In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”. 

Percentages

The term “percentage” was adapted from the Latin word “per centum”, which means “by the hundred”. Percentages are fractions with 100 as the denominator. In other words, it is the relation between part and whole where the value of whole is always taken as 100.

Percentage is a fraction or a ratio in which the value of whole is always 100. For example, if Sam scored 30% marks in his math test, it means that he scored 30 marks out of 100. It is written as 30/100 in the fraction form and 30:100 in terms of ratio.

Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol “%”.

Examples of percentages are:

• 10% is equal to 1/10 fraction
• 20% is equivalent to 1/5 fraction
• 25% is equivalent to 1/4 fraction
• 50% is equivalent to 1/2 fraction
• 75% is equivalent to 3/4 fraction
• 90% is equivalent to 9/10 fraction

Percentages have no dimension. Hence it is called a dimensionless number. If we say, 50% of a number, then it means 50 per cent of its whole.

Calculating percentage means to find the share of a whole, in terms of 100. There are two ways to find a percentage:

• By using the unitary method.
• By changing the denominator of the fraction to 100.

It should be noted that the second method for calculating percentage is not used in situations where the denominator is not a factor of 100. For such cases we use the unitary method.

The percentage formula is used to find the share of a whole in terms of 100. Using this formula, you can represent a number as a fraction of 100. If you observe carefully, all the three ways to get percentage shown above can be easily calculated by using the formula given below:

Percentage = (Value/Total Value) × 100

Example: 2/5 × 100 = 0.4 × 100 = 40 per cent

How to calculate the percentage of a number?

To calculate the percentage of a number, we need to use a different formula such as:

P% of Number = X

where X is the required percentage.

If we remove the % sign, then we need to express the above formulas as;

P/100 x Number = X

Example: Calculate 10% of 80.

Let 10% of 80 = X

10/100 x 80 = X

X = 8

Percentage difference is the change in the value of a quantity over a period of time in terms of percentage. Sometimes we need to know the increase or decrease in some quantity as percentages, which is also referred to as Percentage Change. For example, an increase in population, a decrease in poverty, and so on.

We have the formula to show the change in quantity as a percentage. There are two cases that might arise while calculating percentage difference and those are:

• Calculate percentage increase
• Calculate percentage decrease

Percentage increase refers to the per change change in the value when it is increased over a period of time. For example, population increase, increase in the number of bacteria on a surface, etc. Percentage increase can be calculated by using the following formula:

Percentage Increase = (Increased Value-Original value)/Original value × 100

Percentage decrease refers to the per change change in the value when it is decreased over a period of time. For example, decrease in the level of rainfall, decrease in the number of Covid patients, etc. Percentage decrease can be calculated by using the following formula:

Percentage Decrease = (Original value-Decreased Value)/Original Value × 100

Example: 

Let a bag contain 2 kg of apples and 3kg of grapes. Find the ratio of quantities present, and the percentage occupied by each.

Solution:

The number of apples and grapes in a bag can be compared in terms of their ratio, i.e. 2:3.

The actual interpretation of percentages can be understood as follows:

The same quantity can be represented in terms of the percentage occupied, which can be done as given below.

Total quantity present = 5 kg

Ratio of apples (in terms of total quantity) = 2/5

From the definition of percentage, it is the ratio that is expressed per hundred,

(1/100) = 1%

Thus, Percentage of Apples = (2/5) × 100 = 40%

Percentage of Grapes = (3/5) × 100 = 60%

Example: 

If 16% of 40% of a number is 8, then find the number.

Solution:

Let X be the required number.

Therefore, as per the given question, 

(16/100) × (40/100) × X = 8

So, X = (8 × 100 × 100) / (16 × 40)

= 125

Example:

Which number is 40% less than 90?

Solution:

Required number = 60% of 90

= (90 x 60)/100

= 54

Therefore, the number 54 is 40% less than 90.

Every percentage problem has three possible unknowns or variables :

• Percentage

• Part

• Base

In order to solve any percentage problem, you must be able to identify these variables.

Look at the following examples. All three variables are known:

Example: 70% of 30 is 21

70 is the percentage.

30 is the base.

21 is the part.

Example: 25% of 200 is 50

25 is the percent.

200 is the base.

50 is the part.

Points to Remember:

• To find the percentage of a whole, work out the value of 1% and then multiply it by the percent we need to find.
• An increase or decrease in any quantity can be expressed as a percentage.
• Fractions can be converted into percentages and vice-versa.
• Percentages are reversible. For example, 25% of 40 is the same as 40% of 25.