Percentages are everywhere — in your exam scores, shopping discounts, salary hikes, bank interest, GST bills, and stock market returns. Yet many students and adults struggle with percentage calculations beyond the basics.
This guide covers every type of percentage problem with formulas, worked examples, and mental math shortcuts that will make percentage calculations fast and easy.
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What Is a Percentage?
A percentage is a number expressed as a fraction of 100. "Per cent" literally means "per hundred" in Latin.
- 50% = 50/100 = 0.5
- 25% = 25/100 = 0.25
- 1% = 1/100 = 0.01
- 100% = 100/100 = 1 (the whole thing)
The 3 Core Percentage Formulas
Every percentage problem falls into one of three types:
Formula 1: What is X% of Y?
Result = (X / 100) × Y
Example: What is 15% of ₹2,000? = (15/100) × 2,000 = 0.15 × 2,000 = ₹300
Formula 2: X is what percent of Y?
Result = (X / Y) × 100
Example: 45 is what percent of 180? = (45/180) × 100 = 0.25 × 100 = 25%
Formula 3: Percentage change from X to Y
Change % = ((Y − X) / X) × 100
Example: Price increased from ₹400 to ₹500. What is the percentage increase? = ((500 − 400) / 400) × 100 = (100/400) × 100 = 25%
If Y > X → percentage increase If Y < X → percentage decrease (result will be negative)
7 Tricks to Calculate Percentages Fast (Without a Calculator)
Trick 1: Use the 10% Base Method
Find 10% first, then build from it.
10% of any number = divide by 10
- 10% of 450 = 45
- 10% of 1,300 = 130
- 10% of 75 = 7.5
From there:
- 5% = half of 10%
- 20% = 2 × 10%
- 15% = 10% + 5%
- 30% = 3 × 10%
- 25% = 10% + 10% + 5%
Example: What is 35% of ₹1,200?
- 10% = ₹120
- 30% = ₹360
- 5% = ₹60
- 35% = 360 + 60 = ₹420
This works for any percentage and is faster than manual calculation for mental math.
Trick 2: Swap the Numbers If One Is Easier
X% of Y = Y% of X
This works because both equal X × Y / 100.
- 8% of 25 → easier as 25% of 8 = 8/4 = 2
- 4% of 50 → easier as 50% of 4 = 4/2 = 2
- 12% of 75 → easier as 75% of 12 = 9
- 18% of 50 → easier as 50% of 18 = 9
When one number is a "nice" fraction (25 = 1/4, 50 = 1/2, 20 = 1/5), swap them.
Trick 3: The 1% Method for Any Percentage
Find 1% by dividing by 100, then multiply.
- 1% of 340 = 3.4
- 7% of 340 = 7 × 3.4 = 23.8
- 13% of 340 = 13 × 3.4 = 44.2
This is especially useful for GST calculations (18% or 28% of any amount).
Trick 4: Percentage Change Shortcut
For percentage increase by P%, the new value = Original × (1 + P/100) For percentage decrease by P%, the new value = Original × (1 − P/100)
- Price rises by 20%: New = Original × 1.20
- Price drops by 30%: New = Original × 0.70
- Salary hike of 12%: New salary = Old × 1.12
Example: Your salary is ₹50,000 and you get a 15% hike. New salary = 50,000 × 1.15 = ₹57,500
Trick 5: Reverse Percentage (Finding the Original)
If a price after applying a percentage is known, find the original:
Original = Final / (1 + P/100) for an increase Original = Final / (1 − P/100) for a decrease
Example: A shirt costs ₹840 after a 40% discount. What was the original price? Original = 840 / (1 − 0.40) = 840 / 0.60 = ₹1,400
A common mistake: Students subtract 40% from ₹840 to "reverse" the discount. 840 × 0.40 = 336. 840 + 336 = ₹1,176. Wrong! The discount was 40% of the original price, not the sale price.
Trick 6: Successive Percentage Changes
When two percentage changes are applied one after another, they don't simply add.
Combined effect = A + B + (A × B / 100)
Example: A price first increases by 20%, then decreases by 10%. Combined = 20 + (−10) + (20 × −10 / 100) = 10 − 2 = +8% net increase
This formula is critical for problems involving two successive discounts or two successive markups.
Example: A shirt has two successive discounts: first 30% off, then 20% off. Combined discount = −30 + (−20) + (−30 × −20 / 100) = −50 + 6 = −44% net discount
So the final price = original × 0.56 (not 0.50 as many assume).
Trick 7: Percentage Points vs Percentage
This distinction is critical and often confused.
Percentage point = absolute difference in percentage values Percentage = relative change
Example: Interest rates rise from 5% to 7%.
- Increase in percentage points = 7 − 5 = 2 percentage points
- Increase in percentage = (2/5) × 100 = 40%
Both statements are technically correct but mean very different things. "Rates rose 2 percentage points" ≠ "Rates rose 2%."
Common Real-Life Percentage Problems
Calculating Discount
Discount % = ((MRP − Sale Price) / MRP) × 100
Example: T-shirt MRP ₹1,500, sale price ₹1,050 Discount % = ((1,500 − 1,050) / 1,500) × 100 = (450/1,500) × 100 = 30% off
Calculating GST
GST Amount = Base Price × (GST Rate / 100) Final Price = Base Price + GST Amount = Base Price × (1 + GST Rate/100)
Example: Item costs ₹10,000, GST = 18% GST = 10,000 × 0.18 = ₹1,800 Final price = ₹11,800
Exam Score Percentage
Score % = (Marks Obtained / Total Marks) × 100
Example: You scored 386 out of 500 = (386/500) × 100 = 77.2%
Salary Hike Percentage
Hike % = ((New Salary − Old Salary) / Old Salary) × 100
Example: Salary went from ₹45,000 to ₹52,000 = ((52,000 − 45,000) / 45,000) × 100 = (7,000/45,000) × 100 = 15.6%
Inflation / Depreciation
New Value after X% depreciation = Original × (1 − X/100)^n
Example: Car worth ₹8 lakh depreciates 15%/year for 3 years. = 8,00,000 × (0.85)^3 = 8,00,000 × 0.6141 = ₹4.91 lakh
Percentage Problems in Competitive Exams
Percentages feature heavily in SSC, Banking (IBPS, SBI PO), CAT, GMAT, and school exams (CBSE Class 7–10). Common question types:
Type 1: Population Problems
"A city's population increased 10% last year and 15% this year. If current population is 5,17,000, what was it 2 years ago?"
Use: Original = Current / (1.10 × 1.15) = 5,17,000 / 1.265 = 4,08,696
Type 2: Income-Expenditure Problems
"A person spends 70% of income. If income increases by 20% and expenditure by 10%, find % change in savings."
Let income = 100, spending = 70, savings = 30 New income = 120, new spending = 77, new savings = 43 Change in savings = ((43−30)/30) × 100 = 43.3% increase
Type 3: Mixture/Alloy Problems
Use percentages to find the concentration of one component in a mixture — common in chemistry and aptitude tests.
Quick Reference: Common Percentages as Fractions
Memorise these for faster calculations:
| Percentage | Fraction | Decimal |
|---|---|---|
| 10% | 1/10 | 0.1 |
| 12.5% | 1/8 | 0.125 |
| 16.67% | 1/6 | 0.1667 |
| 20% | 1/5 | 0.2 |
| 25% | 1/4 | 0.25 |
| 33.33% | 1/3 | 0.333 |
| 37.5% | 3/8 | 0.375 |
| 50% | 1/2 | 0.5 |
| 62.5% | 5/8 | 0.625 |
| 66.67% | 2/3 | 0.667 |
| 75% | 3/4 | 0.75 |
Frequently Asked Questions
How do I calculate percentage increase?
((New Value − Old Value) / Old Value) × 100. If the result is positive, it's an increase. Negative means a decrease.
How do I find the original price after a discount?
Original = Sale Price / (1 − Discount%/100). Never subtract the discount % from the sale price directly.
What is 30% of 500?
30/100 × 500 = 150. Or: 10% = 50, so 30% = 150.
What percentage is 60 of 200?
(60/200) × 100 = 30%.
How do I add two successive discounts?
Use: Combined = A + B + (A×B/100) where A and B are both negative (discounts). Example: 20% and 10% = −20 + (−10) + 2 = −28%. Net effective discount is 28%, not 30%.
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