What is a percentage?

Percentage is any number that represents a part of $100$.
Percentage value is the actual number that the percentage represents out of the quantity.
Quantity is the total amount.
You can see the relationship between all these values in the following equation:

$\frac{percentage}{100}=\frac{percentage~value}{total~amount}$

Every percentage question will be solved using this equation (which is an excellent trick!) so it's important that you memorize it!

Let's substitute into the equation.

Let's look at examples:
$25\%$ of students in class 8C study art. There are $40$ students in class 8C.
How many students from class 8C study art?

Solution:
We need to find the percentage value - how many students does the $25\%$ actually represent.
Note - a different way to phrase the same question could look like this: what is $25\%$ of $40$.

Let's work according to the equation:

$\frac{percentage}{100}=\frac{percentage~value}{total~amount}$

The percentage is $25$.
The percentage value is the unknown we are looking for and the total amount is $40$.
Let's substitute and get:
$\frac{25}{100}=\frac{x}{40}$
Let's solve the equation and find the unknown:
$100x=1000$
$x=10$
The answer is $10$ students from class 8'3 study in the art track.

Another question:
Dana invited her entire class to her birthday.
There are $40$ students in the class. $35\%$ of the class are boys. How many girls are in Dana's class?

Solution:
Note - if we want to find the percentage value of $35\%$ out of $40$ we will discover how many boys are in the class
but if we want to know how many girls are in the class we need to find the percentage value of $65\%$ because it represents the number of girls in the class.
Therefore in the equation we will substitute $65$ for the percentage and not the given number $35$.
We get:

$\frac{65}{100}=\frac{x}{40}$

Let's find the unknown:
$100x=2600$
$x=26$

There are $26$ girls in the class.

Let's substitute in the equation!

Examples:

There are $60$ flowers blooming in the garden. $15$ of these flowers are roses. What percentage of the garden's flowers are roses?

Solution:
Note - A similar phrasing for the same question is what percentage is $15$ out of $60$.

In this question we are asked to find the percentage, therefore it will be our unknown. We have the total amount and the percentage value.
Let's substitute in the equation:

$\frac{15}{60}=\frac{x}{100}$

Let's find the unknown:
$60x=1500$
$x=25$
The answer is - the percentage of roses in the garden is $25\%$.

Of course, let's substitute in the equation.

Let's look at examples:
$10\%$ of the contestants in the competition are American. In total, there are $5$ American contestants in the competition.
How many contestants are there in total in the competition?

Solution:
We are given the percentage and the percentage value, and we need to find the total amount, so it is our unknown.
Let's substitute in the formula:

$\frac{10}{100}=\frac{x}{5}$
Let's find the unknown:
$10x=500$
$x=50$
There are a total of $50$ contestants in the competition.

Additional question:
By $16:00$ Danny completed $60\%$ of his total homework for that day. In total, Danny completed $3$ homework tasks by $16:00$.
How many homework tasks did Danny receive in total?

Solution:
We have the percentage and the percentage value and we need to find the total amount.
Let's substitute in the formula:

$\frac{60}{100}=\frac{3}{x}$

$60x=300$
$x=5$
Danny received a total of $5$ homework assignments that day.

What is $25\%$ of $300$?

Solution:
$300$ is the whole.
$25\%$ is the percentage we are asked to find.
The percentage value is what we are asked to calculate, meaning, what is the number that corresponds to $25\%$ of the whole $300$.
The calculation is done by multiplying the two numbers. It's important to remember that to convert a percentage to a number, we need to divide by $100$.

We get:

$300* 25\%$
$\frac{25}{100}*300 = 75$

In other words, the percentage value is $75$.

What is $20\%$ of $400$?

Solution:
$400$ is the whole.
$20\%$ is the percentage we are asked to find.
The percentage value is what we need to calculate, meaning, what is the number that corresponds to $20\%$ of the whole $400$.
The calculation is done by multiplying these two numbers. Remember that to convert a percentage to a number, we need to divide by $100$.

We get:

$400 *20\%=\frac{20}{100}*400=80$

In other words, the percentage value is $80$.

Another example:

Michal bought $80$ cupcakes for her birthday party. She couldn't resist waiting for her friends to arrive, and ate $15$ cupcakes by herself before they came. What percentage of the cupcakes did she eat?

Total number of cakes: $80$

The number of cakes she ate: $15$

The calculation: $\frac{15}{80}$ times $100$ = $18.75\%$

Another example:

Michal's friends wanted to treat her with a birthday gift! They were interested in a dress that cost about $300$ Dollars, but the seller gave them a special discount of about $60$ Dollars. What percentage discount did Michal's friends receive?

• Initial price: $300$.
• Discount: $60$ Dollars.
• Calculation: $\frac{60}{300}$ times $100 = 20\%$.

Additional examples:

Itai found an attractive deal for a summer vacation in Greece! The advertisement promised the first $50$ customers a discount of $480$ Dollars! The original vacation price before the discount is $2400$ Dollars. What percentage discount will Itai receive?
Initial price: $2400$ Dollars
Discount offered to Itai: $480$ Dollars.
Calculation: $\frac{480}{2400}$ times $100$ = $20\%$ discount

Another example:
Dan and Dana wanted to buy Danit a birthday present - a floral dress! The dress costs $450$ Dollars. Dan asked Dana to invest only $135$ Dollars, while he would pay the remainder from his pocket. What percentage of the gift's cost did Dana pay?
Gift price: $450$ Dollars
Dana's required cost: $\frac{135}{450}$ times $100$ = $30\%$