Part 1: Basic Concepts

Definition and Meaning

Percentage means “per hundred” or “out of 100”

• Symbol: %
• 1% = 1/100 = 0.01

Examples:

• 25% means 25 out of 100
• 50% means 50 out of 100 (half)
• 100% means the whole quantity
• 150% means one and a half times the quantity

Converting Between Forms

Fraction to Percentage:

• Multiply by 100
• 3/4 = (3/4) × 100% = 75%
• 7/8 = (7/8) × 100% = 87.5%
• 2/5 = (2/5) × 100% = 40%
• 5/6 = (5/6) × 100% = 83.33%

Decimal to Percentage:

• Multiply by 100
• 0.25 = 0.25 × 100% = 25%
• 0.75 = 0.75 × 100% = 75%
• 0.125 = 0.125 × 100% = 12.5%
• 1.5 = 1.5 × 100% = 150%

Percentage to Fraction:

• Divide by 100 and simplify
• 20% = 20/100 = 1/5
• 75% = 75/100 = 3/4
• 60% = 60/100 = 3/5
• 125% = 125/100 = 5/4

Percentage to Decimal:

• Divide by 100
• 35% = 35/100 = 0.35
• 8% = 8/100 = 0.08
• 250% = 250/100 = 2.5
• 0.5% = 0.5/100 = 0.005

Common Percentage Equivalents

Fractions and their percentage equivalents:

• 1/2 = 50%
• 1/3 = 33.33%
• 2/3 = 66.67%
• 1/4 = 25%
• 3/4 = 75%
• 1/5 = 20%
• 2/5 = 40%
• 3/5 = 60%
• 4/5 = 80%
• 1/8 = 12.5%
• 3/8 = 37.5%
• 5/8 = 62.5%
• 7/8 = 87.5%

Part 2: Finding Percentage of a Number

Basic Formula: x% of y = (x/100) × y

Examples:

Example 1: Find 15% of 200

• 15% of 200 = (15/100) × 200 = 0.15 × 200 = 30

Example 2: Find 25% of 80

• 25% of 80 = (25/100) × 80 = 0.25 × 80 = 20

Example 3: Find 12.5% of 160

• 12.5% of 160 = (12.5/100) × 160 = 0.125 × 160 = 20

Example 4: Find 150% of 40

• 150% of 40 = (150/100) × 40 = 1.5 × 40 = 60

Example 5: Find 0.5% of 2000

• 0.5% of 2000 = (0.5/100) × 2000 = 0.005 × 2000 = 10

Mental Calculation Tricks

10% of any number: Move decimal point one place left

• 10% of 250 = 25
• 10% of 47 = 4.7

5% of any number: Half of 10%

• 5% of 200 = Half of 20 = 10
• 5% of 80 = Half of 8 = 4

25% of any number: Divide by 4

• 25% of 200 = 200/4 = 50
• 25% of 88 = 88/4 = 22

50% of any number: Divide by 2

• 50% of 150 = 150/2 = 75
• 50% of 67 = 67/2 = 33.5

1% of any number: Move decimal point two places left

• 1% of 400 = 4
• 1% of 75 = 0.75

Part 3: Finding What Percentage One Number is of Another

Formula: (Part/Whole) × 100%

Examples:

Example 1: What percentage is 15 of 60?

• (15/60) × 100% = (1/4) × 100% = 25%

Example 2: What percentage is 35 of 140?

• (35/140) × 100% = (1/4) × 100% = 25%

Example 3: What percentage is 72 of 96?

• (72/96) × 100% = (3/4) × 100% = 75%

Example 4: What percentage is 150 of 120?

• (150/120) × 100% = (5/4) × 100% = 125%

Example 5: Express 3 hours as a percentage of a day

• 1 day = 24 hours
• (3/24) × 100% = (1/8) × 100% = 12.5%

Example 6: Express 40 minutes as a percentage of 2 hours

• 2 hours = 120 minutes
• (40/120) × 100% = (1/3) × 100% = 33.33%

Part 4: Finding the Whole When Percentage is Given

Formula: Whole = (Part × 100)/Percentage

Examples:

Example 1: 20% of a number is 15. Find the number.

• Let the number be x
• 20% of x = 15
• (20/100) × x = 15
• x = 15 × (100/20) = 75

Example 2: 25% of a number is 60. Find the number.

• 25% of x = 60
• x = 60 × (100/25) = 240

Example 3: If 15% of students failed, and 51 students passed, how many students appeared?

• If 15% failed, then 85% passed
• 85% of total = 51
• Total = 51 × (100/85) = 60 students

Example 4: A shopkeeper sold 75% of his apples and was left with 60 apples. How many apples did he have initially?

• Remaining apples = 25% of total = 60
• Total = 60 × (100/25) = 240 apples

Example 5: After spending 80% of his salary, a man saves ₹4000. What is his salary?

• Savings = 20% of salary = ₹4000
• Salary = 4000 × (100/20) = ₹20,000

Part 5: Percentage Increase and Decrease

Percentage Increase

Formula: Percentage Increase = (Increase/Original Value) × 100%

Examples:

Example 1: Price increased from ₹500 to ₹600. Find percentage increase.

• Increase = 600 – 500 = ₹100
• Percentage increase = (100/500) × 100% = 20%

Example 2: Population increased from 50,000 to 65,000. Find percentage increase.

• Increase = 65,000 – 50,000 = 15,000
• Percentage increase = (15,000/50,000) × 100% = 30%

Example 3: A number increased from 80 to 120. Find percentage increase.

• Increase = 120 – 80 = 40
• Percentage increase = (40/80) × 100% = 50%

Percentage Decrease

Formula: Percentage Decrease = (Decrease/Original Value) × 100%

Examples:

Example 1: Price decreased from ₹800 to ₹600. Find percentage decrease.

• Decrease = 800 – 600 = ₹200
• Percentage decrease = (200/800) × 100% = 25%

Example 2: Weight reduced from 75 kg to 60 kg. Find percentage decrease.

• Decrease = 75 – 60 = 15 kg
• Percentage decrease = (15/75) × 100% = 20%

Example 3: Production decreased from 1000 units to 850 units. Find percentage decrease.

• Decrease = 1000 – 850 = 150 units
• Percentage decrease = (150/1000) × 100% = 15%

Finding New Values After Percentage Change

After increase: New Value = Original × (100 + Percentage Increase)/100

After decrease: New Value = Original × (100 – Percentage Decrease)/100

Examples:

Example 1: Increase 250 by 20%

• New value = 250 × (100 + 20)/100 = 250 × 1.2 = 300

Example 2: Decrease 400 by 15%

• New value = 400 × (100 – 15)/100 = 400 × 0.85 = 340

Example 3: A salary of ₹25,000 is increased by 12%. Find new salary.

• New salary = 25,000 × (100 + 12)/100 = 25,000 × 1.12 = ₹28,000

Part 6: Successive Percentage Changes

When two percentage changes are applied one after another

Formula: If two changes of a% and b% are applied: Net change = a + b + (ab/100) %

Examples:

Example 1: A number is increased by 20% and then decreased by 10%. Find net change.

• Net change = 20 + (-10) + (20 × (-10))/100
• = 20 – 10 – 2 = 8% increase

Example 2: Price increased by 25% and then decreased by 20%. Find net change.

• Net change = 25 + (-20) + (25 × (-20))/100
• = 25 – 20 – 5 = 0% (no change)

Example 3: A number is decreased by 30% and then increased by 40%. Find net change.

• Net change = (-30) + 40 + ((-30) × 40)/100
• = -30 + 40 – 12 = -2% (2% decrease)

Step-by-step Method

Example: A price of ₹1000 is increased by 15% and then decreased by 20%.

Step 1: After 15% increase

• New price = 1000 × 1.15 = ₹1150

Step 2: After 20% decrease on ₹1150

• Final price = 1150 × 0.80 = ₹920

Net change: 920 – 1000 = -₹80 (8% decrease)

Part 7: Applications in Business

Profit and Loss Percentages

Cost Price (C.P.): Price at which article is bought Selling Price (S.P.): Price at which article is sold

Profit = S.P. – C.P. (when S.P. > C.P.) Loss = C.P. – S.P. (when C.P. > S.P.)

Profit % = (Profit/C.P.) × 100% Loss % = (Loss/C.P.) × 100%

Examples:

Example 1: An article bought for ₹500 is sold for ₹600. Find profit%.

• Profit = 600 – 500 = ₹100
• Profit% = (100/500) × 100% = 20%

Example 2: An article bought for ₹800 is sold for ₹680. Find loss%.

• Loss = 800 – 680 = ₹120
• Loss% = (120/800) × 100% = 15%

Example 3: If S.P. = ₹1200 and profit% = 20%, find C.P.

• S.P. = C.P. + Profit
• 1200 = C.P. + 20% of C.P.
• 1200 = C.P. × 1.2
• C.P. = 1200/1.2 = ₹1000

Discount Calculations

Marked Price (M.P.): Listed price of an article Discount: Reduction in marked price Selling Price = Marked Price – Discount

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

Examples:

Example 1: M.P. = ₹500, Discount = 20%. Find S.P.

• Discount amount = 20% of 500 = ₹100
• S.P. = 500 – 100 = ₹400

Example 2: S.P. = ₹800, Discount% = 20%. Find M.P.

• S.P. = M.P. – 20% of M.P. = 0.8 × M.P.
• 800 = 0.8 × M.P.
• M.P. = 800/0.8 = ₹1000

Example 3: M.P. = ₹1500, S.P. = ₹1200. Find discount%.

• Discount = 1500 – 1200 = ₹300
• Discount% = (300/1500) × 100% = 20%

Commission and Brokerage

Commission: Percentage of sales value paid to agent Brokerage: Percentage charged by broker

Examples:

Example 1: An agent sells goods worth ₹50,000 and gets 5% commission. Find commission.

• Commission = 5% of 50,000 = ₹2,500

Example 2: A broker charges 2% brokerage on ₹80,000 transaction. Find brokerage.

• Brokerage = 2% of 80,000 = ₹1,600

Part 8: Tax Calculations

Sales Tax and VAT

Examples:

Example 1: An article costs ₹800 + 12% sales tax. Find total cost.

• Sales tax = 12% of 800 = ₹96
• Total cost = 800 + 96 = ₹896

Example 2: Total bill including 18% GST is ₹1180. Find original amount.

• Total = Original + 18% of Original = 1.18 × Original
• 1180 = 1.18 × Original
• Original = 1180/1.18 = ₹1000

Income Tax

Example: A person earning ₹5,00,000 annually pays 20% income tax. Find tax and net income.

• Income tax = 20% of 5,00,000 = ₹1,00,000
• Net income = 5,00,000 – 1,00,000 = ₹4,00,000

Part 9: Simple and Compound Interest

Simple Interest as Percentage

Formula: S.I. = (P × R × T)/100 where P = Principal, R = Rate%, T = Time

Examples:

Example 1: Find S.I. on ₹2000 at 8% per annum for 3 years.

• S.I. = (2000 × 8 × 3)/100 = ₹480

Example 2: At what rate will ₹5000 amount to ₹6000 in 4 years?

• S.I. = 6000 – 5000 = ₹1000
• 1000 = (5000 × R × 4)/100
• R = (1000 × 100)/(5000 × 4) = 5%

Compound Interest

Formula: A = P(1 + R/100)^T C.I. = A – P

Example: Find C.I. on ₹8000 at 10% per annum for 2 years.

• A = 8000(1 + 10/100)² = 8000 × (1.1)² = 8000 × 1.21 = ₹9680
• C.I. = 9680 – 8000 = ₹1680

Part 10: Population and Growth Problems

Population Growth/Decline

Formula: Final Population = Initial × (1 ± Growth Rate/100)^Time

Examples:

Example 1: A city’s population is 1,00,000. If it grows at 5% annually, find population after 2 years.

• Population = 1,00,000 × (1 + 5/100)² = 1,00,000 × (1.05)² = 1,10,250

Example 2: A machine worth ₹80,000 depreciates at 10% annually. Find value after 3 years.

• Value = 80,000 × (1 – 10/100)³ = 80,000 × (0.9)³ = ₹58,320

Example 3: Population decreased from 50,000 to 40,500 in 2 years. Find annual rate of decrease.

• 40,500 = 50,000 × (1 – R/100)²
• (1 – R/100)² = 40,500/50,000 = 0.81
• 1 – R/100 = 0.9
• R = 10%

Part 11: Examination and Result Analysis

Pass/Fail Percentages

Examples:

Example 1: In an exam, 75% students passed. If 150 students failed, find total students.

• Failed students = 25% of total = 150
• Total = 150 × (100/25) = 600 students

Example 2: Out of 800 students, 640 passed. Find pass percentage.

• Pass percentage = (640/800) × 100% = 80%

Example 3: To pass, a student needs 40% marks. A student got 150 marks and failed by 30 marks. Find maximum marks.

• Passing marks = 150 + 30 = 180
• 40% of maximum = 180
• Maximum = 180 × (100/40) = 450 marks

Grade Analysis

Example: Class results: A grade (20%), B grade (35%), C grade (30%), D grade (10%), F grade (5%). If 120 students got C grade, find total students.

• C grade students = 30% of total = 120
• Total = 120 × (100/30) = 400 students

Part 12: Sports and Statistics

Batting Averages and Strike Rates

Example 1: A batsman scored 450 runs in 600 balls. Find strike rate.

• Strike rate = (Runs/Balls) × 100 = (450/600) × 100 = 75%

Example 2: Team A won 15 matches out of 25 played. Find win percentage.

• Win percentage = (15/25) × 100% = 60%

Election Results

Example: In an election with 80% voter turnout, candidate A got 45% of valid votes, B got 35%, and 20% were invalid. If A won by 40,000 votes, find total voters.

• Valid votes = 80% of total voters
• A’s votes = 45% of valid votes = 45% × 80% = 36% of total
• B’s votes = 35% × 80% = 28% of total
• Difference = 36% – 28% = 8% of total = 40,000
• Total voters = 40,000 × (100/8) = 5,00,000

Part 13: Practice Problems

Basic Percentage Problems:

1. Convert 7/8 to percentage
2. Find 24% of 350
3. What percentage is 45 of 180?
4. If 30% of a number is 120, find the number

Increase/Decrease Problems:

1. Increase 640 by 15%
2. Price decreased from ₹500 to ₹425. Find percentage decrease
3. A number increased by 25% becomes 375. Find the original number
4. After two successive increases of 10% and 20%, a number becomes 396. Find original number

Business Problems:

1. C.P. = ₹400, S.P. = ₹500. Find profit%
2. M.P. = ₹2000, Discount = 15%. Find S.P.
3. An article is sold for ₹1440 after giving 10% discount. Find M.P.
4. A shopkeeper marks his goods 40% above C.P. and gives 20% discount. Find profit%

Mixed Problems:

1. In an election, 75% people voted. Winner got 60% of valid votes. If he won by 7200 votes, find total voters
2. A student needs 40% to pass. He got 178 marks and failed by 22 marks. Find maximum marks
3. Population of a town increases by 10% annually. If current population is 66,000, what was it 2 years ago?
4. A sum becomes ₹7260 in 2 years at 10% compound interest annually. Find the principal

Solutions:

Basic Percentage Problems:

1. 7/8 = 87.5%
2. 24% of 350 = 84
3. (45/180) × 100% = 25%
4. Number = 120 × (100/30) = 400

Increase/Decrease Problems:

1. 640 × 1.15 = 736
2. Decrease = (75/500) × 100% = 15%
3. Original = 375 ÷ 1.25 = 300
4. Original = 396 ÷ (1.1 × 1.2) = 300

Business Problems:

1. Profit% = (100/400) × 100% = 25%
2. S.P. = 2000 × 0.85 = ₹1700
3. M.P. = 1440 ÷ 0.9 = ₹1600
4. Net effect = 40 – 20 – (20×40)/100 = 12% profit

Mixed Problems:

1. Total voters = 7200 ÷ (0.75 × 0.2) = 48,000
2. Maximum marks = 200 ÷ 0.4 = 500
3. Population 2 years ago = 66,000 ÷ (1.1)² = 54,545
4. Principal = 7260 ÷ (1.1)² = ₹6000