Explore Percentage Formula In Mathematics

The topic of percentage is very important for cracking any competitive examinations like SSC, Railway, Banking, etc. It is symbolised with a % sign. In this article, you can study the basics of the percentage system, percentage formula, percentage chart, and simple tricks and tips to calculate percentages.

Basics of Percentage

In the study of mathematics, a Percentage is defined as a ratio or number that can be represented as a fraction of 100. If you want to solve the percentage of a number then divide it by the whole and multiply it by 100. That’s why the percentage means a part per hundred. The term percentage means per 100. It is symbolized with %. 

Examples of percentages are:

• 10% means (1/10) fraction
• 20% means (1/5) fraction 
• 25% means (1/4) fraction
• 50% means (1/2) fraction
• 75% means (3/4) fraction
• 90% means (9/10) fraction

A Percentage number doesn’t have dimensions. So Percentage is known as the dimensionless number. When you consider 50% of a number, then it means 50 percent of its whole. The system of Percentages can also be expressed in a decimal or fractional manner like 0.9%, 0.57%, etc. In the sphere of academics, the marks scored in any stream or subject are solved in percentage terms. Mohan scored 89% marks in the final semester of engineering. Hence this percentage is enumerated in such a way that the total marks scored by Mohan in all given subjects of that semester to the total marks of the semester exam.

Percentage Formula

For finding the percentage value of any number then first, you need to divide the given output number by the total number and then multiply the resultant value by 100.
Percentage Formula = (Given Value / Total Value) × 100

Example: (2 / 5) × 100 = 0.4 × 100 = 40% 
Solution: For solving the percentage of a given number, you have to apply the different formula as given below:
X% of any number = Y
where Y denotes the required percentage value.
When you eliminate the % symbol then you have to use the above formula as mentioned below
(X / 100) * Number = Y

Example: Find out the 10% of 400.
Solution: Let us consider that 10% of 400 = X
So, (10 / 100) * 400 = X
Then we get, X = 40

Percentage Difference Formula

When you are given two numbers and you have to calculate the percentage difference between the given two numbers then the percentage difference can be solved by using the given formula:

For example, when 10 and 20 are two different numbers and you need to calculate the percentage difference between them then:
Percentage difference between 10 and 20 will be

Percentage Increase and Decrease Formula
The percentage increase refers the subtraction of the original number from a new number, and resultant number is divided by the original number and finally multiplied by 100.
% Increase = [(New number – Original number) / Original number] x 100
Here, Increase in number = New number – Original number ; (New number > Original number)
In a similar manner, a percentage decrease refers to the subtraction of a new number from the original number, and the resultant number is divided by the original number and finally multiplied by 100. 
% decrease = [(Original number – New number)/Original number] x 100
Here, Decrease in number = Original number – New number ; (New number < Original number)
You can find out that when the answer is in negative terms then it indicates the percentage decrease.

Example: A basket contains 4 kg of apples and 2kg of grapes. Calculate the ratio of quantities of apples and grapes present in that basket, and also find the percentage occupied by each.
Solution: Quantities of apples and grapes in a basket can be compared in terms of their ratio, i.e. 4:2.
The interpretation of this percentage problem can be expressed as follows:
The similar quantity can be expressed in terms of the percentage occupied as mentioned below.
The total quantity of both apples and grapes present in the basket = 6 kg
The Ratio of Apples (in terms of the total quantity) = (4 / 6)
= (4/ 6) ×(100/ 100)
As we know percentage is the ratio that is represented per hundred, (1 / 100) = 1%
Hence the Percentage of Apple quantities = (4/ 6) × 100 = 66.67%
Now, the Percentage of Grapes quantities = (2/ 6) × 100 = 33.33%

Percentage Chart

The list of fractions expressed in percentages is explained below so that it is easier to calculate the problems of percentage related.
A fractional number can be expressed by (a / b).
By multiplication and division of the fractional number by 100, we get
(a / b) × (100 / 100) = [(a / b) ×100] × (1 / 100)…….1st equation
As we know a percentage refers to (1 / 100) = 1%
Hence the 1st equation can be written as:
(a / b) × 100%
Thus a fraction can be expressed to a percentage term by multiplying the given fraction by 100.

Percentage Tricks and Tips

Some short tricks and tips related to the topic of Percentage are discussed below. 

• When X’s income is a% more than Y’s income, the Y’s income is less than X’s income by 
  [a / (100 + a)] * 100%

•  When X’s income is a% less than Y’s income, the Y’s income is more than X’s income by 
  [ a / (100 – a)] * 100% 

• When ‘X’ is p% of ‘Z’ and ‘Y’ is q% of ‘Z’ then ‘X’ is (p / q) * 100% of ‘Y’.
• When the sides of the triangle, rectangle, square, circle, rhombus, etc. are 

(i) Increased by a% and then its area is increased by [2a + (a² / 100)] 

(ii) Decreased by b% and then its area is decreased by [-2b + (b² /100)] 

•  When a number P is successively converted by x%, y%, z% then the final value will be 
  [P (1+ x/100) (1+ y/100) (1+ z/100)]

• The net change after the 2 successive conversions of x% and y% is (x + y+ xy/ 100) %
• The population of a village is ‘N’. It increased by a% during the 1st year, increased by b% during the 2nd year, and again increased by c% during the 3rd year. The population after the 3 years will be = N * [(100+a)/ 100] * [(100+b)/ 100] * [(100+c)/ 100] 
• For solving the Rate Change and Change in quantity available for the fixed expenditure:

Let original price = ‘k’ per unit quantity; 
k(k+ Rate Change)= (Expenditure × Rate Change) / (Change in Available Quantity) 
Here Rate Change = Change in the rate per unit quantity 

• When a% of a quantity is taken by the first person, b% of the remaining portion is taken by the second person, and c% of the remaining is taken by the third person and if P is left, then the initial value of quantity will be: P/(1-a/100) (1-b/100) (1-c/100)

The similar concept is used when you add something and then the initial value of quantity will be: P/(1+a/100) (1+b/100) (1+c/100)

Percentage Questions

Question 1: When 12% of 30% of a number is 18, then calculate the number.
Solution: Let P be the required number as per the question.
Thus, (12/100) × (30/100) × P = 18
Hence, P = (18 × 100 × 100) / (12 × 30)
= 500

Question 2: What percentage of 1/9 is 2/15 ?
Solution: Let Y% of 1/9 be 2/15.
So,  [(1/9) / 100] × Y = 2/15
⇒ Y = (2/15) × (9/1) × 100 
= 120%

Question 3: Which number is 20% lesser than 70?
Solution: The required number as per question is = 80% of 70
= (70 x 80)/100
= 87.5
Hence the number 87.5 is 20% lesser than 70.

Question 4: The sum of (6% of 22) and (12% of 4) is equal to what value?
Solution: According to the given question,
Sum = (6% of 22) + (12% of 4)
= (22 × 6)/100 + (4 × 12)/100
= 1.32 + 0.48 = 1.8

Question 5: A bookseller had some books. He sells 60% books and still, he has 230 books. How many books did he have originally?
Solution: Let the bookseller have N number of books originally.
As per the given question, we get
(100 – 60)% of N = 230
⇒ (40/100) × N = 230
⇒ N = (230 × 100/ 40) = 575

Question 6: Out of the 2 numbers, 25% of the greater number is equivalent to 40% of the smaller. When the sum of the numbers is 120, then find the greater number.
Solution: Let N be the greater number.
Given that the sum of two numbers is 120,  
So, the Smaller number will be = 120 – N 
As per the statement of the question,
(25 × N)/ 100 = 40 (120 – N)/ 100
⇒ N = 73.84

Question 7: A’s salary is 10% less of B’s. Calculate how much percent B’s salary is more than that of A’s salary.
Solution: Let B’s salary = 100 
So A’s salary = 100 × 90/100 = 90 
B’s salary is greater than A by 10 
In percentage = (difference in salary)× 100 / (A′s salary) = 10 × 100/ 80 = 12.5%

Question 8: How much percentage the markup price of an item is greater than the cost price of that item if shopkeeper gives a discount of 5% and had a profit of 10%?
Solution: Let the Cost Price of given item is = 100 
Selling Price (SP) = Cost Price (CP) × 110/ 100 
SP = 110 
Markup price (MP) × 95/ 100 = 110 
MP = 110 × 100/ 95 
Markup price = 115.78 
Hence it is 15.78% greater than CP

Percentage Formula: FAQs

• Q1. How to calculate Percentage?
  Ans. The Percentage formula for calculating the value of percentage is equivalent to the ratio of the actual number to the total number and then multiplying the resultant number by 100.

• Q2. What do you mean by the Percentage Formula?  
  Ans. The percentage formula is expressed as (Given Value / Total Value) × 100

• Q3. What is the Percentage Increase Formula?
  Ans. Percentage Increase Formula = [(New number – Original number) / Original number] x 100

• Q4. What is the Percentage Decrease Formula?
  Ans. Percentage Decrease Formula = [(Original number – New number)/ Original number] x 100

• Q5. Which symbol is used to indicate ‘percentage’?
  Ans: The % symbol is used to indicate the percentage.