Percentage is one of the most important topics in competitive mathematics. Questions based directly or indirectly on percentages are regularly asked in examinations such as SSC CGL, SSC CHSL, SSC MTS, SSC GD, RRB NTPC and Railway Group D.
More importantly, percentage is not an isolated chapter. A good understanding of percentage makes several other arithmetic topics easier, including Profit and Loss, Discount, Simple Interest, Compound Interest, Data Interpretation and Population Growth.
This chapter explains the fundamental concepts, important formulas, calculation methods and common question types that students should master before attempting advanced percentage problems.
What is Percentage?
The word percentage means "per hundred".
Therefore,
Percentage = (Part / Total) × 100
For example, suppose a student answers 36 questions correctly out of 40 questions.
Percentage of correct answers:
(36 / 40) × 100 = 90%
Therefore, the student answered 90% of the questions correctly.
Understanding Percentage Through Fractions
Competitive examination students should learn common fraction-to-percentage conversions because they significantly reduce calculation time.
1/2 = 50%
1/3 = 33⅓%
2/3 = 66⅔%
1/4 = 25%
3/4 = 75%
1/5 = 20%
2/5 = 40%
3/5 = 60%
4/5 = 80%
1/6 = 16⅔%
5/6 = 83⅓%
1/8 = 12.5%
3/8 = 37.5%
5/8 = 62.5%
7/8 = 87.5%
1/10 = 10%
Students should try to remember these values instead of calculating them repeatedly during an examination.
How to Find a Percentage of a Number
To calculate x% of a number:
x% of N = (x/100) × N
Example 1
Find 25% of 480.
25% of 480
= 25/100 × 480
= 120
Answer: 120
Example 2
Find 15% of 600.
15% of 600
= 15/100 × 600
= 90
Answer: 90
Example 3
Find 12.5% of 640.
Since 12.5% = 1/8,
640 × 1/8 = 80
Answer: 80
This illustrates why remembering fraction-percentage equivalents can save considerable time.
Finding What Percentage One Number is of Another
Use:
Required Percentage = (First Number / Second Number) × 100
Example
What percentage of 250 is 50?
= (50/250) × 100
= 20%
Answer: 20%
Percentage Increase
When a quantity increases from an original value to a new value:
Percentage Increase = (Increase / Original Value) × 100
Example
The price of an item increases from ₹800 to ₹920.
Increase = 920 - 800 = ₹120
Percentage increase:
(120/800) × 100 = 15%
Answer: 15%
An important point is that percentage change is calculated using the original value as the base.
Percentage Decrease
Percentage Decrease = (Decrease / Original Value) × 100
Example
The price of an item decreases from ₹500 to ₹425.
Decrease = 500 - 425 = ₹75
Percentage decrease:
(75/500) × 100 = 15%
Answer: 15%
Successive Percentage Change
This is an important concept for SSC and Railway examinations.
If a quantity changes successively by a% and b%, the net percentage change can be calculated as:
Net Change = a + b + (ab/100)
Use a positive sign for an increase and a negative sign for a decrease.
Example 1: Two Successive Increases
A number increases by 20% and then by 10%.
Net increase:
20 + 10 + (20 × 10)/100
= 30 + 2
= 32%
Answer: 32% increase
Example 2: Increase Followed by Decrease
A price increases by 20% and then decreases by 10%.
Take the second change as -10.
Net change:
20 - 10 + (20 × -10)/100
= 10 - 2
= 8%
Answer: 8% increase
Equal Increase and Decrease
A common mistake is to assume that an increase of x% followed by a decrease of x% returns a quantity to its original value.
It does not.
If a quantity is increased by x% and subsequently decreased by x%, the net decrease is:
x²/100 %
Example
A number is increased by 20% and then decreased by 20%.
Net decrease:
20²/100 = 4%
Answer: 4% decrease
Check it directly.
Suppose the original number is 100.
After a 20% increase:
100 → 120
After a 20% decrease:
120 → 96
Therefore, the final number is 4% less than the original number.
Reverse Percentage
Sometimes the final value and percentage change are given, while the original value has to be determined.
Example
After a 25% increase, a salary becomes ₹25,000. Find the original salary.
The new salary represents 125% of the original salary.
Original salary:
25000 × 100/125
= ₹20,000
Answer: ₹20,000
Comparing Two Quantities
Suppose A is 25% more than B.
If B = 100, then A = 125.
Now, to determine by what percentage B is less than A:
Difference = 25
Required percentage:
(25/125) × 100 = 20%
Therefore:
If A is 25% more than B, B is 20% less than A.
The percentages are different because their base values are different.
Important Conversion Formula
If A is x% more than B, then B is less than A by:
[x / (100 + x)] × 100%
For example, if A is 50% more than B:
50/150 × 100
= 33⅓%
Therefore, B is 33⅓% less than A.
Expenditure and Consumption Questions
Percentage is frequently combined with expenditure problems.
Expenditure = Price × Consumption
Suppose the price of a commodity increases. If a person's total expenditure must remain unchanged, consumption must decrease.
If price increases by x%, required reduction in consumption is:
[x / (100 + x)] × 100%
Example
The price of rice increases by 25%. By what percentage should consumption be reduced so that expenditure remains unchanged?
Required reduction:
25/125 × 100
= 20%
Answer: 20%
Population Questions
Population questions often involve successive percentage changes.
Example
The population of a town is 50,000. It increases by 10% in the first year and 20% in the second year. Find the population after two years.
After the first year:
50,000 × 1.10 = 55,000
After the second year:
55,000 × 1.20 = 66,000
Answer: 66,000
Notice that simply adding 10% and 20% would give an incorrect result.
The actual overall increase is:
10 + 20 + (10 × 20)/100
= 32%
32% of 50,000 = 16,000
Final population = 66,000.
Marks-Based Percentage Questions
Percentage questions involving examination marks are particularly common.
Example
A candidate needs 40% marks to pass an examination. He obtains 180 marks and fails by 20 marks. Find the maximum marks.
Passing marks:
180 + 20 = 200
Therefore, 40% of total marks = 200.
Total marks:
200 × 100/40
= 500
Answer: 500
Election-Based Percentage Questions
Election problems frequently combine percentage with subtraction.
Example
In an election between two candidates, the winner receives 60% of the valid votes and wins by 4,000 votes.
The loser receives:
100% - 60% = 40%
Difference:
60% - 40% = 20%
Therefore:
20% = 4,000
100% = 20,000
Answer: 20,000 valid votes
Important Percentage Shortcuts
Students preparing for competitive examinations should remember the following relationships:
10% of a number = divide it by 10
20% of a number = divide it by 5
25% of a number = divide it by 4
50% of a number = divide it by 2
75% of a number = multiply it by 3/4
12.5% of a number = divide it by 8
6.25% of a number = divide it by 16
33⅓% of a number = divide it by 3
66⅔% of a number = multiply it by 2/3
These shortcuts can considerably reduce calculation time.
Common Mistakes Students Should Avoid
1. Using the Wrong Base
If a price rises from ₹400 to ₹500, the increase is ₹100.
Percentage increase:
100/400 × 100 = 25%
The denominator must be the original value.
2. Adding Successive Percentages Directly
A 20% increase followed by a 10% increase does not mean a 30% increase.
The actual increase is 32%.
3. Assuming Equal Increase and Decrease Cancel Each Other
A 20% increase followed by a 20% decrease results in a 4% overall decrease.
4. Confusing "More Than" With "Less Than"
If A is 25% more than B, it does not mean B is 25% less than A.
B is actually 20% less than A.
Practice Questions
1. Find 35% of 480.
2. 72 is what percentage of 240?
3. A number increases from 500 to 650. Find the percentage increase.
4. A price decreases from ₹1,200 to ₹960. Find the percentage decrease.
5. A number is increased by 20% and then increased by 25%. Find the net percentage change.
6. A number is increased by 30% and then decreased by 20%. Find the net percentage change.
7. A salary of ₹24,000 increases by 15%. Find the new salary.
8. After a 20% discount, an article costs ₹800. Find its original price.
9. A student obtains 360 marks out of 450. Find the percentage of marks obtained.
10. A candidate needs 35% marks to pass. He gets 150 marks and fails by 25 marks. Find the maximum marks.
11. The population of a village is 20,000. It increases by 10%. Find the new population.
12. A number is reduced by 25%. What percentage increase is required to restore it to its original value?
13. A is 20% more than B. By what percentage is B less than A?
14. The price of sugar rises by 20%. By what percentage should consumption be reduced to maintain the same expenditure?
15. In an election, a candidate receives 55% of the votes and wins by 2,000 votes. Find the total number of votes, assuming there are only two candidates and all votes counted are valid.
Answers
1. 168
2. 30%
3. 30%
4. 20%
5. 50% increase
6. 4% increase
7. ₹27,600
8. ₹1,000
9. 80%
10. 500
11. 22,000
12. 33⅓%
13. 16⅔%
14. 16⅔%
15. 20,000
How to Prepare Percentage for SSC and Railway Exams
Start by mastering fraction-percentage conversions. After that, practise basic percentage calculations before moving to percentage increase and decrease.
Once these concepts are comfortable, practise successive percentage change, expenditure, population, marks and election questions.
Percentage should not be studied merely as an individual chapter. Students who become comfortable with percentages generally find Profit and Loss, Discount, Interest and Data Interpretation much easier.
Regular calculation practice is more useful than memorising a large number of unrelated tricks.
Frequently Asked Questions
Is percentage important for SSC examinations?
Yes. Percentage is an important arithmetic concept for SSC examinations and is also used in several related chapters.
Is percentage important for Railway examinations?
Yes. Percentage and percentage-based arithmetic are useful for RRB NTPC, Group D and other Railway examination preparation.
What should I learn before solving advanced percentage questions?
Students should first understand fractions, ratios, basic multiplication and division.
Which percentage values should I memorise?
Common conversions such as 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8 and 6.25% = 1/16 are particularly useful.
How can I become faster at percentage calculations?
Practise converting percentages into simple fractions and solve calculations mentally whenever possible.
Conclusion
Percentage is a foundation chapter of competitive arithmetic. Students who understand the concept rather than relying entirely on shortcuts can solve both direct and application-based questions more confidently.
For SSC and Railway preparation, focus particularly on percentage increase and decrease, successive changes, marks, expenditure, population, elections and comparisons between two quantities.
After mastering these concepts, practise mixed percentage questions regularly and analyse every mistake before moving to more advanced arithmetic chapters.