The three questions behind most percentage problems
Almost every percentage question is a version of one of three: find a percentage of a number, find what percentage one number is of another, or find the change between two values. Learn the common fractions and many calculations become instant: 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 16.67% = 1/6, 14.29% = 1/7, 11.11% = 1/9, 10% = 1/10, 8.33% = 1/12 and 6.25% = 1/16.
Core methods
- X% of N: N × X ÷ 100. Use 10% and 1% building blocks: 35% of 240 = 24 × 3 + 12 = 84.
- Percentage change: (new − old) ÷ old × 100. The old value is always the base.
- Successive changes: multiply the factors. +20% then −10% is 1.2 × 0.9 = 1.08, a net 8% increase.
- Percentage point vs percent: a rise from 20% to 25% is 5 percentage points, but a 25% increase.
- Reverse percentage: if a value after a 20% increase is 60, the original is 60 ÷ 1.2 = 50.
Worked examples
Example 1. A price rises from 80 to 92. The rise is 12, and 12 ÷ 80 = 15%.
Example 2. A number is increased by 25% and then decreased by 20%. Net change: 1.25 × 0.8 = 1.0, so no change at all.
Example 3. In an election between two candidates, one gets 55% of the votes cast and wins by 2,400 votes. How many votes were cast? The margin is 55% − 45% = 10% of the votes, so the total is 24,000.
Example 4. A town's population grows by 10% in the first year and falls by 10% in the second. Starting from 50,000: 55,000 after year one, then 49,500. Net change: 1.1 × 0.9 = 0.99, a 1% fall.
Example 5. If A's salary is 25% more than B's, B's salary is what percentage less than A's? Take B as 100, A as 125. B is 25 less than A, and 25 ÷ 125 = 20%.
Notice Example 5: the percentage difference depends on which value is the base. This is the trap that catches most candidates.
Common traps
- Adding percentages for successive changes instead of multiplying factors.
- Using the wrong base: "A is 25% more than B" and "B is 25% less than A" are not the same.
- Confusing percent and percentage points.
- Forgetting that a 100% rise doubles a quantity and that a fall of more than 100% is impossible for positive quantities.
Speed tips
- Convert percentages to fractions where possible; 16.67% is 1/6.
- For successive changes, use the shortcut a + b + ab ÷ 100: 20 and −10 give 20 − 10 − 2 = 8.
- Estimate first when options are far apart.
More in mental maths techniques.
A practice routine
- Day 1: fractions to percentages, ten conversions, then ten "X% of N" questions.
- Day 2: percentage change and the base trap, ten questions.
- Day 3: successive changes and population, ten questions.
- Day 4: a timed set of 15 mixed questions at 45 seconds each.
- Day 5: review errors; add traps to your error log.
Where percentage appears in exams
Percentage underpins profit and loss, interest, data interpretation and mixtures, and appears directly in SSC CGL quantitative aptitude, banking quantitative aptitude, CAT quantitative aptitude, CLAT quantitative techniques and GATE General Aptitude.
Where to go next
Once percentage feels easy, move to profit and loss, data interpretation and averages. Practice tests below are published by institutes and educators on TestMint; each shows its creator, question count and duration.
