Why it matters
Percentage is the base of profit-loss, interest, DI and mixtures. Mastering fraction equivalents alone can save 5 to 10 minutes in the exam. Target: 30 to 45 seconds per question.
Percentage-fraction table (memorise)
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/2 | 50% | 1/3 | 33 1/3% (33.333...%) |
| 1/4 | 25% | 1/5 | 20% |
| 1/6 | 16 2/3% (16.666...%) | 1/7 | 14 2/7% (14.285...%) |
| 1/8 | 12.5% | 1/9 | 11 1/9% (11.111...%) |
| 1/10 | 10% | 1/11 | 9 1/11% (9.090...%) |
| 1/12 | 8 1/3% (8.333...%) | 1/13 | 7 9/13% (7.692...%) |
| 1/14 | 7 1/7% (7.142...%) | 1/15 | 6 2/3% (6.666...%) |
| 1/16 | 6.25% | 1/20 | 5% |
| 1/25 | 4% | 2/3 | 66 2/3% (66.666...%) |
| 3/4 | 75% | 3/8 | 37.5% |
| 5/8 | 62.5% | 7/8 | 87.5% |
| 2/7 | 28 4/7% (28.571...%) | 4/9 | 44 4/9% (44.444...%) |
Key formulas
- x% of y = y% of x (8% of 25 = 25% of 8 = 2).
- Successive change: a% then b% gives net a + b + ab/100 (use minus sign for decreases). +20% then -20% gives -4%.
- If A is r% more than B, then B is r/(100 + r) x 100% less than A. (25% more means 20% less.)
- If price rises by r%, consumption must fall by r/(100 + r) x 100% to keep expenditure fixed.
- Population after n years: P(1 + r/100)^n; n years ago: P/(1 + r/100)^n.
- Expenditure = price x consumption, so % change = a + b + ab/100 with a, b as the two changes.
Election and examination formulas
- Winner's margin = (winner% - loser%) of valid votes.
- Pass marks problems: the difference of two percentages equals the difference of marks. If 30% fails by 30 and 45% passes by 45, then 15% = 75 marks, maximum = 500.
Worked example
A man spends 30% of his income on rent and 25% of the rest on food, and saves Rs. 21,000. Income?
Savings fraction = 0.70 x 0.75 = 0.525. Income = 21,000/0.525 = Rs. 40,000.
Worked example 2
The price of petrol rises by 25%. By what % should a car owner cut consumption to keep spending the same?
25/125 x 100 = 20%.
Base-100 trick: when no actual value is given, assume the starting quantity is 100. Every percentage becomes a plain number and the answer drops out directly.
Common traps
- Adding percentages taken on different bases ("30% of income, 25% of the remainder").
- Percentage points vs percent change.
- Using total votes instead of valid votes for the margin.
- Successive change on a decrease: -10% then -10% is -19%, not -20%.
Increase-decrease pairs (memorise)
| A is more than B by | B is less than A by |
|---|---|
| 20% | 16.67% (1/6) |
| 25% | 20% (1/5) |
| 50% | 33.33% (1/3) |
| 100% | 50% (1/2) |
Worked example 3 (election)
A candidate gets 55% of the valid votes and wins by 2,000 votes. Margin = 55% - 45% = 10% of valid votes, so valid votes = 2,000/0.10 = 20,000.
Time target: 30 seconds for one-step questions, 60 seconds for Mains multi-step questions.