Percentage

“Percentage” is from Latin “per centum,” i.e., “by the hundred.” Percentages are of the type 100/x where x is an integer larger than 100. I.e., it is part to whole ratio where the value of “whole” is always taken to be 100.

For example, if marks of a student in math are 15 out of 50. The corresponding percentage would be found by taking “marks obtained” as a fraction of “total marks” and then multiplying by 100. i.e., percentage of marks = 15 / 50 × 100 = 30%. Learn more about percentages and the procedure of converting them into fractions and decimals.

The percentage is a fraction or ratio where the value of the whole (denominator) is always 100. For example, if Sam got 30% marks in his maths test, then it means that he got 30 marks out of 100. It can be written as 30/100 in the fractional form and 30:100 in the ratio form. Here “%” is the symbol for percentage and is pronounced as “percent” or “percentage”. The percentage symbol can always be replaced with “divided by 100” in order to represent it in terms of fraction or decimal form.

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 means to find the fraction out of the whole, i.e., in relation to 100. We have two ways of calculating percentage:

• By changing the denominator of the fraction to 100: Here, we just convert the equivalent fraction of a certain fraction in such a way that the new denominator is 100. The numerator alone is the percentage. For example:
  4/25 = 4/25 × 4/4 = 16/100 = 16%
• By using the unitary method: Here, we just multiply the fraction by 100 to get the percentage. For example, the percentage equivalent to the fraction 4/25 is:
  4/25 × 100 = 400/25 = 16%

It should be kept in mind that the first way of determining the percentage is not suggested in situations where the denominator is not a factor of 100. In such a situation, we use the unitary method. Let’s discuss how to get the percentage using the two above methods in detail.

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

Since the sum of the total number of items is 100, the percentages were quite simply calculated as shown above. What if the total number of items does not equal 100? Let us go on.

For example, Emma has a bracelet with 8 red beads and 12 blue beads. Here, the total number of beads is 8 + 12 = 20 (which is not equal to 100). In this case, the percentage can be calculated as shown in the table below (by the unitary method).

It is not necessary to follow this method, but following this, we get the percentage as

But here also, the percentages can be calculated by making the denominators 100. 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%

See the following example which shows the advantage of unitary method over the other method.

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

Solution: The student has scored 35/40 marks. But in the current scenario, the denominator is not a multiple of 100. Hence, converting percentage in the unitary method is useful here.

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

The formula for percentage is used to find the fraction of a whole in relation to 100. You can express a number as a fraction of 100 by this formula. If you observe closely, all three ways to find the percentage mentioned above can easily be calculated by the formula given 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 by “/100”. The following points should be kept in mind while converting percentages to decimals and vice versa.

• to convert percentages to decimals, simply replace % by “divided by 100”. For example, 40% = 40/100 = 0.4.
• to convert decimals to percentages, simply multiply by 100. For example, 0.4 = 0.4 × 100 = 40%.
  

Percentage change is change in the value of quantity over a time interval in percentage terms. For example, increase in population, decrease in poverty, etc. We do have a formula to express change in quantity in percentage terms. There are two cases that can arise while finding the percentage change and those are:

• Calculate percentage increase
• Calculate percentage decrease

Percentage Increase

Percentage increase is the percentage change in the value along with increasing it over a time interval. For example, population increase, the number of bacteria on a surface increase, etc. Percentage increase can be found using the below formula:

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

Example: A jacket’s price is increased from $100 to $150. Then by what percentage the price is increased?

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

Percentage Decrease

Percentage decrease is the percentage change in the value if it is reduced over an interval of time. E.g., reduction in the level of rainfall, reduction in Covid patients, etc. Percentage decrease can be calculated by the following formula:

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

Example: Rainfall has decreased by 127 mm to 103 mm. Then what is the percentage decrease in this?

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

Important Points on Percentages:

• To calculate the percentage of a number out of the total number, just use the formula number / total number × 100.
  
• Either an increase or a decrease in any amount can be expressed as a percentage. This is called percentage change.
• Fractions can also be converted to percentages and vice versa. Multiply by 100 to convert the fractions into percentages. Divide by 100 to convert percentages to fractions.
• Percentages are reversible. For example, 50% of 60 equals 60% of 50.