. The question is, what is Milan's salary now?

This type of example makes it very tempting to add up the percentages as in the previous case. After all, we are still - seemingly - counting on the same basis, Milan's salary. So we calculate 30% + 10% and add that to his base salary. Milan would then earn 14,000 crowns. But it's not like that. Let's go through it.

In the first year, he was paid 10,000. In the second year, he got 10,000 + 30%. That's 13,000, as you've already calculated. And now, the next year, he gets a ten percent raise. But watch out! The base is no longer the 10,000 he had the first year. The base is now thirteen thousand! And as we know, we can only add percentages if the percentages are calculated on the same base. And in our case, the basis is different. The first time it was 10,000 and now it's 13,000. So last year, Milan got 13,000 + 10%, which is 14,300.

The 14,000 solution was both wrong and we cheated poor Milan out of 300 crowns. Don't be mean to Milan!

Buy a math book at a discount!

The classic example is also at a discount. Imagine that the shop has a maths textbook that you would really like from Father Christmas for your birthday. After all, what else would you want too, right? :-)

But before Christmas they make the book more expensive, by 20 %. You tell yourself that you will wait and buy it after Christmas after all. You did the right thing because the store discounted the book after the holidays by 20 %. The question is - is the book worth the same as before the delay? Or does it cost more/less?

Again, some people's common sense dictates that the book will cost the same, but that's not true. Let's do the math. Suppose the book cost 300 crowns. A 20% increase in price means that the book was more expensive by $\frac{20}{100}\cdot300=60$ crowns. So the book cost 360 crowns.

But the 20% discount is no longer calculated on a base of 300, but on a base of 360! This means that the book will be cheaper by $\frac{20}{100}\cdot360=72$ crowns. After the discount, the book costs 288 crowns.

(But in truth, who would buy a maths textbook when there's a Maths Half Price for you ;-))

Promile

Promiles are most commonly used in measuring a driver's blood alcohol content. Some people mistakenly believe that a promile is a thousandth of a percent, but this is a mistake, beware. A promile is a thousandth of a whole. Otherwise, promiles are treated exactly the same as percentages. If you want to find x promile of 5000, you calculate it as $\frac{x}{1000}\cdot 5000$. Instead of 100, 1000 is in the denominator of the fraction. So three per mille out of 5000 is $\frac{3}{1000}\cdot5000=3\cdot5=15$.

Per mil is denoted by the symbol ‰. Thus, $x ‰ = \frac{x}{10} %$.

More in a separate article on promiles.

Typography of percentages

There is a difference between writing "100%" (with a space) and "100%" (without a space). The version with a space means "one hundred percent", whereas the version without a space means "one hundred percent". Please keep this in mind when writing and don't get too excited in case the newspaper gets it wrong :-).

Percentage point

In addition to normal percentages, we can still have percentage points. What is the difference between the two? Imagine that the Firefox browser was used by 20 % people two years ago, but this year 30% of people are using it. What is the increase between those two years? It would be a mistake to say that the increase is 10%. If Firefox was used by 20% of people first and then 30%, that means that half as many people are using it now as before. That's a 50% increase.

But then again, it's a bit silly to say there's a 50% increase when it would be clearer for everyone to say that the percentage has gone up by ten. That is what we have percentage points for. In this case, we can say that the increase in Firefox usage is ten percentage points. We can use percentage points when we want to express a simple difference between two percentages.