Percentage – Complete Notes for Competitive Exams
Percentage is one of the most important arithmetic topics for competitive examinations. Questions based on percentage are commonly connected with marks, salaries, population, expenditure, profit and loss, discounts, elections, production, consumption and data interpretation.
The word percentage simply means a value expressed out of 100. Once the basic relationship between percentage, fraction and decimal is understood, many seemingly lengthy questions can be solved quickly.
These notes are written from scratch for ExamMitra learners preparing for SSC, Railway, Banking, Assam competitive examinations and other objective examinations.
Table of Contents
- 1. What is Percentage?
- 2. Basic Percentage Formula
- 3. Percentage to Fraction
- 4. Percentage to Decimal
- 5. Important Fraction-Percentage Conversions
- 6. Finding Percentage of a Number
- 7. Finding the Original Number
- 8. Percentage Increase
- 9. Percentage Decrease
- 10. Successive Percentage Changes
- 11. Reverse Percentage
- 12. Percentage Comparison
- 13. More Than and Less Than
- 14. Percentage in Marks
- 15. Population Problems
- 16. Salary and Income Problems
- 17. Income, Expenditure and Savings
- 18. Price and Consumption Problems
- 19. Percentage Connection with Profit and Loss
- 20. Percentage Connection with Discount
- 21. Election and Vote Problems
- 22. Exam-Oriented Percentage Problems
- 23. Percentage Shortcuts
- 24. Important Formulas
- 25. Solved Examples
- 26. Common Mistakes
- 27. Quick Revision Sheet
- 28. Preparation Strategy
- 29. Frequently Asked Questions
1. What is Percentage?
Percentage means "per hundred". It represents a quantity in relation to 100.
Therefore:
25% means 25 out of 100.
25% = 25/100 = 1/4
Percentage is represented using the symbol %.
2. Basic Percentage Formula
The most important percentage formula is:
A student scores 72 marks out of 100. What is the percentage?
Percentage = (72/100) × 100
= 72%
A student scores 360 marks out of 500.
Percentage = (360/500) × 100
= 72%
3. Percentage to Fraction
To convert a percentage into a fraction, divide it by 100 and simplify.
40% = 40/100 = 2/5
75% = 75/100 = 3/4
20% = 20/100 = 1/5
4. Percentage to Decimal
To convert a percentage into a decimal, divide by 100.
25% = 0.25
50% = 0.50 = 0.5
12.5% = 0.125
75% = 0.75
To convert a decimal into a percentage, multiply it by 100.
0.4 = 40%
0.65 = 65%
0.125 = 12.5%
5. Important Fraction-Percentage Conversions
Memorising common conversions can save considerable time in objective examinations.
| Fraction | Percentage | Decimal |
|---|---|---|
| 1/2 | 50% | 0.50 |
| 1/3 | 33.33% approximately | 0.333... |
| 2/3 | 66.67% approximately | 0.666... |
| 1/4 | 25% | 0.25 |
| 3/4 | 75% | 0.75 |
| 1/5 | 20% | 0.20 |
| 2/5 | 40% | 0.40 |
| 3/5 | 60% | 0.60 |
| 4/5 | 80% | 0.80 |
| 1/8 | 12.5% | 0.125 |
| 3/8 | 37.5% | 0.375 |
| 5/8 | 62.5% | 0.625 |
| 7/8 | 87.5% | 0.875 |
| 1/10 | 10% | 0.10 |
| 1/20 | 5% | 0.05 |
6. Finding Percentage of a Number
30% of 250
= 30/100 × 250
= 3/10 × 250
= 75
Using Fractions for Faster Calculation
25% of 480
25% = 1/4
480 ÷ 4 = 120
12.5% of 640
12.5% = 1/8
640 ÷ 8 = 80
7. Finding the Original Number
Sometimes the percentage and resulting value are given, while the original number must be found.
40% of a number is 160. Find the number.
Number = (160 × 100) / 40
= 16000/40
= 400
8. Percentage Increase
If a quantity increases from an old value to a new value:
A price rises from 500 to 600.
Increase = 600 - 500 = 100
Percentage increase = (100/500) × 100
= 20%
Finding the New Value After an Increase
Increase 800 by 15%.
New value = 800 × 115/100
= 920
9. Percentage Decrease
A price falls from 800 to 680.
Decrease = 800 - 680 = 120
Percentage decrease = (120/800) × 100
= 15%
Finding the New Value After a Decrease
Reduce 900 by 20%.
New value = 900 × 80/100
= 720
10. Successive Percentage Changes
When a quantity changes more than once, the percentages should not normally be added directly.
Two Successive Increases
A number increases by 10% and then by 20%.
Net increase = 10 + 20 + (10 × 20)/100
= 30 + 2
= 32%
Two Successive Decreases
A price decreases by 10% and then by 20%.
Net decrease = 10 + 20 - 2
= 28%
One Increase and One Decrease
The sign of the final result indicates whether there is an overall increase or decrease.
A number increases by 20% and then decreases by 10%.
Net change = 20 - 10 - (20 × 10)/100
= 10 - 2
= 8% increase
11. Reverse Percentage
Reverse percentage problems ask you to find the original value after a percentage increase or decrease.
After an Increase
After a 25% increase, a salary becomes 50,000. Find the original salary.
Original salary = 50,000 × 100 / 125
= 40,000
After a Decrease
After a 20% decrease, a number becomes 480. Find the original number.
Original number = 480 × 100 / 80
= 600
12. Percentage Comparison
To find how much one quantity is more or less than another:
The most important point is identifying the reference value. In questions asking "A is what percent more than B?", B is the base.
A = 150 and B = 100.
A is what percent more than B?
Difference = 150 - 100 = 50
Percentage = 50/100 × 100
= 50%
13. More Than and Less Than
These questions require careful attention because the base changes when the comparison is reversed.
A is 25% more than B.
Let B = 100.
Then A = 125.
Therefore B is less than A by:
25/125 × 100 = 20%
14. Percentage in Marks
Percentage questions involving marks are very common in competitive-exam aptitude sections.
A candidate obtains 420 marks out of 600.
Percentage = 420/600 × 100
= 70%
Finding Required Marks
How many marks are required to score 65% in an examination of 800 marks?
Required marks = 65/100 × 800
= 520 marks
Marks Increased by Percentage
A student scores 360 marks. His marks increase by 10%. Find the new marks.
Increase = 10% of 360 = 36
New marks = 360 + 36
= 396
15. Population Problems
Population questions generally involve percentage increase or decrease over one or more years.
A town has a population of 20,000. It increases by 10% in one year.
Increase = 10% of 20,000 = 2,000
New population = 22,000
Population After Two Years
Initial population = 10,000.
Increase in first year = 10%.
Increase in second year = 10%.
First year = 10,000 × 110/100 = 11,000
Second year = 11,000 × 110/100 = 12,100
Thus two successive 10% increases result in a 21% overall increase, not 20%.
16. Salary and Income Problems
Percentage concepts are frequently used to calculate salary increases, reductions, bonuses and changes in income.
A person's salary is 30,000. It increases by 12%.
Increase = 12/100 × 30,000
= 3,600
New salary = 33,600
17. Income, Expenditure and Savings
In these questions, remember:
A person earns 40,000 and spends 75% of his income.
Expenditure = 75% of 40,000 = 30,000
Savings = 40,000 - 30,000
= 10,000
Change in Income and Expenditure
When both income and expenditure change, calculate the new values separately before finding the new savings.
Income = 50,000 and expenditure = 40,000.
Income rises by 10% and expenditure rises by 5%.
New income = 55,000
New expenditure = 42,000
New savings = 13,000
18. Price and Consumption Problems
A common question asks how consumption should change when the price changes so that total expenditure remains constant.
The price of an item rises by 25%. By what percentage should consumption decrease to keep expenditure unchanged?
```Required decrease:
= 25 / 125 × 100
= 20%
```If price decreases by r%, the corresponding increase in consumption required to keep expenditure constant is:
19. Percentage Connection with Profit and Loss
Percentage is the foundation of profit and loss calculations.
An article costs 800 and is sold for 920.
Profit = 920 - 800 = 120
Profit% = 120/800 × 100
= 15%
20. Percentage Connection with Discount
An article marked at 2,000 is sold for 1,700.
Discount = 2,000 - 1,700 = 300
Discount% = 300/2,000 × 100
= 15%
21. Election and Vote Problems
Percentage questions involving elections usually require careful attention to total votes, valid votes, invalid votes and winning margins.
In an election, a candidate receives 60% of the valid votes. If there are 20,000 valid votes, how many votes did the candidate receive?
Votes = 60/100 × 20,000
= 12,000
Invalid Votes
Suppose 5% of 30,000 votes are invalid.
Invalid votes = 5/100 × 30,000 = 1,500
Valid votes = 30,000 - 1,500
= 28,500
22. Exam-Oriented Percentage Problems
Type 1: Find a Percentage
What percentage of 250 is 50?
= 50/250 × 100
= 20%
Type 2: Find the Number
35% of a number is 140. Find the number.
Number = 140 × 100/35
= 400
Type 3: Increase
Increase 500 by 18%.
New value = 500 × 118/100
= 590
Type 4: Decrease
Decrease 750 by 16%.
New value = 750 × 84/100
= 630
Type 5: Reverse Percentage
A number becomes 720 after a 20% increase. Find the original number.
Original = 720 × 100/120
= 600
Type 6: Successive Changes
A value increases by 15% and then decreases by 10%.
Net change = 15 - 10 - 15 × 10/100
= 5 - 1.5
= 3.5% increase
23. Percentage Shortcuts
Shortcut 1: 10%
Example: 10% of 850 = 85
Shortcut 2: 5%
Example: 5% of 600 = 30.
Shortcut 3: 25%
Example: 25% of 720 = 720 ÷ 4 = 180.
Shortcut 4: 50%
Example: 50% of 940 = 470.
Shortcut 5: 12.5%
Example: 12.5% of 800 = 800 ÷ 8 = 100.
Shortcut 6: 1%
This is useful for constructing other percentages quickly.
17% of 500 = 10% + 5% + 2%
= 50 + 25 + 10
= 85
24. Important Formulas
25. Solved Examples
What is 35% of 240?
35/100 × 240
= 84
What percentage of 400 is 72?
72/400 × 100
= 18%
45% of a number is 270. Find the number.
Number = 270 × 100/45
= 600
Increase 1,200 by 15%.
Increase = 180
New value = 1,380
Decrease 2,500 by 12%.
Decrease = 300
New value = 2,200
A number increases by 20% and then by 30%. Find the net increase.
Net increase = 20 + 30 + 20 × 30/100
= 50 + 6
= 56%
A price is reduced by 20% and then by 10%. Find the total decrease.
Net decrease = 20 + 10 - 20 × 10/100
= 30 - 2
= 28%
A candidate obtains 540 marks out of 750. Find the percentage.
= 540/750 × 100
= 72%
A population of 50,000 increases by 8%. Find the new population.
Increase = 4,000
New population = 54,000
A person's income is 40,000 and expenditure is 70% of income. Find savings.
Expenditure = 28,000
Savings = 40,000 - 28,000
= 12,000
A number is 25% more than another number. If the smaller number is 400, find the larger number.
Increase = 25% of 400 = 100
Larger number = 500
A number is 20% less than another number. If the larger number is 500, find the smaller number.
Decrease = 20% of 500 = 100
Smaller number = 400
A product costs 800. Its price is increased by 25%. Find the new price.
New price = 800 × 125/100
= 1,000
After a 25% discount, an item sells for 900. Find the marked price.
Selling price = 75% of marked price.
Marked price = 900 × 100/75
= 1,200
A quantity increases by 10% and then decreases by 10%. Find the net change.
Net change = 10 - 10 - 10 × 10/100
= -1%
Therefore, there is a 1% decrease.
26. Common Mistakes
Mistake 1: Using the Wrong Base
In percentage comparison, identify the quantity against which the comparison is being made.
Mistake 2: Adding Successive Percentages Directly
A 20% increase followed by a 20% increase is not a 40% increase. The actual overall increase is 44%.
Mistake 3: Confusing Percentage with Percentage Points
A change from 40% to 50% is an increase of 10 percentage points, but the relative percentage increase is 25%.
Mistake 4: Forgetting the Original Value
Percentage increase and decrease are normally calculated using the original value as the base.
Mistake 5: Reversing Increase and Decrease
If a number increases by 25%, reducing the new number by 25% does not return it to the original value.
27. Quick Revision Sheet
28. Preparation Strategy
Step 1: Learn the Basic Meaning
Understand that percentage means a quantity out of 100. This makes the fraction and decimal relationships much easier.
Step 2: Memorise Common Conversions
Learn common fraction-percentage-decimal equivalents so you can recognise them immediately during an exam.
Step 3: Master Increase and Decrease
Practise questions involving price, salary, population, marks and income.
Step 4: Practise Successive Changes
These questions are important because the second percentage is applied to the already changed value.
Step 5: Connect Percentage With Other Chapters
Percentage is closely related to profit and loss, discount, ratio, average, simple interest, compound interest, data interpretation and mixture-based calculations.
Step 6: Practise Under Time Pressure
Start with untimed practice for accuracy. Later, solve mixed sets with a timer to improve speed.
29. Frequently Asked Questions
What is percentage?
Percentage is a way of expressing a quantity in relation to 100. For example, 30% means 30 out of 100.
What is the basic percentage formula?
Percentage = (Part / Total) × 100.
How do I calculate a percentage of a number?
Multiply the number by the percentage and divide by 100. For example, 20% of 500 = 20/100 × 500 = 100.
How do I calculate percentage increase?
Percentage increase = (Increase / Original Value) × 100.
How do I calculate percentage decrease?
Percentage decrease = (Decrease / Original Value) × 100.
What is successive percentage change?
Successive percentage change occurs when a quantity is changed by one percentage and then changed again using the new value as the base.
Is percentage important for SSC and Railway examinations?
Yes. Percentage is a fundamental arithmetic topic and is also useful in many related chapters such as profit and loss, discount, interest, ratio and data interpretation.
What should I memorise for percentage questions?
Focus on common fraction-percentage conversions, basic multiplication and division, percentage increase and decrease formulas, successive changes and common arithmetic shortcuts.
Final Revision Advice
Percentage questions become much easier when you stop treating each question as a separate formula. Most problems are based on a small number of ideas: finding a part of a quantity, comparing two quantities, changing a quantity by a percentage, or finding the original value from a changed value.
Build a strong foundation with fractions and decimals first. Then practise percentage increase and decrease, successive changes, marks, population, income and expenditure, profit and loss, discounts and other application-based questions.
For competitive examinations, aim for both accuracy and speed. A good understanding of the base quantity is often more important than memorising numerous shortcuts.
Comments
Leave a Comment
Share your question, feedback or study-related thoughts.