Percentages

In short

A percentage expresses a quantity as parts per hundred, so any ratio or fraction can be rewritten as x%, meaning x/100. It is the single most widely used tool in quantitative aptitude because profit-loss, interest, discount, data interpretation, mixtures and growth questions are all percentage questions in disguise. Every campus aptitude test in India - TCS NQT, Infosys, Wipro Elite, Capgemini, Accenture, Cognizant, bank PO and SSC papers - carries direct percentage sums plus percentage-based data interpretation. On the job it drives GST and TDS working, salary hikes, attrition and shrinkage, target achievement and variance reporting in Excel.

Explained

Meaning, conversions and the base

• Percentage means "per hundred": x% = x/100. So 37% of a quantity = 0.37 × quantity.
• Fraction to percentage: multiply by 100 (3/8 = 37.5%). Percentage to fraction: divide by 100 and reduce (62.5% = 5/8).
• Decimal to percentage: shift the decimal two places right (0.045 = 4.5%); the reverse for percentage to decimal.
• The base is the quantity the percentage is taken of - "25% of 80" has base 80. Almost every error in this topic is a wrong base.
• Percentage of a percentage: 20% of 30% = 0.20 × 0.30 = 6%, not 50%.
• Percentage points vs percentage change: an interest rate moving from 6% to 9% rises by 3 percentage points but by 50% in relative terms. Regulators and news reports use "basis points" (1 bp = 0.01 percentage point), so 25 bp = 0.25 percentage point - the usual RBI repo-rate step.
• Memorising the fraction table (below) converts most sums into one-line mental arithmetic.

Increase, decrease and reverse working

• Increase by r%: new = old × (1 + r/100). Decrease by r%: new = old × (1 − r/100).
• Multiplying factor is the fast form: +15% → ×1.15; −15% → ×0.85; +100% → ×2; −50% → ×0.5.
• Percentage change = (new − old)/old × 100; positive is a rise, negative a fall. The old value is always the base.
• Reverse percentage (finding the original): if the final value after an r% rise is F, then original = F/(1 + r/100). After an r% fall, original = F/(1 − r/100). Never subtract r% from the final value.
• A is x% more than B → A = B(1 + x/100), so B = A/(1 + x/100); B is then (x/(100 + x)) × 100 % less than A - the two percentages are not equal.
• A is x% less than B → A = B(1 − x/100), and B is (x/(100 − x)) × 100 % more than A.
• Equivalent-fraction shortcut: "x% more" with x = 1/n in fraction form means new/old = (n+1)/n. 16⅔% more = 7/6 times; 20% less = 4/5 times.

Successive changes and net effect

• Two successive changes of a% and b% multiply, not add: net factor = (1 + a/100)(1 + b/100).
• Net percentage change = a + b + ab/100, with a decrease entered as a negative number.
• Equal rise and fall: +x% then −x% gives a net loss of x²/100 %, never zero (e.g. +20% then −20% → −4%).
• Three or more changes: keep multiplying factors - 1.10 × 0.90 × 1.05 = 1.0395, a 3.95% net rise.
• Order does not matter for the net result because multiplication is commutative; it matters only for intermediate values.
• Price-consumption rule: if price rises by r%, consumption must fall by (r/(100 + r)) × 100 % to keep expenditure unchanged; if price falls by r%, consumption may rise by (r/(100 − r)) × 100 %.
• Expenditure = price × quantity, so percentage changes in price and quantity combine by the same net-change formula.

Standard question types in Indian aptitude papers

• Marks and pass percentage: a student scoring m marks fails by d marks when the pass mark is p% → total = (m + d)/(p/100). Two-candidate versions give a pair of equations.
• Election problems: winner gets w% of valid votes, loser (100 − w)%; the victory margin is (2w − 100)% of valid votes. Invalid votes are removed from total votes first.
• Population or salary growth: value after n years at r% per year = P(1 + r/100)ⁿ; with different yearly rates, multiply each factor separately. Negative r for decline or depreciation.
• Alcohol/water or mixture replacement: removing and replacing a fraction k of a solution n times leaves pure quantity = P(1 − k)ⁿ.
• Percentage error: (|measured − true|/true) × 100; in aptitude papers, "a student multiplied by 3/5 instead of 5/3" type sums compare the wrong and correct answers.
• Income, savings and expenditure chains: express each item as a percentage of the one above it and multiply the factors.
• Percentage share in data interpretation: share of a part = part/total × 100; percentage contribution to growth and percentage distribution in pie charts both use this.

Where percentages sit inside other topics

• Profit and loss: profit% and loss% are taken on cost price; discount% on marked price - different bases in the same sum.
• Simple interest is a fixed percentage of the principal each year; compound interest is the successive-change formula applied to money.
• CAGR (compound annual growth rate) = [(final/initial)^1/n − 1] × 100 % - the single equivalent annual percentage behind n years of uneven growth.
• Ratio and proportion: a ratio a : b written as percentages of the total is a/(a+b) × 100 and b/(a+b) × 100.
• Averages and weighted averages: a weighted average of percentages must use the actual bases as weights, not a plain mean of the percentages.

Indian practice: where the numbers come from

• GST is levied as a percentage of taxable value under the CGST/SGST Acts, 2017; since 22 September 2025 the main rates are nil, 5% and 18%, with a 40% de-merit rate (the earlier 12% and 28% slabs were folded into these) and special rates such as 3% on gold. Tax is split equally between CGST and SGST for intra-state supply (IGST for inter-state); the GST Council revises rates from time to time, so treat any specific rate in a question as given data.
• Income tax is slab-based with surcharge and a 4% health and education cess computed on tax, and TDS is deducted at prescribed percentages - rates change with each Finance Act.
• EPF contribution is 12% of basic wages plus dearness allowance from both employee and employer under the EPF Act, 1952.
• RBI announces the repo rate and CRR/SLR as percentages, and moves the repo rate in basis points.
• Reservation, attendance and pass criteria: most Indian universities need 75% attendance and a 40% or 50% minimum in a subject; CGPA-to-percentage conversion is institution-specific (many use percentage ≈ CGPA × 9.5, an AICTE/CBSE-style convention, not a universal rule).

Key points

• Percentage - parts per hundred; x% = x/100, a pure number with no unit.
• Base - the quantity a percentage is taken of; percentage change always uses the old value as base.
• Multiplying factor - +r% → (1 + r/100), −r% → (1 − r/100); chain them for successive changes.
• Net change of two changes - a + b + ab/100, decreases taken as negative.
• Equal up-and-down - +x% then −x% always loses x²/100 %.
• Reverse percentage - divide by the factor; never subtract the same percentage back.
• x% more than B means B is x/(100 + x) × 100 % less than A - asymmetric by design.
• Price up r% - consumption must drop r/(100 + r) × 100 % to hold expenditure constant.
• Percentage points - the plain difference of two percentages; 100 basis points = 1 percentage point.
• Growth over n periods - P(1 + r/100)ⁿ; decline uses a negative r.
• CAGR - [(F/P)^1/n − 1] × 100 %, the smoothed annual rate.
• Percentage error - error relative to the true value, expressed per hundred.
• Pass-mark type - total = (marks scored + marks short)/(pass fraction).
• Election margin - (2w − 100)% of valid votes when the winner polls w%.
• Fraction table - 1/8 = 12.5%, 1/6 = 16⅔%, 1/3 = 33⅓%, 3/8 = 37.5%, 5/6 = 83⅓% - learn these to skip long division.
• Percentage of a percentage - multiply the decimals; 20% of 30% = 6%.
• Weighted average of percentages - weight by the real bases, never average the percentages.
• Successive discounts - two discounts of a% and b% equal a single discount of a + b − ab/100 %.

Formulas and examples

Example 1 - successive change and reverse working. A laptop priced at ₹60,000 is raised 25% for the festive catalogue, then sold at a 20% discount on the raised price.

• Raised price = 60,000 × 1.25 = ₹75,000.
• Selling price = 75,000 × 0.80 = ₹60,000.
• Net change = +25 − 20 + (25 × −20)/100 = 5 − 5 = 0%, so the final price equals the original ₹60,000.
• If instead the net effect were a 4% fall to ₹57,600, the original would be 57,600/0.96 = ₹60,000 - division, not subtraction.

Example 2 - price and consumption. Cooking-gas price rises from ₹800 to ₹960 per cylinder. By what percentage must a family cut usage to keep the monthly gas bill unchanged?

• Price rise = (960 − 800)/800 × 100 = 20%.
• Required cut = (20/(100 + 20)) × 100 = 100/6 = 16⅔%.
• Check: a family using 6 cylinders spent 6 × 800 = ₹4,800; cutting 16⅔% leaves 5 cylinders → 5 × 960 = ₹4,800. Bill unchanged.

Example 3 - pass marks. A candidate scores 190 marks and fails by 35 marks; the pass mark is 45% of the total.

• Pass marks = 190 + 35 = 225.
• Total = 225 ÷ 0.45 = 500 marks.
• The candidate's percentage = 190/500 × 100 = 38%, i.e. 7 percentage points short of 45%.

Example 4 - salary chain and CAGR. A graduate joins at ₹4,20,000 CTC; appraisals give +12%, then +8%, then −5% (a role change with lower variable pay).

• Final CTC = 4,20,000 × 1.12 × 1.08 × 0.95 = 4,20,000 × 1.14912 = ₹4,82,630 (to the nearest rupee).
• Net change = (1.14912 − 1) × 100 = +14.912% over three years.
• CAGR = (1.14912)^1/3 − 1 ≈ 0.0474 → about 4.74% per year, well below the plain average of (12 + 8 − 5)/3 = 5%.

Example 5 - percentage share with GST. An invoice shows a taxable value of ₹25,000 at an 18% GST slab.

• GST = 25,000 × 0.18 = ₹4,500, split as CGST ₹2,250 (9%) and SGST ₹2,250 (9%) for an intra-state supply.
• Invoice total = ₹29,500; tax as a share of the invoice total = 4,500/29,500 × 100 = 15.25%, not 18% - the base changed from taxable value to total.
• Working backwards from a tax-inclusive price of ₹29,500 at 18%: taxable value = 29,500/1.18 = ₹25,000.

Where it is used

• Finance and accounting: GST and TDS computation, interest and EMI schedules, margin and variance analysis in Tally and SAP; roles in Infosys BPM, Genpact, WNS, EXL.
• Banking and NBFC: interest spreads, NPA ratios, LTV percentages and credit-score cut-offs; RBI reporting uses percentages and basis points throughout.
• Retail and e-commerce: markdown and successive-discount pricing, conversion rate, return rate and GMV growth at Reliance Retail, Flipkart, Myntra.
• BPO/ITES workforce management: shrinkage, attrition, occupancy, SLA and quality scores are all percentage metrics reported daily in Excel dashboards.
• Manufacturing and quality: yield, rejection and scrap percentages, OEE, PPM defect rates and percentage error in measurement and calibration.
• Analytics and consulting: percentage contribution, growth decomposition and CAGR in client decks; business analyst and MIS executive roles ask these in the interview itself.

Interview questions

• Q: Why does a 20% rise followed by a 20% fall not return to the original value? A: The fall is applied to the raised value, a bigger base, so the loss exceeds the gain; the net is −x²/100 = −4%.
• Q: A price rose 25%. By what percentage must it fall to come back to the original? A: By 25/125 × 100 = 20%, because the fall is calculated on the higher price.
• Q: What is the difference between percentage and percentage points? A: Percentage points are the arithmetic difference between two percentages; percentage change is that difference relative to the starting percentage - 6% to 9% is +3 points but +50%.
• Q: If A's salary is 25% more than B's, how much less is B's than A's? A: 25/125 × 100 = 20% less; the two directions are never equal because the bases differ.
• Q: Two successive discounts of 20% and 10% - what single discount is equivalent? A: 20 + 10 − (20 × 10)/100 = 28%.
• Q: How do you extract the base price from a GST-inclusive amount? A: Divide by (1 + rate/100) - ₹29,500 at 18% gives ₹25,000; subtracting 18% of the inclusive amount is wrong.
• Q: Is the average of 40% and 70% equal to 55%? A: Only if both groups are the same size; otherwise weight by the actual counts - 50 and 150 students give 62.5%.
• Q: What does CAGR tell you that an average annual growth rate does not? A: CAGR is the constant rate that would carry the initial value to the final value over n periods, so it accounts for compounding; a simple average of yearly rates overstates it.
• Q: Price falls 10% and consumption rises 10% - what happens to expenditure? A: It falls by 1%, since 0.90 × 1.10 = 0.99.
• Q: A number is increased by 10% and the result decreased by 10% - express the final value as a fraction of the original. A: 99/100, i.e. 0.99 of the original.
• Q: 60% of students passed and 80 failed - how many students were there? A: The 80 failures are 40%, so the total is 80/0.40 = 200.
• Q: Which is larger: 1/6 or 16%? A: 1/6 = 16⅔%, so 1/6 is larger - knowing the fraction-percentage table avoids such traps.

Common mistakes

• Adding successive percentages instead of multiplying the factors, so +10% then +10% is reported as +20% rather than +21%.
• Subtracting the same percentage to reverse an increase instead of dividing by the multiplying factor.
• Using the new value as the base in percentage change, which understates a rise and overstates a fall.
• Confusing percentage points with percentage change in interest-rate, margin and vote-share questions.
• Mixing bases within one sum - profit on cost price, discount on marked price, tax on taxable value - and comparing them as if they shared a base.
• Taking a plain average of percentages when the groups have different sizes, instead of a weighted average.
• Rounding intermediate steps (especially the CAGR root or a recurring fraction like 16⅔%) and then presenting the answer to the rupee.

Related topics

• Profit, loss and discount · Simple and compound interest · Ratio, proportion and partnership · Averages, mixtures and alligation · Data interpretation · Simplification and approximation