Percentage

A percentage is simply a fraction or ratio where the whole is always 100. For example, if Sam scored 30% in his math test, it means he scored 30 out of 100. In fraction form, this is written as \(\frac{30}{100}\), and in ratio form, it is 30:100. The symbol “%” is read as “percent” or “percentage” and can be written as a fraction or a decimal.
Understanding percentages helps with many things, such as using a percentage calculator, learning how to find and calculate percentages, understanding percentage change, percentage difference, and percentage increase, and converting fractions such as \(\frac{16}{20}\), \(\frac{17}{20}\), \(\frac{15}{20}\), \(\frac{14}{20}\), and \(\frac{18}{20}\) into percentages. You can also learn how to find the percentage of a number using simple percentage formulas.

To become better at percentage calculation, we need to know all percentage formulas. The most basic formula compares the actual value to the total value and then multiplies the result by 100. This gives us the percentage, and it can be written as,

\(\ \text{Percentage} = \left( \frac{\text{Actual Value}}{\text{Total Value}} \right) \times 100 \ \)

For example, \(\ \frac{3}{5} \times 100 = 0.6 \times 100 = 60\% \ \)

Calculating a percentage means finding how much a part is out of 100. We can do this in two simple ways, which are as follows:

Step 1: Changing the denominator to 100:
We convert the given fraction into an equivalent fraction with 100 as the denominator. Then the numerator becomes the percentage. For example, \(\ \frac{3}{20} = \frac{3}{20} \times \frac{5}{5} = \frac{15}{100} = 15\% \ \)

Step 2: Using the unitary method:
We multiply the fraction by 100 to get the percentage. For example, \(\ \frac{3}{20} \times 100 = \frac{300}{20} = 15\% \ \). Remember, the first method works only when the denominator can be easily changed to 100. If not, the unitary method is the better choice. Now, let’s look at how to find percentages using both methods in detail.

When different values added up to 100, each value itself represents its percentage of the total. This makes it easy to convert them into fractions and percentages because the total is already 100. Here’s a new example to understand this better:
For example,
 

In this case, since the total number is 100, each value directly becomes its percentage.

When the total adds up to 100, finding percentages is easy, as each value directly represents its percentage. But what do we do when the total is not 100? For example.
Emma has a bracelet with 8 red beads and 12 blue beads. The total number of beads is 8 + 12 = 20, which is not equal to 100, so we need to calculate the percentages differently.
 

Even when the total is not 100, we can still convert the fractions so that their denominators become 100.
 

Percentage of red beads:
 

\(\ \frac{8}{20} \times \frac{5}{5} = \frac{40}{100} = 40\% \ \)

Percentage of blue beads:

\(\ \frac{12}{20} \times \frac{5}{5} = \frac{60}{100} = 60\% \ \)

Now, for example 
How do we find the percentage of marks for a student who scored 35 out of 40 in math?
 

The fraction is \(\frac{35}{40}\), but 40 is not a factor of 100. So, it’s easier to use the unitary method. Percentage of marks = \(\frac{35}{40}\) \(× 100 = 87.5%\)
 

Percentage difference tells us how much a value has changed compared to another value, expressed as a percentage. To find the percentage difference between two numbers, follow these steps:
 

Step 1: Find the difference between the two values (subtract them and ignore any negative sign).

Step 2: Find the average of the two values (add them and divide by 2).

Step 3: Divide the difference by the average.

Step 4: Convert the result into a percentage by multiplying it by 100.

\(\ \text{Percentage Difference} = \left| \frac{\text{First Value} - \text{Second Value}}{\frac{\text{First Value} + \text{Second Value}}{2}} \right| \times 100\% \ \)

Percentage Increase and Decrease Formula

When working with percentage difference, we usually come across two situations:
 

• Percentage increase
• Percentage decrease

Here, subtract the original value from the new value, then divide that result by the original value. Finally, multiply by 100 to convert it into a percentage.
 

\(\ \text{Percentage Increase} = \frac{\text{Rise in the Number}}{\text{Original Number}} \times 100\% \ \)

Increase in value = New number - Original number

Similarly, to find the percentage decrease, subtract the new number from the original number. Then divide that result by the original value, and multiply by 100 to express it as a percentage.

\(\ \text{Percentage Decrease} = \frac{\text{Decrease in the Number}}{\text{Original Number}} \times 100\% \ \)

Decrease in value = Original number - New number

Remember, if the new value is higher than the original value, it represents a percentage increase. If the new value is lower, it represents a percentage decrease.

 

A ratio compares two numbers.

A fraction represents a part of a whole.

A percent shows how many parts out of 100.

A decimal is another way to express a fraction.
 

Let us know some of the important definitions related to the percentage.
 

• It is the price printed or tagged on a product.
  
   
• It is also referred to as the list price or tag price.
  
   
• If no discount is applied, the selling price will be the same as the marked price.
  
   
• When a product is sold for more than its cost price, the difference is called profit.
  
   
• When a product is sold for less than its cost price, the difference between the two amounts is known as a loss.
   

A quick guide to help students understand and calculate percentages easily.
 

• Convert percentages to decimals or fractions to make calculations easier.
  
   
• Remember that a percentage always means “out of 100”.
  
   
• Break difficult percentages into smaller, easy parts for quick mental math.
  
   
• Learn common percentages in daily life like discounts, taxes, and profit/loss.
  
   
• Parents should encourage children to relate percentages to real-life examples, such as shopping discounts or exam scores.
  
   
• Teachers must teach students to break complex percentages into smaller parts.

Percentages are used in daily life for calculating discounts, interest rates, and profit or loss. Here are some of the real applications of percentage.

Shopping discounts and sales: Stores use percentages to show price reductions.


Interest rates in banks: Both savings (earnings) and loans (repayments) are calculated in percentages.


Exams and results: Marks are often expressed as a percentage of the total score.


Tax calculation: Income tax, GST, or VAT are all charged based on a percentage of income or price.


Statistics in news and reports: Percentages are used to represent unemployment rates, literacy rates, population growth, etc.