Percentage: shortcuts and formulas
One rule per row, each checked against a real TNPSC question
| Concept | Rule | Worked check |
|---|---|---|
| Fractions worth knowing (1) | 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/8 = 12.5% | Clothing 75 of total 600 = 1/8 = 12.5% |
| Fractions worth knowing (2) | 1/3 = 33 1/3%, 1/6 = 16 2/3%, 1/9 = 11 1/9%, 1/18 = 5 5/9% | 15 paise of Rs 2.70 = 15/270 = 1/18 = 5 5/9% |
| Small percentages as fractions | 0.5% = 1/200; 1% = 1/100; 2% = 1/50 | Cloth shrinks 0.5% -> 1/200 |
| x% of y | x% of y = x × y / 100 = y% of x | If x% of y = 100 and y% of z = 200, then z = 2x |
| What per cent is a of b | (a / b) × 100 | 3% is what % of 5%? 3/5 × 100 = 60% |
| Percentage of a percentage | Multiply the decimals one after another | 10% of 20% of 25% of 8000 = 0.1 × 0.2 × 0.25 × 8000 = 40 |
| Whole from a part | Whole = part × 100 / per cent | 80% = 576 marks -> maximum = 576 × 100/80 = 720 |
| r% more / r% less than N | N × (100 + r)/100 or N × (100 − r)/100 | 15% less than 240 = 240 × 0.85 = 204 |
| Original value before a rise (reverse %) | Original = final × 100 / (100 + r) | Price with 8% VAT Rs 9,720 -> 9,720 × 100/108 = Rs 9,000 |
| Two successive changes a% and b% | Net % = a + b + (a × b)/100 (decrease = negative) | Length −50%, breadth +80%: −50 + 80 − 40 = −10% area |
| Up r% then down r% (or the reverse) | Always a loss of r²/100 % | +10% then −10% -> 1% loss; −50% then +50% -> 25% loss |
| Successive discounts | Single discount = a + b − (a × b)/100 | 25% and 20% -> 25 + 20 − 5 = 40% |
| Price rises r%: cut consumption to keep spending same | Cut = r / (100 + r) × 100 % | Tea up 20% -> 20/120 = 16 2/3%; bread up 25%: 300 loaves -> 240 |
| Value falls r%: rise needed to get back | Rise = r / (100 − r) × 100 % | Salary cut 10% -> raise 10/90 = 11 1/9% |
| A is r% less than B: how much is B more than A | r / (100 − r) × 100 % | 20% less -> 25% more; 25% less -> 33 1/3% more |
| Price up, sales down: change in income | Use a + b + ab/100 with sales as the negative change | Price +30%, sales −40%: 30 − 40 − 12 = 22% decrease |
| Area changes by p%: change in side | New side = old side × √(1 + p/100) | Area +69% -> √1.69 = 1.3 -> side +30% |
| Savings when income and spending both change | Take income = 100; savings = income − expenditure | Spends 75%; income +20%, spending +10%: savings 25 -> 37.5 = +50% |
| Pass % of a mixed group | Weight each group's % by its share, then add | Boys:girls 5:4, fail 18% and 9% -> fail (90 + 36)/9 = 14%, pass 86% |
| Growth over n years (population) | P × (1 + r/100)^n | 24,000 at 5% for 3 years = 24,000 × 1.157625 = 27,783 |
| Value n years ago (growth reversed) | P ÷ (1 + r/100)^n | Tree 847 cm, grows 10% a year -> 847 ÷ 1.21 = 700 cm two years ago |
| Depreciation over n years | V × (1 − r/100)^n | Rs 70,000 at 4% for 2 years = 70,000 × 0.9216 = Rs 64,512 |
| Numerator and denominator both change | New fraction = old × (100 + a)/(100 − b) | Num +15%, den −8% gives 15/16 -> old = 15/16 × 92/115 = 3/4 |
Remember
- Up-and-down by the same r% is never 'no change': it is always a loss of r²/100 % (TNPSC offers 'No change' as the trap).
- Consumption cut uses (100 + r) below; the 'raise back' after a fall uses (100 − r) below. 20 -> 16 2/3; 10 -> 11 1/9.
- For any % story, start with 100: price, income, population. The answer falls out as a plain difference.
Asked in TNPSC
- If price of a tea increased by 20%, by what percentage must the consumption be reduced to keep the expense the same (2022, CCS1P/22)
- Raman's salary was decreased by 50% and subsequently increased by 50%. How much percent does he lose? (2017, STGS/17)
- The salary of a person was reduced by 10%. By what percent should his reduced salary be raised so as to bring it at par with his original salary? (2018, JACGS/18)
- If A's salary is 20% less than B's salary by how much percent is B's salary more than A's? (2017, ADIGS/17)
- The population of a city was 24,000 in the year 1997. It increased at the rate of 5% per annum. Find the population at the end of the year 2000. (2023, IOFGS/2023)
Built from: Samacheer Class 7 Mathematics term 3, lesson 2 (Percentage and Simple Interest); Samacheer Class 8 Mathematics, lesson 4 (Life Mathematics); TNPSC past papers with the Commission's keys
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