The five percentage calculations you actually need
Almost every real-world percentage question falls into one of five types:
- "What is X% of Y?" The straightforward percentage of a number. Used for discounts, taxes, tips, interest calculations.
- "X is what percent of Y?" Finding the percentage relationship between two numbers. Used for grade calculations, market share, progress tracking.
- "What is the percentage change from X to Y?" Measuring increase or decrease. Used for inflation, growth rates, price comparisons.
- "X is P% of what?" Reverse percentage: find the whole when you know a part and its percentage. Used for reverse GST/VAT, finding original prices after a discount.
- "What is X increased/decreased by P%?" Apply a percentage change to get the new value. Used for salary increments, price adjustments, budget planning.
These are the five modes in the calculator. Once you know which type of question you're asking, the answer is mechanical.
Mode A: Percentage of a number
Formula: Result = (Percentage ÷ 100) × Number
Examples:
- 18% GST on ₹5,000: (18 ÷ 100) × 5,000 = ₹900
- 15% service charge on a ₹2,400 bill: (15 ÷ 100) × 2,400 = ₹360
- 20% discount on ₹1,200: (20 ÷ 100) × 1,200 = ₹240 savings → final price ₹960
Mode B: What percent is X of Y?
Formula: Result = (X ÷ Y) × 100
Examples:
- 48 out of 60 in an exam: (48 ÷ 60) × 100 = 80%
- ₹2,400 savings on a ₹15,000 salary: (2,400 ÷ 15,000) × 100 = 16%
- 3 defective items in a batch of 500: (3 ÷ 500) × 100 = 0.6%
This mode is also the basis of reverse tax calculations. If you know the tax-inclusive price and the tax rate, you can find the base price:
- Price including 18% GST = ₹5,900
- Base price = 5,900 ÷ 1.18 = ₹5,000
- GST paid = 5,900 − 5,000 = ₹900 (which is 18% of ₹5,000 ✓)
Mode C: Percentage change
Formula: Change = ((New − Original) ÷ |Original|) × 100
Examples:
- Revenue grew from ₹8L to ₹9.6L: ((9.6 − 8) ÷ 8) × 100 = +20% increase
- Stock fell from ₹450 to ₹360: ((360 − 450) ÷ 450) × 100 = −20% decrease
- Population grew from 1.38B to 1.44B: ((1.44 − 1.38) ÷ 1.38) × 100 = +4.35%
Note: the denominator is always the original (starting) value, not the new value. A common error is calculating relative to the new value, which gives a different (and wrong) answer.
Mode D: Find the whole (reverse percentage)
Formula: Whole = Part ÷ (Percentage ÷ 100)
This is the reverse of Mode A. You know a part and what percentage it represents; you want the total.
Examples:
- GST of ₹900 is 18% of the base price: 900 ÷ 0.18 = ₹5,000
- You saved ₹2,400 which is 16% of your salary: 2,400 ÷ 0.16 = ₹15,000
- A discount of ₹240 is 20% of the original price: 240 ÷ 0.20 = ₹1,200
This is also called "reverse percentage" or "find the original value." The common use case in India is reverse GST: you have the tax-inclusive price and want to back out the base amount.
Mode E: Apply a percentage change
Formula: Result = Original × (1 + Percentage ÷ 100)
Use a positive percentage for an increase, a negative percentage for a decrease.
Examples:
- Salary of ₹50,000 with an 8% increment: 50,000 × 1.08 = ₹54,000
- Product price of ₹1,200 after a 15% discount: 1,200 × 0.85 = ₹1,020
- Revenue of ₹8L after a 20% decline: 8,00,000 × 0.80 = ₹6,40,000
The difference between Mode C and Mode E: Mode C takes two actual values and tells you the percentage change between them. Mode E takes an original value and a percentage, and gives you the new value after applying that change.
Percentage points vs percentages
These are frequently confused:
- If interest rates go from 6% to 8%, they increased by 2 percentage points (absolute difference)
- They also increased by 33.3% (relative change: (8−6) ÷ 6 × 100)
The distinction matters enormously in financial and political contexts. "Inflation fell from 7% to 5%" is a 2 percentage point drop, but a 28.6% reduction in the inflation rate.
Compounding vs simple percentage
Percentage change calculations assume simple (non-compounding) changes. For compound growth over multiple periods, you need the CAGR (Compound Annual Growth Rate) formula:
CAGR = (Final ÷ Initial)^(1/years) − 1
For example, if ₹1L grew to ₹2.59L over 10 years, the CAGR = (2.59)^(0.1) − 1 = 10%. Using Mode C naively would give ((2.59−1) ÷ 1) × 100 = 159%, which is the total return, not the annualized return.
Mental math shortcuts
- 10%: Move the decimal left by one place. 10% of ₹3,750 = ₹375.
- 5%: Half of 10%. 5% of ₹3,750 = ₹187.5.
- 15%: 10% + 5%. 15% of ₹3,750 = ₹375 + ₹187.5 = ₹562.5.
- 25%: Divide by 4.
- 33.3%: Divide by 3.
- 1%: Move the decimal left by two places.