Percentage

1. WHAT IS PERCENTAGE?

The word percentage means "per hundred". It is used to express a quantity as a fraction of 100.

Percentage = (Part ÷ Whole) × 100

The symbol used for percentage is %.

Example: A student scores 45 marks out of 60. Find the percentage.

Percentage = (45 ÷ 60) × 100
= 75%

Answer = 75%

2. CONVERTING FRACTION INTO PERCENTAGE

Percentage = Fraction × 100

Example: Convert 3/5 into percentage.

= 3/5 × 100
= 60%

Example: Convert 7/8 into percentage.

= 7/8 × 100
= 87.5%

3. CONVERTING PERCENTAGE INTO FRACTION

x% = x/100

Example: Convert 35% into fraction.

35% = 35/100
= 7/20

4. PERCENTAGE INTO DECIMAL

To convert a percentage into decimal, divide it by 100.

25% = 25/100 = 0.25
8% = 8/100 = 0.08
125% = 125/100 = 1.25

5. DECIMAL INTO PERCENTAGE

Multiply the decimal by 100 and attach the percentage sign.

0.45 × 100 = 45%
0.72 × 100 = 72%
1.25 × 100 = 125%

6. IMPORTANT BASIC FORMULAS

Percentage = (Part / Whole) × 100

Part = (Percentage × Whole) / 100

Whole = (Part × 100) / Percentage

Example: Find 25% of 480.

= 25/100 × 480
= 120

Quick Tip: 10% means one-tenth, 25% means one-fourth, 50% means one-half, 75% means three-fourths and 100% means the whole.

7. PERCENTAGE INCREASE

When a quantity increases from an old value to a new value, the percentage increase is calculated using the original value as the base.

Percentage Increase = [(New Value − Original Value) ÷ Original Value] × 100

Example: The price of a book increases from ₹200 to ₹250. Find the percentage increase.

Increase = 250 − 200 = ₹50
Percentage increase = 50/200 × 100
= 25%

8. PERCENTAGE DECREASE

Percentage Decrease = [(Original Value − New Value) ÷ Original Value] × 100

Example: A price falls from ₹800 to ₹680.

Decrease = 800 − 680 = ₹120
Percentage decrease = 120/800 × 100
= 15%

9. FINDING NEW VALUE AFTER INCREASE

New Value = Original Value × (100 + Increase%) / 100

Example: Increase ₹600 by 20%.

New value = 600 × 120/100
= ₹720

10. FINDING NEW VALUE AFTER DECREASE

New Value = Original Value × (100 − Decrease%) / 100

Example: Reduce ₹900 by 15%.

New value = 900 × 85/100
= ₹765

11. SUCCESSIVE PERCENTAGE CHANGE

When a quantity changes more than once, the percentages are applied successively. They should not simply be added in every situation.

For successive increases of a% and b%:
Net Increase = a + b + ab/100

Example: A number increases by 20% and then by 10%. Find the total percentage increase.

Net increase = 20 + 10 + (20×10)/100
= 30 + 2
= 32%

For successive decreases of a% and b%:
Net Decrease = a + b − ab/100

Example: A price decreases by 20% and then by 10%.

Net decrease = 20 + 10 − 2
= 28%

12. INCREASE FOLLOWED BY DECREASE

Net Change = a − b − ab/100

Here a is the percentage increase and b is the percentage decrease.

Example: A number is increased by 20% and then decreased by 10%.

Net change = 20 − 10 − (20×10)/100
= 10 − 2
= 8% increase

Important: A 20% increase followed by a 20% decrease does NOT bring a quantity back to its original value.

13. PROFIT AND LOSS PERCENTAGE

Percentage is widely used in commercial arithmetic to calculate profit and loss.

Profit = Selling Price − Cost Price
Profit% = (Profit / Cost Price) × 100

Loss = Cost Price − Selling Price
Loss% = (Loss / Cost Price) × 100

Example: An article is bought for ₹500 and sold for ₹600.

Profit = 600 − 500 = ₹100
Profit% = 100/500 × 100
= 20%

14. DISCOUNT

Discount is the reduction given on the marked price of an article.

Discount = Marked Price − Selling Price
Discount% = (Discount / Marked Price) × 100

Example: A shirt marked at ₹1200 is sold for ₹960. Find the discount percentage.

Discount = 1200 − 960 = ₹240
Discount% = 240/1200 × 100
= 20%

15. MARKS AND EXAMINATION

Percentage Marks = (Marks Obtained / Maximum Marks) × 100

Example: A student obtains 420 marks out of 500.

Percentage = 420/500 × 100
= 84%

16. POPULATION CHANGE

New Population after increase = Original Population × (100 + r)/100

New Population after decrease = Original Population × (100 − r)/100

Example: A town has a population of 50,000. It increases by 8%.

Increase = 50,000 × 8/100 = 4,000
New population = 54,000

17. EXPENDITURE AND SAVINGS

Percentage can be used to compare expenditure, income and savings.

Savings = Income − Expenditure
Savings% of Income = (Savings / Income) × 100

Example: A person's monthly income is ₹30,000 and expenditure is ₹24,000.

Savings = ₹30,000 − ₹24,000 = ₹6,000
Savings% = 6000/30000 × 100
= 20%

18. REVERSE PERCENTAGE

Sometimes the final value and percentage change are known, and the original value has to be found.

Original Value after an increase of r%:
Original = Final × 100/(100+r)

Original Value after a decrease of r%:
Original = Final × 100/(100−r)

Example: After a 20% increase, the price of an article is ₹720. Find its original price.

Original = 720 × 100/120
= ₹600

19. PERCENTAGE SHORTCUTS

Percentage	Easy Fraction
50%	1/2
25%	1/4
20%	1/5
12.5%	1/8
10%	1/10
5%	1/20
2.5%	1/40

Fast Calculation: To find 15% of a number, find 10% and 5% separately and add them.

20. COMPLETE FORMULA SHEET

Topic	Formula
Basic Percentage	Part/Whole × 100
Find Part	Percentage × Whole / 100
Find Whole	Part × 100 / Percentage
Increase %	Increase/Original × 100
Decrease %	Decrease/Original × 100
New Value after Increase	Original × (100+r)/100
New Value after Decrease	Original × (100−r)/100
Profit %	Profit/Cost Price × 100
Loss %	Loss/Cost Price × 100
Discount %	Discount/Marked Price × 100
Successive Increase	a+b+ab/100
Successive Decrease	a+b−ab/100
Increase then Decrease	a−b−ab/100

21. IMPORTANT SOLVED EXAMPLES

Q1. Find 35% of 800.
= 35/100 × 800
= 280

Q2. 72 is what percentage of 240?
= 72/240 × 100
= 30%

Q3. A number is increased from 400 to 460. Find the percentage increase.
Increase = 60
Percentage = 60/400 × 100
= 15%

Q4. A number is decreased by 25%. If the original number is 640, find the new number.
New number = 640 × 75/100
= 480

Q5. A product costing ₹800 is sold at a profit of 15%. Find the selling price.
Profit = 800 × 15/100 = ₹120
Selling Price = 800 + 120
= ₹920

22. PRACTICE QUESTIONS

Find 18% of 750.
45 is what percentage of 180?
Convert 7/20 into percentage.
Convert 62.5% into a fraction.
A number increases from 500 to 575. Find the percentage increase.
A price falls from ₹1200 to ₹900. Find the percentage decrease.
Increase ₹1500 by 12%.
Decrease ₹2400 by 15%.
An article costs ₹800 and is sold for ₹920. Find the profit percentage.
An article marked at ₹2000 is sold for ₹1700. Find the discount percentage.
A student scores 432 marks out of 600. Find the percentage.
A population of 40,000 increases by 5%. Find the new population.
A number is increased by 20% and then by 10%. Find the net percentage increase.
A number is decreased by 20% and then by 10%. Find the net percentage decrease.
After a 25% increase, a number becomes 500. Find the original number.

23. ANSWERS

1. 135    |    2. 25%    |    3. 35%
4. 5/8    |    5. 15%    |    6. 25%
7. ₹1680    |    8. ₹2040
9. 15%    |    10. 15%
11. 72%    |    12. 42,000
13. 32%    |    14. 28%
15. 400

24. FINAL REVISION NOTES

Remember these five rules:

1. Percentage always means per hundred.
2. For percentage calculation, identify the correct base value.
3. Increase and decrease percentages are normally calculated using the original value as the base.
4. Successive percentage changes must be applied one after another.
5. In reverse percentage problems, work backward from the final value.

EXAM ALERT: Never calculate a percentage change using the wrong base. For example, if a price changes from ₹500 to ₹600, the increase percentage is calculated on ₹500, not ₹600.