The percentage is one of the most used mathematical concepts, in fact, almost all people calculate a percentage at least once in their life.
On many occasions when we go to the market, the shopping center or when we pay the electricity bill, we apply the idea of percentage to calculate a discount or a tax.
What is percentage?
Percentages are a way of representing the parts of a whole number. Just like fractions and decimal numbers.
A percentage is a fraction whose denominator is always 100.
Percentages consist of a number accompanied by the symbol %, which is pronounced “percent”, and which means “per 100”.
When we talk about 55% we mean 55 out of 100. The same goes for any other percentage, be it 10%, 50%, 75%, 120% or any other.
The percentage as a fraction.
In this section we explain how you can write the percentage as a fraction. We also tell you how to go from a fraction to a percentage.
From percentage to fraction
When we refer to 15%, we are really talking about a fraction whose numerator is 15 and the denominator is 100.
15% = 15100
As you already know, if the numerator and denominator of a fraction have common divisors other than 1, then it is possible to simplify it.
In the case of fifteen% The simplification would be the following:
15% =15100 =320
5 is a common divisor of 15 and 100.
Let’s see how to write 20%, 73% and 120% in fraction form.
20% in fraction form:
20% = 20100 = 420= fifteen
4 and 5 are the common divisors of 20 and 100.
73% in fraction form:
73% = 73100
73 and 100 have no common divisors other than 1, so it is not possible to simplify this fraction.
120% in fraction form:
120% = 120100 = 1210= 65
10 and 2 are common divisors of 120 and 100.
From fraction to percentage
When we have a fraction it is possible to determine the percentage that is associated with it. Let’s see:
If we want to know what is the percentage associated with the fraction 720 , then we multiply the denominator and the numerator by the same number so that we obtain 100 in the denominator. This is:
720= 7x520x5= 35100= 35%
In this way we obtain that 35% is equivalent to 720
In the case of the fraction 1725 We multiply the denominator and the numerator by 4, so that the result in the denominator is 100.
1725= 17x425x4= 28100= 68%
According to the calculations carried out, the fraction 1725 It is equivalent to 68%
Finally, if we want to know what percentage the fraction 15250 is equivalent to, we proceed as follows: Since the denominator is greater than 100, we divide the numerator and the denominator by 5, which is a common divisor of 15 and 250. We obtain the following:
15250= 15:5250:5= 350
Now we multiply the numerator and the denominator by 2, so that we get 100 in the denominator.
15250=15:5250:5= 350= 3 x 250 x 2= 6100= 6%
Graphic representation of a percentage.
As we have already explained, a percentage is a fraction in which the denominator is 100. For example, if we want to graphically represent 12% which is equal to 12100, then we divide a unit into 100 equal parts and select 12 of those parts. Let’s see:
In this graph we have shaded 12 of the 100 parts into which the unit has been divided. This is one way to visualize the 12%.
If we now want to visualize the meaning of 100%, we carry out a procedure similar to the previous one. We divide a unit into 100 equal parts and in this case we select the 100 parts.
When we refer to 100% we are talking about everything or totality. For example, if in a class there are 25 students of which 13 are girls and 12 boys, then we can affirm that 25 is 100% of the students in the class. Another case that is interesting to visualize is when the percentages are greater than 100%. For example, to represent 120% we require more than one unit. Let’s see:
In this case we have selected a complete unit ( 100100) and 20 of the 100 parts ( 20100) from another unit. This means that 120100= 120% is larger than the whole or unit.
Decimal expression of a percentage
As we have already seen, the percentage is a fraction with a denominator equal to 100. Furthermore, we know that Every fraction can be expressed as a decimal numberhence The percentage can be written as a decimal expression.
Let’s look at some examples:
From percentage to decimal
To find out what the decimal expression associated with 28% is, we do the following:
First we write the percentage, in this case 28%, as a fraction
28% = 28100
Then, we divide the numerator by the denominator. In this case it would be:
28 : 100 =0.28
28% = 28100=0.28
In this way we obtain that the decimal associated with 28% is 0.28 in decimal form.
If we want to express 67% in the form of a decimal number, we carry out a process similar to the previous one.
We write 67% in fraction form:
67% = 67100
Now we divide 67 : 100 = 0.67
67% = 67100=0.67
67% expressed as a decimal is 0.67
From decimal to percentage
To express 0.52 as a percentage, the first thing we must do is write the decimal as a fraction. To do this, follow the following steps:
Write the number without the comma in the numerator. In the denominator write the unit followed by as many zeros as there are decimals in the number. In this case, since there are two decimals, we divide by 100. Let’s see:
0.52 = 52100
Next, we apply the definition of percentage.
0.52 = 52100=52%
To write 1.25 as a percentage we carry out a procedure similar to the previous one.
We write 1.25 in fraction form.
1.25 = 125100
Then we apply the definition of percentage.
1.25 = 125100=125%
How to calculate the percentage that a quantity represents.
In this section we explain how to know what percentage a certain amount represents of another amount that has been given.
That is to say, Let’s calculate what percentage it represents of a quantity Dadaist.
To do this we must follow three easy steps:
Step 1: We represent the expression we are working with in fraction form.
In this step we simplify the fraction, only if possible.
xy
Step 2: We calculate the decimal expression of the fraction obtained in step 1.
This is the quotient obtained when performing the operation.
x ÷ y
Step 3: We multiply the decimal expression that we obtained in step 2 by 100.
Let’s look at some examples:
Example 1: What percentage does 20 out of 60 represent?
Step 1: We wrote representation in the form of a fractionand we are left with 20 out of 60 that can be expressed like this:
2060= 26= 13
As you can see, we have also simplified the fraction.
Step 2: Now we calculate the decimal expression of this fraction. To do this, we divide the numerator by the denominator, and we get the following:
13=1 ÷ 3 = 0.333…
Step 3: Finally, We multiply the decimal expression of the fraction by 100. We are left like this:
0.333 x 100 = 33.33%
We can say that 20 represents 33.33% of 60.
Example 2: What percentage is 5 out of 25?
Step 1: We write the given expression in fraction form, and simplify, leaving it like this:
525 = fifteen
Step 2: We calculate the decimal expression of the previous fraction, and we have the following:
fifteen=1 ÷ 5 = 0.2
Step 3: We multiply the previous decimal expression by 100, in this way:
0.2 x 100 = 20%
We are left with 5 being 20% of 25.
Example 3: There are 36 people enrolled in the spinning class at a gym, of which 27 are women. What percentage does the number of women represent of the total number of students enrolled in the class?
This is a simple problem that We can solve by calculating what percentage 27 (number of women) represents of 36 (total number of enrolled in the class).
Step 1: Let’s start by writing the expression in fraction form:
2736 = 3. 4
Step 2: We calculate the decimal expression of the fraction, and we are left with:
3. 4=3 ÷ 4 = 0.75
Step 3: We multiply the decimal expression by 100, and we have:
0.75 x 100 = 75%
For this reason, women represent 75% of the total number of spin class enrollees.
Example 4:In a box of chocolates there are 50 chocolates. Of these 50 chocolates, 15 are filled with peanuts, and they are my favorites. What percentage of the chocolates in the box are not filled with peanuts?
To solve this problem First we have to calculate exactly how many chocolates are not filled with peanuts.
So, we subtract the ones that do have peanuts from the total number of chocolates in the box:
50 – 15 = 35
We know now that There are 35 chocolates that are not filled with peanuts.
Now let’s see What percentage do these 35 peanut-free chocolates represent of the total of 50 chocolates in the box?
Step 1: Let’s represent the previous expression in fraction form and simplify it:
3550 = 710
Step 2: We calculate the decimal expression of the fraction:
710=7 ÷ 10 = 0.7
Step 3: We multiply the decimal expression that we found in the previous step by 100:
0.7 x 100 = 70%
Finally, we know that 70% of the chocolates are not filled with peanuts.
Calculate the percentage of a number.
To calculate the percentage of a number we will use the fractional expression of the percentage we want to calculate, we will also use the idea of proportionality and the decimal expression of percentage.
So we have these two ways to calculate the percentage of a number:
Way 1: We will calculate the percentage of a number with the fractional expression of the percentage and the idea of proportionality.
For example, if we want to calculate 40% of 120, the first thing we do is express the percentage in the form of a fraction, and we are left with:
40100
Then, we will call the quantity that represents the percentage we want to calculate and establish the same relationship as with the previous quantities:
x120
Now we apply the proportionality that exists between these two expressions, because 40 is like 100 it’s 120. We have the following left:
40100=x120
From this expression we obtain that:
x = 40×120100
Now we can calculate the percentage we want:
x = 40×120100
x = 4800100
x = 48
We can conclude that 40% of 120 is 48.
Form 2: We use the fractional expression of the percentage and the decimal expression of the percentage.
We are going to work with the same case that we used in form 1, so you can see that we will obtain the same result. Then we will calculate 40% of 120. The first thing we will do is express the percentage we want to calculate in the form of a fraction:
40% = 40100
Now we are going to calculate the decimal expression of the percentage. To do this, we divide the numerator by the denominator, and we have:
40% = 40100=0.4
We already have the decimal expression of the percentage. Finally, we multiply the amount at which we want to calculate that percentage by the decimal expression of the percentage:
120 x 0.4 = 48
We conclude that 40% of 120 is 48.
We are going to solve a couple of exercises to calculate the percentage of a number.
Exercise 1: Calculate 35% of 225.
Notice how we calculate it with both ways.
Way 1:
We express 35% in fractional form, and apply that same relationship between the quantity that represents the percentage we want to calculate ()…