Percentage is one of the most useful topics in mathematics. We use it in school exams, government job exams, banking, shopping, business, marks calculation, profit and loss, discount, tax, population, salary, interest, and data interpretation. The word percentage means per hundred. So, when we say 25 percent, it means 25 out of 100. It is written as 25%.
This topic becomes easy when we understand one simple idea: every percentage is a fraction with denominator 100. For example, 40% means 40/100. In decimal form, it is 0.40. In simplified fraction form, it is 2/5. In the same way, 75% means 75/100, which is 3/4.
The existing rule is: to express x% as a fraction, write x/100 and simplify it if possible. Thus, 3% = 3/100 = 0.03, and 48% = 48/100 = 12/25 = 0.48. This is the base of almost every percentage question.
What is percentage?
Percentage is a way of comparing a number with 100. If a student scores 80 marks out of 100, we say the student scored 80%. If a student scores 45 marks out of 50, then first we convert the marks into out of 100. The percentage is 45/50 × 100 = 90%.
Percentage helps us compare values easily. Suppose one exam is of 50 marks and another exam is of 200 marks. Direct marks may confuse us. But percentage changes both results into a common base of 100, so comparison becomes easy.
Basic percentage formula
The basic formula is:
Percentage = Value / Total value × 100
Here, value means the part we are talking about, and total value means the whole amount.
Example: A student gets 72 marks out of 90. Find the percentage.
Percentage = 72 / 90 × 100
Percentage = 80%
So, the student scored 80% marks.
How to convert percentage into fraction
To convert a percentage into a fraction, divide it by 100 and simplify.
| Percentage | Fraction form | Simplified form |
|---|---|---|
| 10% | 10/100 | 1/10 |
| 20% | 20/100 | 1/5 |
| 25% | 25/100 | 1/4 |
| 40% | 40/100 | 2/5 |
| 50% | 50/100 | 1/2 |
| 75% | 75/100 | 3/4 |
| 80% | 80/100 | 4/5 |
Example: Convert 64% into a fraction.
64% = 64/100
Divide numerator and denominator by 4.
64/100 = 16/25
So, 64% = 16/25.
How to convert fraction into percentage
To express a/b as a percentage, multiply it by 100.
a/b as percentage = a/b × 100%
Example: Convert 1/4 into percentage.
1/4 × 100 = 25%
So, 1/4 = 25%.
Example: Convert 3/5 into percentage.
3/5 × 100 = 60%
So, 3/5 = 60%.
How to convert decimal into percentage
To convert a decimal into percentage, multiply it by 100.
Example: Convert 0.35 into percentage.
0.35 × 100 = 35%
Example: Convert 1.25 into percentage.
1.25 × 100 = 125%
A percentage can be more than 100%. It means the value is more than the original whole.
How to convert percentage into decimal
To convert percentage into decimal, divide by 100.
Example: 18% = 18/100 = 0.18
Example: 125% = 125/100 = 1.25
Finding percentage of a number
To find the percentage of a number, convert the percentage into a fraction or decimal and multiply by the number.
Example: Find 20% of 450.
20% = 20/100 = 1/5
20% of 450 = 450 × 1/5 = 90
So, 20% of 450 is 90.
Example: Find 35% of 800.
35% of 800 = 35/100 × 800
35 × 8 = 280
So, 35% of 800 is 280.
Finding the whole when percentage is given
Sometimes the question gives a percentage value and asks for the total value.
Example: 30% of a number is 150. Find the number.
Let the number be x.
30% of x = 150
30/100 × x = 150
x = 150 × 100 / 30 = 500
So, the number is 500.
Percentage increase
Percentage increase is used when a value becomes greater than before. It is common in price increase, salary increase, population increase, production increase, and marks improvement.
Percentage increase = Increase / Original value × 100
Example: The price of a book increases from Rs. 200 to Rs. 250. Find the percentage increase.
Increase = 250 – 200 = 50
Percentage increase = 50 / 200 × 100 = 25%
So, the price increased by 25%.
Percentage decrease
Percentage decrease is used when a value becomes smaller than before. It is common in discount, population decrease, depreciation, marks decrease, and reduction in expenditure.
Percentage decrease = Decrease / Original value × 100
Example: The price of a bag decreases from Rs. 1200 to Rs. 900. Find the percentage decrease.
Decrease = 1200 – 900 = 300
Percentage decrease = 300 / 1200 × 100 = 25%
So, the price decreased by 25%.
Price increase and required reduction in consumption
This is an important formula already included in the old content. If the price of a commodity increases by R%, then the reduction in consumption required so that expenditure does not increase is:
Required reduction = R / (100 + R) × 100%
Example: The price of rice increases by 25%. By what percentage should a family reduce consumption so that total expenditure remains the same?
Required reduction = 25 / (100 + 25) × 100
Required reduction = 25 / 125 × 100 = 20%
So, the family should reduce rice consumption by 20%.
Price decrease and possible increase in consumption
If the price of a commodity decreases by R%, then the increase in consumption possible without increasing expenditure is:
Possible increase = R / (100 – R) × 100%
Example: The price of sugar decreases by 20%. By what percentage can consumption increase without changing the total expenditure?
Possible increase = 20 / (100 – 20) × 100
Possible increase = 20 / 80 × 100 = 25%
So, consumption can increase by 25%.
Successive percentage change
When two percentage changes happen one after another, we should not simply add them in every case. Use the formula:
Net change = a + b + ab/100
Here, a and b are percentage changes. Increase is taken as positive and decrease is taken as negative.
Example: A value increases by 20% and then decreases by 10%. Find the net change.
Net change = 20 – 10 + (20 × -10)/100
Net change = 10 – 2 = 8%
So, the final value is 8% more than the original value.
Example: A price decreases by 10% and then again decreases by 20%.
Net change = -10 – 20 + (-10 × -20)/100
Net change = -30 + 2 = -28%
So, the final price is 28% less than the original price.
Percentage and marks
Percentage questions based on marks are very common in school and competitive exams.
Example: A candidate scores 384 marks out of 600. Find the percentage.
Percentage = 384 / 600 × 100 = 64%
Example: A student needs 40% marks to pass an exam of 750 marks. How many marks are required to pass?
Required marks = 40/100 × 750 = 300
So, 300 marks are required to pass.
Percentage and population
The old content also included population increase. If the present population is P and it increases at the rate of R% per annum, then:
Population after n years = P × (1 + R/100)^n
Population n years ago = P / (1 + R/100)^n
Example: The present population of a town is 50,000. It increases by 10% every year. Find the population after 2 years.
Population after 2 years = 50000 × (1 + 10/100)^2
= 50000 × 1.1 × 1.1
= 60,500
So, the population after 2 years will be 60,500.
Example: The present population of a town is 72,600. It increased by 10% per year for 2 years. Find the population 2 years ago.
Population 2 years ago = 72600 / (1.1 × 1.1)
= 72600 / 1.21 = 60,000
So, the population 2 years ago was 60,000.
Percentage and depreciation
Depreciation means decrease in the value of an asset over time. Machines, vehicles, furniture, and electronic items usually depreciate every year.
If the present value of a machine is P and it depreciates at R% per annum, then:
Value after n years = P × (1 – R/100)^n
Value n years ago = P / (1 – R/100)^n
Example: A machine costs Rs. 80,000 and depreciates by 10% per year. Find its value after 2 years.
After first year, value = 80000 × 90/100 = 72000
After second year, value = 72000 × 90/100 = 64800
So, the value after 2 years is Rs. 64,800.
A is R% more than B
This is another important formula from the old content. If A is R% more than B, then B is less than A by:
R / (100 + R) × 100%
Example: A is 25% more than B. By what percentage is B less than A?
B is less than A by 25 / (100 + 25) × 100
= 25 / 125 × 100 = 20%
So, B is 20% less than A.
A is R% less than B
If A is R% less than B, then B is more than A by:
R / (100 – R) × 100%
Example: A is 20% less than B. By what percentage is B more than A?
B is more than A by 20 / (100 – 20) × 100
= 20 / 80 × 100 = 25%
So, B is 25% more than A.
Percentage and profit-loss
Percentage is used in profit and loss questions. Profit percentage and loss percentage are usually calculated on cost price.
Profit percentage = Profit / Cost price × 100
Loss percentage = Loss / Cost price × 100
Example: A man buys a chair for Rs. 800 and sells it for Rs. 1000. Find the profit percentage.
Profit = 1000 – 800 = 200
Profit percentage = 200 / 800 × 100 = 25%
So, the profit is 25%.
Percentage and discount
Discount is a reduction in marked price. Discount percentage is calculated on marked price.
Discount percentage = Discount / Marked price × 100
Example: The marked price of a shirt is Rs. 1500 and the shopkeeper gives a discount of 20%. Find the selling price.
Discount = 20% of 1500 = 300
Selling price = 1500 – 300 = 1200
So, the selling price is Rs. 1200.
Percentage and salary
Salary increment questions are also based on percentage increase.
Example: A person gets a salary of Rs. 30,000 per month. His salary increases by 12%. Find the new salary.
Increase = 12% of 30000 = 3600
New salary = 30000 + 3600 = 33600
So, the new salary is Rs. 33,600.
Percentage and data interpretation
In government job exams, percentage is used in data interpretation questions based on tables, charts, and reports.
Example: In a village, there are 1200 people. 45% are women. Find the number of women.
Women = 45% of 1200 = 45/100 × 1200 = 540
So, there are 540 women.
Example: A company has 800 employees. 35% work in the sales department. Find the number of sales employees.
Sales employees = 35% of 800 = 280
So, 280 employees work in sales.
Government job exam type examples
These examples are similar to the type of percentage questions asked in SSC, Railway, Banking, Police, State PSC, and other competitive exams.
Example 1: Exam marks
A candidate scored 420 marks out of 600. What is his percentage?
Percentage = 420 / 600 × 100 = 70%
Answer: 70%
Example 2: Passing marks
In an exam, a candidate needs 33% marks to pass. If the total marks are 900, find the passing marks.
Passing marks = 33/100 × 900 = 297
Answer: 297 marks
Example 3: Failed by marks
A student got 240 marks and failed by 30 marks. If passing percentage is 45%, find the total marks.
Passing marks = 240 + 30 = 270
45% of total marks = 270
Total marks = 270 × 100 / 45 = 600
Answer: 600 marks
Example 4: Population increase
The population of a town is 40,000. It increases by 5% in one year. Find the new population.
Increase = 5% of 40000 = 2000
New population = 40000 + 2000 = 42000
Answer: 42,000
Example 5: Price rise and consumption
The price of petrol increases by 25%. By what percentage should consumption be reduced so that expenditure remains the same?
Reduction = 25 / 125 × 100 = 20%
Answer: 20%
Example 6: Successive discount
A shopkeeper gives two successive discounts of 10% and 20%. Find the single equivalent discount.
Net change = -10 – 20 + 2 = -28%
Answer: 28% discount
Example 7: Comparison of two numbers
Ravi’s income is 25% more than Mohan’s income. By what percentage is Mohan’s income less than Ravi’s income?
Required percentage = 25 / 125 × 100 = 20%
Answer: 20%
Example 8: Voters question
In an election, 60% of voters voted. If the total number of voters is 25,000, find the number of voters who did not vote.
Voted people = 60% of 25000 = 15000
Not voted = 25000 – 15000 = 10000
Answer: 10,000
Example 9: Salary reduction and increase
A salary is reduced by 20%. By what percentage should the reduced salary be increased to get the original salary?
Required increase = 20 / (100 – 20) × 100
= 20/80 × 100 = 25%
Answer: 25%
Example 10: Production target
A factory produces 12,000 items in a month. Next month production increases by 15%. Find the new production.
Increase = 15% of 12000 = 1800
New production = 12000 + 1800 = 13800
Answer: 13,800 items
Quick percentage tricks
Some percentages can be calculated quickly by using fractions.
| Percentage | Easy meaning |
|---|---|
| 1% | 1/100 |
| 5% | 1/20 |
| 10% | 1/10 |
| 12.5% | 1/8 |
| 20% | 1/5 |
| 25% | 1/4 |
| 33.33% | 1/3 approximately |
| 50% | 1/2 |
| 66.66% | 2/3 approximately |
| 75% | 3/4 |
Example: Find 25% of 640.
25% means 1/4.
640 / 4 = 160
So, 25% of 640 is 160.
Example: Find 12.5% of 400.
12.5% means 1/8.
400 / 8 = 50
So, 12.5% of 400 is 50.
Common mistakes in percentage
The first mistake is calculating percentage on the wrong base. For example, profit percentage is calculated on cost price, not selling price. Discount percentage is calculated on marked price, not selling price.
The second mistake is adding successive percentage changes directly. An increase of 20% followed by a decrease of 20% does not make zero change. If a value is 100, after 20% increase it becomes 120. After 20% decrease on 120, it becomes 96. So, the net result is 4% decrease.
The third mistake is confusing “more than” and “less than” comparison. If A is 25% more than B, then B is not 25% less than A. B is 20% less than A because the base changes.
Practice questions
- Find 15% of 600.
- Convert 3/8 into percentage.
- Convert 0.72 into percentage.
- A student scores 360 out of 500. Find the percentage.
- The price of an item increases from Rs. 400 to Rs. 500. Find the percentage increase.
- The value of a machine decreases from Rs. 50,000 to Rs. 40,000. Find the percentage decrease.
- 25% of a number is 80. Find the number.
- A price is increased by 10% and then decreased by 10%. Find the net percentage change.
- If A is 50% more than B, by what percentage is B less than A?
- If A is 25% less than B, by what percentage is B more than A?
Answers
- 90
- 37.5%
- 72%
- 72%
- 25%
- 20%
- 320
- 1% decrease
- 33.33%
- 33.33%
Frequently asked questions
What is the meaning of percentage?
Percentage means per hundred. It is used to express a number as a part of 100.
What is the formula of percentage?
The formula is: Percentage = Value / Total value × 100.
How do we convert percentage into fraction?
Divide the percentage by 100 and simplify the fraction. For example, 60% = 60/100 = 3/5.
How do we convert fraction into percentage?
Multiply the fraction by 100. For example, 2/5 × 100 = 40%.
Can percentage be more than 100?
Yes. A percentage more than 100 means the value is more than the original whole. For example, 150% means 1.5 times the original value.
What is the difference between percentage increase and percentage decrease?
Percentage increase shows how much a value has increased compared with the original value. Percentage decrease shows how much a value has decreased compared with the original value.
Why is percentage important for government exams?
Percentage is used in many competitive exam topics such as profit and loss, simple interest, compound interest, discount, population, data interpretation, marks, salary, and ratio comparison. A strong percentage base makes these topics easier.
What is the easiest way to calculate percentage?
For common percentages, use fractions. For example, 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10, and 5% = 1/20.
More mixed examples for government exams
In competitive exams, percentage questions are often mixed with other topics. The question may look like marks, salary, price, ratio, population, or discount, but the basic idea remains the same. First find the original value, then find the required percentage or changed value.
Example 11: Number increased by percentage
A number is increased by 30% and becomes 260. Find the original number.
Let the original number be x.
After 30% increase, the value becomes 130% of x.
130% of x = 260
x = 260 × 100 / 130 = 200
Answer: 200
Example 12: Number decreased by percentage
A number is decreased by 15% and becomes 170. Find the original number.
After 15% decrease, the value becomes 85% of the original number.
85% of x = 170
x = 170 × 100 / 85 = 200
Answer: 200
Example 13: Percentage based on ratio
The ratio of boys and girls in a class is 3 : 2. Find the percentage of girls in the class.
Total parts = 3 + 2 = 5
Girls part = 2
Percentage of girls = 2/5 × 100 = 40%
Answer: 40%
Example 14: Water and milk mixture
A mixture contains 80 litres of milk and water. If milk is 75% of the mixture, find the quantity of water.
Milk = 75% of 80 = 60 litres
Water = 80 – 60 = 20 litres
Answer: 20 litres
Example 15: Expenditure and saving
A man earns Rs. 40,000 per month and saves 20% of his income. Find his monthly expenditure.
Saving = 20% of 40000 = 8000
Expenditure = 40000 – 8000 = 32000
Answer: Rs. 32,000
Example 16: Percentage of valid votes
In an election, 20,000 votes were cast. 5% votes were declared invalid. A candidate got 60% of the valid votes. Find the votes received by the candidate.
Invalid votes = 5% of 20000 = 1000
Valid votes = 20000 – 1000 = 19000
Candidate votes = 60% of 19000 = 11400
Answer: 11,400 votes
Example 17: Department selection
In a recruitment exam, 12,000 candidates appeared. 18% qualified for the next round. Find the number of candidates who did not qualify.
Qualified candidates = 18% of 12000 = 2160
Not qualified = 12000 – 2160 = 9840
Answer: 9,840 candidates
Example 18: Government exam cut-off
The maximum marks in an exam are 500. The cut-off is 62%. How many marks are needed to clear the cut-off?
Required marks = 62% of 500
= 62/100 × 500 = 310
Answer: 310 marks
How to decide which formula to use
If the question asks “what percent,” use Value / Total × 100. If the question says “increased from old value to new value,” use Increase / Original × 100. If the question says “decreased from old value to new value,” use Decrease / Original × 100. If the question asks for a price-consumption adjustment, use R / (100 + R) × 100 for price increase and R / (100 – R) × 100 for price decrease.
For “A is more than B” and “B is less than A” type questions, remember that the base changes. This is why the reverse percentage is not the same. If A is 25% more than B, then B is 20% less than A. If A is 20% less than B, then B is 25% more than A.
Revision method for percentage
For easy learning, revise percentage in four rounds. In the first round, learn conversion between percentage, fraction, and decimal. In the second round, solve direct questions such as 20% of 500 and 35% of 800. In the third round, practice increase, decrease, discount, marks, and salary questions. In the fourth round, solve government exam type mixed questions where percentage is combined with ratio, profit-loss, population, and data interpretation.
While solving, always write the base value. Most wrong answers come from choosing the wrong base. In profit-loss, the base is cost price. In discount, the base is marked price. In marks, the base is total marks. In population increase or decrease, the base changes every year when the change is successive.
Extra FAQs
What does 100% mean?
100% means the complete value or the whole amount. If a student gets 100 marks out of 100, it means the student scored 100%.
What does 0% mean?
0% means nothing out of the total. If 0% discount is given, there is no discount.
Is percentage always calculated on the original value?
For increase and decrease, percentage change is calculated on the original value. But in successive percentage change, the second change is calculated on the new value after the first change.
Why are successive discounts not added directly?
Successive discounts are not added directly because the second discount is calculated on the reduced price, not on the original marked price.
Which percentage questions are most important for competitive exams?
The most important types are percentage conversion, marks percentage, percentage increase and decrease, successive change, price and consumption, population, depreciation, comparison, discount, and data interpretation.
Conclusion
Percentage is a basic but powerful topic. It means a value out of 100 and helps us compare marks, prices, population, salary, discount, profit, loss, and many other quantities. To solve percentage questions, first identify the original value or total value. Then apply the correct formula. For government job exams, practice percentage conversion, increase and decrease, successive percentage change, price-consumption questions, population, depreciation, and comparison questions. Once these types are clear, percentage becomes one of the easiest scoring topics in mathematics.