Percentages

We see the term “percentage” or “percent” on daily basis. For example, when your tablet battery charge is at 15%, you know it is time to plug it; when a store is offering 20% off, you know it is a good opportunity to buy and save some money; or when a food label says it is 100% bio, you know it is a healthy product.

These are only some examples of how percentages are used in real life, and we are about to study many more.

What is a percentage?

Formally, a percentage is a fraction having 100 as its denominator. It indicates how many parts of 100 are considered in a certain situation. To indicate a percentage, we usually write a number followed by the symbol %. The symbol % is read as “percent”.

For example, a survey is made to 100 people to know the time at which they normally eat lunch. If 46% of the people answer that they eat lunch at 12:00 p.m. Then, this means that 46 of those 100 people eat lunch at 12:00 p.m. By the way, 46% should be read as “46 percent” and represents 46100 of the surveyed people.

Maybe you are wondering why are we introducing a new symbol for a fraction? Or, what happens if we don’t have a total of 100 things, but less or more?

Well, the answer to the first question is that percentages are so commonly used in real life that they need a simpler and more particular notation.

The second question will be answered in what follows with a series of fun examples.

Real-life percentage examples.

Example 1: In a zoo, there are 100 different species of animals, and only 10 percent of those species are reptiles.

First of all, we can rewrite that expression as “10% of the species are reptiles”. Now, how many species is 10%?

Well, 10% means that 10 out of the 100 species of animals are reptiles; that is 10100 species.

Moreover, at this zoo, 30% of the species are birds, and the rest are mammals. This means that 30 of the 100 species are birds.

On the other hand, since there are 100 species in all, 30 are birds, and 10 are reptiles, then “the rest”: 100-30-10=60 are mammals. In other words, 60% of the species are mammals.

The zoo’s publicist thinks it is a good idea to take a picture having one animal of each species on it. Let’s assume, for the purposes of this example, that these animals are very friendly among them, so gathering them won’t be a problem.

But, how on earth are they going to make fit 100 animals in a pic? That doesn’t sound any easy, even for a zookeeper!

Luckily, the total species of the zoo can be represented by a smaller group having the same percentage of each class: reptiles, birds, and mammals.

They only need to use equivalent fractions:

$$10\%=\frac{10}{100}=\frac1{10}$$

$$30\%=\frac{30}{100}=\frac3{10}$$

$$60\%=\frac{60}{100}=\frac6{10}$$

Thus, they can gather a group of 10 animals: 1 reptile, 3 birds, and 6 mammals; which will also represent 10%, 30%, and 60% of their class, respectively.

This is of course a much more manageable amount of animals to make fit in a picture and, being very lucky, the photographer could even make them smile at the camera!

The previous example gives us an idea of how to represent percentages when we have a group of fewer than 100 elements. Let’s study that situation with the following example.

Example 2: Ann and Louise went to a cake shop. Each ate a slice of a delicious chocolate cake. The cake was cut into 8 equal slices. 

What percentage of the cake Ann and Louise ate?

Since each girl ate $$\frac18$$ of the cake, they ate $$\frac28$$ of the cake between the two of them.

To represent this amount as a percentage, we find a fraction equivalent to $$\frac28$$ having 100 as denominator:
$$\frac28=\frac14=\frac14\times\frac{25}{25}=\frac{25}{100}$$

This means that the girls ate 25% of the cake.

As we see, it is also possible to represent percentages of a totally different from 100.

In general, a formula to find percentages is:
$$\frac{Part}{Total}=Percentage$$

There are three components in that formula: “part”, “total” and “percentage”.

Therefore, we can solve three different problems using it, depending on which of the three components we need to find. We will study those three different problems next.

Percentage, part, and total.

In the following example, we find out what percentage represents a part of a total.

Example 1: The image shows the content of a crayon box. What percentage of the box represents the amount of green crayons?

We can see 4 green crayons in the image, and a total of 20 crayons.

$$\frac{Part}{Total}=\frac4{20}=\frac4{20}\times\frac55=\frac{20}{100}$$

That is, 20% of the crayon box are green crayons.

In the next example, we will find the amount (part) that represents a certain percentage.

Example 2: Mathews works 25% of the hours of each day. How many hours does Mathews work each day?

Each day has 24 hours, which represents the total hours. We want to find the part of 24 that represents the 25%. Substituting in the formula:

$$\frac{Part}{Total}=Percentage$$,

we get that:

$$\frac{Part}{24}=25\%=\frac{25}{100}$$

Solving for “Part” we get that:

$$\frac{Part}{24}=\frac{25}{100}\times24=6$$

That is, Mathews works 6 hours each day.

Now, we will find the original or total amount, knowing the percentage that a given part represents.

Example 3: Mary bought a bag of assorted flavor candies for Halloween. She read on the bag that 10% of its content were strawberry-flavored candies. Mary opened the bag and found 25 strawberry candies in it. How many candies did the bag contain?

We know that 25 of the total candies represent 10%. Thus,

$$\frac{25}{Total}=10\%$$

Solving for “Total” and knowing that

$$10\%=\frac{10}{100}=\frac1{10}$$,

we get that

$$Total=25\times10=250$$

Therefore, the bag contained 250 candies.

Percent change: increase and decrease.

Very often percentages are used to compare changes in a quantity. If the resulting new quantity, after the change, is greater than the original one, we have a percent increase. If it is smaller, we have a percent decrease.

Example 1: Last year there were 120 kids at a summer camp. This year, there is an increase of 24 kids. What was the percent increase?

That this year there is an increase of 24 kids, means that there are 24 more kids this year than last year. To calculate the percent change, we can use proportions.

We know that 120 kids represented the 100%, we want to know which percentage represents 24.

$$120\rightarrow100\\24\;\;\rightarrow\%Change$$

It follows that:

$$\frac{\%\;Change}{100}=\frac{24}{120}$$

Or, equivalently, $$\%\;Change=\frac{24}{100}\times100=20$$. This means that the percent change (increase) from last year to this year was 20%.

From this example we get the general formula for percent change:

$$\%\;Change=\frac{Original\;in\;value}{Original\;value}\times100$$

Let’s use this formula in a different example where we will have a percent decrease.

Example 2: David has a $25 coupon to be used at a tech store. He uses the coupon to buy a tablet that costs $250. What is the percent decrease in the price of the tablet after using the coupon?

The original price of the tablet is $250, and the change in the price is $25. We use the formula of percent change to get that:

$$\%\;Change=\frac{\$25}{\$250}\times100=10$$

This means that the percent decrease in the price of the tablet is 10%.

Multi steps problems with percentages.

Lastly, we will study some problems where it is needed to find several percentages in order to get the solutions.

Example 1: David continues shopping, and he finds a clothes store offering some discounts. He buys the items in the image. Before the discount, the shirt cost $31.50, the shorts cost $33,60, and the shoes cost $89. How much did David pay for the three items?

We first calculate the discount of each item, using the formula

$$\frac{Part}{Total}=Percentage$$

Or, equivalently,

$$Part=Percentage\times Total$$

• The 50% of the shirt is:
  $$Percentage\times total=50\%\times\$31.50=\frac{50}{100}\times\$31.50=\$15.75$$
• The 40% of the shorts is:
  $$Percentage\times total=40\%\times\$33.60=\frac{40}{100}\times\$33.60=\$13.44$$
• The 15% of the shoes is:
  $$Percentage\times total=15\%\times\$89=\frac{15}{100}\times\$89=\$13.35$$

The final price of each item is:

Final price = Original price – Discount.

Then,

• The final price of the shirt is $31.50 – $15.75 = $15.75. Of course, half the original price!
• The final price of the shorts is $33.60 – $13.44 = $20.16
• The final price of the shoes is $89 – $13.35 = $75.65

Therefore, David payed $15.75 + $20.16 + $75.65 = $111.56 for the three items.
Well, along with the tablet, it seems David didn’t save a lot after all!

Example 2: Clara loves to read. She owns 125 books, 40% of them are fantasy books, and 36% of her fantasy books are about wizards. How many wizard books does Clara have?

We first find out how many of the 125 books are fantasy books. It is the 40% of them:

$$Fantasy\;books=125\times\frac{40}{100}=50$$

We know that 36% of those 50 books are about wizards. Thus, we need to find the 36% of 50:

$$Wizard\;books=50\times\frac{36}{100}=18$$

Therefore, Clara has 18 books that tell stories about wizards.

We hope you have learned a lot about percentages with us today. Most importantly, we expect you to practice this concept using it on daily basis. We are sure you won’t have trouble finding the best discounts from now on!