Percentage

“Percentage” was derived from Latin “per centum,” meaning “by the hundred.” Percentages are ratios of the form 100/x where x is a whole number greater than 100. That is, it is the ratio of part to whole where the value of “whole” is always assumed to be 100.

For instance, suppose marks of a student in maths are 15 out of 50. The respective percentage would be obtained by taking “marks obtained” as a fraction of “total marks” and multiplying it by 100. i.e., percentage of marks = 15 / 50 × 100 = 30%. Discover more about percentages and the process of changing them into fractions and decimals.

The percentage is a ratio or fraction in which the whole value (denominator) is always 100. For instance, if Sam obtained 30% marks in his mathematics test, then it implies that he obtained 30 marks out of 100. It can be expressed as 30/100 in the fraction form and 30:100 in the form of ratio. Here “%” is the symbol of percentage and it is read as “percent” or “percentage”. This percentage sign can always be substituted with “divided by 100” to represent it as a fraction or decimal equivalent.

Examples of Percentage

• 10% = 10/100 ( = 1/10 (or) 0.1)
• 25% = 25/100 ( = 1/4 (or) 0.25)
• 12.5% = 12.5/100 ( = 1/8 (or) 0.125)
• 50% = 50/100 ( = 1/2 ( or )0.5)

Calculate percentage is determining the portion out of the total, i.e., in terms of 100. We have two methods of finding percentage:

• By changing the denominator of the fraction to 100: In this method, we simply find the equivalent fraction of a given fraction so that the new denominator is 100. The numerator itself becomes the percentage. For instance:
  4/25 = 4/25 × 4/4 = 16/100 = 16%
• By using the unitary method: In this approach, we simply multiply the fraction by 100 to obtain the percentage. For instance, the percentage that is equivalent to the fraction 4/25 is:
  4/25 × 100 = 400/25 = 16%

It must be remembered that the first method for finding the percentage is not implied in cases when the denominator isn’t a factor of 100. In that case, we apply the unitary method. Let’s explore how to obtain the percentage applying the two above methods in a detailed manner.

Finding Percentage When the Total is 100

When we have two or more values adding up to 100, the percentage of those individual values to the total value is that number itself. For instance, Sally purchased tiles of three colors for her home. The purchase details are provided in the table below.

Finding Percentage When the Total is NOT 100

As the total number of items sums up to 100, the percentages were quite easily calculated as demonstrated above. What if the total number of items is not equal to 100? Let us proceed.

For instance, Emma has a bracelet consisting of 8 red beads and 12 blue beads. In this case, the total number of beads is 8 + 12 = 20 (which is not equal to 100). In this scenario, the percentages can be calculated as given in the table below (using the unitary method).

But here, the percentages can be calculated by making the denominators 100 too. Then we get

• Percentage of red beads = 8/20 × 5/5 = 40/100 = 40%
• Percentage of blue beads = 12/20 × 5/5 = 60/100 = 60%

Observe the following example which illustrates the superiority of unitary method in comparison to the other method.

Example: How to find the percentage of marks of a student who has obtained 35 out of 40 in math?

Solution: The student has obtained 35/40 marks. However, in the present case, the denominator is not a factor of 100. Therefore, converting percentage in the unitary method is beneficial here.

Percentage of marks = 35/40 × 100 = 87.5%.

The formula for percentage is utilized to determine the portion of a whole in relation to 100. You can express a number as a portion of 100 using this formula. If you look closely, all three methods to obtain the percentage given above can be simply calculated using the formula below:

Percentage = (Value/Total Value)×100

Example: There are 10 girls in a class of 40 children. Then what is the percentage of girls?

Solution: The number of girls here = 10.

Total number of children = 40.

By the formula of percentage,

percentage of girls = 10/40 × 100 = 25%.

As we have already seen, the % symbol can always be replaced with “/100”. The following points should be taken care of while converting percentages into decimals and vice versa.

• to convert percentages into decimals, just replace % with “divided by 100”. For instance, 40% = 40/100 = 0.4.
• to convert decimals into percentages, just multiply by 100. For instance, 0.4 = 0.4 × 100 = 40%.

Percentage change is variation in the value of quantity within a time period in percentage terms. For instance, rise in population, fall in poverty, etc. We do have a formula for representing change in quantity in percentage terms. There are two situations that can occur while determining the percentage change and those are:

• Calculate percentage increase
• Calculate percentage decrease

Percentage Increase

Percentage increase is the percentage change in the value while increasing it over a time interval. For instance, increase in population, increase in the number of bacteria on a surface, etc. Percentage increase can be determined using the following formula:

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

Example: The price of a jacket is raised from $100 to $150. Then by what percentage the price is raised?

Solution: Percentage increase = (150 – 100) / 100 × 100 = 50%.

Percentage Decrease

Percentage decrease is the percentage change in the value if it is lowered over a time period. E.g., decrease in the amount of rainfall, decrease in Covid patients, etc. Percentage decrease can be determined using the following formula:

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

Example: The rainfall has fallen by 127 mm to 103 mm. Then what is the percentage decrease corresponding to this?

Solution: Percentage decrease = (127 – 103) / 127 × 100 = 18.9% (Approximately).

Important Points on Percentages:

• In order to find the percentage of a number out of the total number, simply apply the formula number / total number × 100.
• Either an increase or a decrease in any quantity can be presented as a percentage. This is known as percentage change.
• Fractions can also be changed into percentages and vice-versa. To change the fractions to percentages, multiply by 100. To change percentages to fractions, divide by 100.
• Percentages are reversible. For instance, 50% of 60 is equivalent to 60% of 50.