The only 3 formulas you need
Nearly every percentage question is one of three types. Finding the part: Part = (Percent ÷ 100) × Whole — e.g. 20% of 250 = 0.20 × 250 = 50. Finding the percentage: Percent = (Part ÷ Whole) × 100 — e.g. 45 marks out of 60 = (45 ÷ 60) × 100 = 75%. Finding the whole: Whole = Part ÷ (Percent ÷ 100) — e.g. 30 is 15% of what? 30 ÷ 0.15 = 200.
The trick is identifying which value is missing before touching the calculator. Ask: 'do I know the whole?' If yes and you need the part, multiply. If you have part and whole, divide. If you have the part and the rate, divide by the rate. Three patterns cover discounts, tips, taxes, scores, and salary hikes alike.
Shopping: discounts and stacked offers
A ₹1,999 shirt at 30% off: discount = 0.30 × 1,999 = ₹599.70, so you pay ₹1,399.30. Easy. But stacked offers trip people up: 'extra 10% off the already-discounted price' does not mean 40% off. After the 30% cut the price is ₹1,399.30, and 10% off that is ₹139.93 more — final price ₹1,259.37, an effective discount of 37%, not 40%.
Percentages multiply, they never add, when applied in sequence. Two successive 20% discounts equal a 36% total cut (0.8 × 0.8 = 0.64 of the original), and a 20% hike followed by a 20% cut leaves you 4% below where you started. Sellers count on shoppers adding instead of multiplying — now you know better.
- Single discount: price × (1 − discount rate)
- Stacked discounts multiply: 30% then 10% = 37% effective, not 40%
- 20% up then 20% down = 4% net loss (1.2 × 0.8 = 0.96)
- Always compute on the running price, not the original tag
Food bills: tips, tax, and splitting
A ₹1,200 bill with 10% tip and 5% GST: tax = 0.05 × 1,200 = ₹60, tip = 0.10 × 1,200 = ₹120 (tip on the pre-tax amount is the standard), total ₹1,380. Splitting among 4? ₹345 each. The common error is tipping on the tax-inclusive total — harmless once, but it adds up over a year of dining out.
For quick mental tips, use the 10% anchor: 10% of ₹1,200 is ₹120 (move the decimal one place), so 15% is ₹120 + ₹60 (half again) = ₹180, and 20% is simply double the 10% figure. This anchor method handles any bill in seconds without a calculator.
Percent change vs percentage points
If your investments grow from ₹50,000 to ₹65,000, the percent change is ((65,000 − 50,000) ÷ 50,000) × 100 = 30%. The direction matters: going from ₹65,000 back to ₹50,000 is a 23% fall, not 30% — because the base changed. Always divide by the starting value, never the ending one.
And do not confuse percent change with percentage points. If a home-loan rate rises from 8% to 9%, that is a 1 percentage-point increase but a 12.5% relative increase (1 ÷ 8). Politicians, advertisers, and headlines routinely blur this line; checking which one is meant takes five seconds and often reverses the conclusion.
Mental-math shortcuts that work
Three shortcuts handle most everyday cases. Flip it: 18% of 50 is the same as 50% of 18, which is 9 — swapping makes one side friendly. Build from 10% and 1%: 10% of 640 is 64 and 1% is 6.4, so 12% is 64 + 6.4 + 6.4 = 76.8. And for 'X is what % of Y', simplify the fraction first: 30 of 120 is 1/4, which is 25%.
Practice these on bills and price tags for a week and they become automatic. The calculator remains the tool for exact figures — EMIs, tax filings, investment returns — but estimation is the skill that tells you whether the calculator's answer is sane before you act on it.
- Flip: a% of b = b% of a — pick the easier direction
- Build from 10% and 1% anchors for any rate
- Simplify part/whole to a fraction before converting
- Estimate first, calculate exact second — catch input errors
Salary hikes, CAGR, and real growth
A 10% hike on a ₹60,000 monthly salary adds 0.10 × 60,000 = ₹6,000, taking you to ₹66,000 — but two consecutive 10% hikes do not total 20%. Year two compounds on the new base: 66,000 × 1.10 = ₹72,600, a total gain of 21% (1.10² = 1.21). Employers quoting '10% every year for two years' are describing 21% growth, and your rent negotiations should treat it that way.
Investments need the reverse lens: CAGR. If ₹50,000 grows to ₹65,000 over 3 years, the annualised return is (65,000 ÷ 50,000)^(1/3) − 1 = 1.30^0.333 − 1 ≈ 9.1% per year — not the 10% you get from dividing 30% by 3. Simple averages overstate multi-year growth because they ignore compounding; CAGR is the only honest yearly figure, which is why mutual funds must report it.
- ₹60,000 + 10% = ₹66,000; two 10% hikes = ₹72,600 (21%, not 20%)
- ₹50,000 → ₹65,000 in 3y is 30% total but only ~9.1% CAGR
- Never average yearly percentages — compound or use CAGR instead
- Apply to appraisals, rent escalation, and SIP return claims alike
Four worked everyday problems
GST: a ₹18,000 laptop with 18% GST adds 0.18 × 18,000 = ₹3,240, for a total of ₹21,240. Reverse it — a ₹21,240 price inclusive of 18% means the base is 21,240 ÷ 1.18 = ₹18,000 exactly. Dividing by (1 + rate) unpicks any tax-inclusive price, from restaurant bills to freelance quotes.
Scores and tips round out the set: 132 marks out of 150 is (132 ÷ 150) × 100 = 88%, while a 15% tip on an ₹860 bill is 0.15 × 860 = ₹129 — round to ₹130 and move on. Notice each uses a different one of the three core formulas (whole from inclusive total, percent from part and whole, part from percent and whole), which is precisely why identifying the missing value first solves 95% of problems.
- Add GST: base × (1 + rate); strip GST: inclusive ÷ (1 + rate)
- Exam score: (marks ÷ total) × 100 — 132/150 = 88%
- Tip: rate × pre-tax bill — 15% of ₹860 = ₹129
- Name the missing value (part, percent, or whole) before calculating
Percentage traps even smart people fall for
Trap one is averaging percentages with different bases: 50% off one store plus 30% off another is not 40% off overall unless both bills are identical — percentages only average cleanly over equal bases. Trap two is the base switch: a stock that falls 50% needs a 100% gain to recover (₹100 → ₹50 needs +₹50, which is 100% of ₹50), a fact loss-recovery emails hope you never compute.
Trap three is confusing '200% more' with '200% of': a tip rising from ₹100 to ₹300 is 300% of the original but 200% more than it — the word 'more' subtracts the base. When stakes are high — loan offers, medical risks, salary comparisons — always convert back to absolute rupees first, then decide. Percentages persuade; absolute numbers clarify.
- Never average percentages across unequal bases — weight by the base
- A 50% loss needs a 100% gain to break even; losses hurt more than gains help
- '200% more' = 3× total; '200% of' = 2× total — read the wording
- Convert to rupees before deciding anything expensive