Concept 1
Percentage means per hundred
x% means x/100. The quantity represented by 100% is the base. Convert percent to decimal by dividing by 100; convert a decimal to percent by multiplying by 100.
| Percent | Fraction | Decimal |
|---|---|---|
| 50% | 1/2 | .5 |
| 25% | 1/4 | .25 |
| 12.5% | 1/8 | .125 |
| 66⅔% | 2/3 | .666… |
Concept 2
The three core questions
Find the part
Part = rate × whole. 18% of 250 = 45.
Find the rate
Rate = part/whole ×100. 36 out of 240 = 15%.
Find the whole
Whole = part/rate. If 35% is 84, the whole is 240.
Concept 3
Change uses the original value
Percentage change = (new−original)/original ×100. A rise from 80 to 100 is 25%; the reverse fall is 20% because the base changes.
Successive changes multiply: a 20% rise then 10% fall gives 1.20×.90=1.08, an 8% net rise. Equal rise and fall percentages do not cancel.
Concept 4
Reverse by dividing by the multiplier
If the value after a 25% rise is 500, the original is 500/1.25=400. If a price after a 20% discount is ₹960, the marked price is 960/.80=₹1,200.
Applications
Where exams use percentages
Price and consumption
For fixed expenditure, a 25% price rise requires consumption ×1/1.25=.8, a 20% reduction.
Marks
If 180 marks are 72%, maximum marks are 180/.72=250.
Population
Repeated 10% growth for two years is ×1.1²=1.21, or 21% overall.
Percentage points
40% to 50% is 10 points but a 25% relative rise.
Worked methods
Examples with reasoning
Convert 7/16 to a percentage
7÷16=.4375; multiply by 100 to obtain 43.75%.
A salary rises 15% to ₹34,500
The final salary is 115% of the original. Original=34,500÷1.15=₹30,000.
A value falls 30%, then rises 20%
Use signed multipliers: .70×1.20=.84. The final value is 84% of the original, a 16% net fall.
Pass marks are 40%; a learner gets 144 and misses by 16
Pass marks=160. If 160 is 40% of the maximum, maximum marks=160÷.40=400.
A is 25% more than B
If B=100, A=125. Therefore B is 25/125×100=20% less than A. The reverse comparison uses a different base.
Exam traps
Errors that change the answer
Wrong base
Always identify which quantity represents 100% before calculating.
Equal changes cancel
A 20% rise and 20% fall give 1.2×.8=.96: a 4% fall, not zero.
Adding successive rates
Changes compound; multiply their factors instead of adding blindly.
Reverse by subtraction
To recover an original, divide the final value by its multiplier.
Different units
Convert quantities to the same unit before forming the percentage fraction.
Points versus percent
A direct gap between rates is measured in percentage points; relative change uses the earlier rate as base.
Original practice
Check understanding after learning
Rapid revision
The one-minute sheet
- x%=x/100; first identify the base represented by 100%.
- Part=rate×whole; rate=part/whole; whole=part/rate.
- Percentage change=difference/original×100.
- Use 1+r/100 for an increase and 1−r/100 for a decrease.
- Successive changes multiply; reverse percentages divide.
- For signed successive changes a and b, net%=a+b+ab/100.
- Percentage points measure the direct gap between two percentage rates.
Source and authorship ledger
Official foundations
- SSC CGL official syllabus: Percentage in Quantitative Aptitude and Mathematical Abilities
- NCERT official mathematics exemplar directory
- NCERT Class VII Comparing Quantities exemplar
Explanations, worked examples and practice questions are original course material. Official references are credited and not reproduced. Reviewed 24 September 2026.
Ratio and proportion: equivalent ratios, division and applications.