Active and passive voice: rules for every tense
How the verb changes in each tense, pronoun changes, modals and imperative sentences, with SSC examples.
Study notes
Percentage: formulas, shortcuts and solved examples
Fraction–percent table, successive change, population and depreciation, and the shortcuts that save time in SSC questions.
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What a percentage means
A percentage is a fraction with 100 as the denominator: x% = x/100. So 25% of 360 = 360 × 25/100 = 90.
Fractions you must know by heart
| Fraction | Percent | Fraction | Percent |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% |
| 1/3 | 33.33% | 1/11 | 9.09% |
| 1/4 | 25% | 1/12 | 8.33% |
| 1/5 | 20% | 1/15 | 6.66% |
| 1/6 | 16.66% | 1/16 | 6.25% |
| 1/7 | 14.28% | 1/20 | 5% |
| 1/8 | 12.5% | 1/25 | 4% |
Shortcut: 37.5% = 3/8, 62.5% = 5/8, 87.5% = 7/8, 83.33% = 5/6, 66.66% = 2/3. Turning a percent into a fraction usually makes the arithmetic one step.
Increase, decrease and percentage change
Percentage change = (Change ÷ Original value) × 100
New value after r% increase = Original × (100 + r)/100
New value after r% decrease = Original × (100 − r)/100
New value after r% increase = Original × (100 + r)/100
New value after r% decrease = Original × (100 − r)/100
Always divide by the original value, not the new one. A rise from 80 to 100 is 25%, but a fall from 100 to 80 is 20%.
Successive percentage change
Two changes a% and b% in a row give a net change of: a + b + (a × b)/100
(use a minus sign for a decrease)
(use a minus sign for a decrease)
Example: a 20% rise followed by a 10% fall → 20 − 10 + (20 × −10)/100 = 8% net rise.
Equal rise and fall of x% always gives a net loss of x²/100 %. A 10% rise then a 10% fall = 1% loss.
If A is r% more than B
B is less than A by: r/(100 + r) × 100 %
If A is r% less than B, B is more than A by: r/(100 − r) × 100 %
If A is r% less than B, B is more than A by: r/(100 − r) × 100 %
Example: A’s salary is 25% more than B’s. B’s salary is less than A’s by 25/125 × 100 = 20%.
Price and consumption
If the price of an item rises by r%, to keep spending the same, consumption must fall by r/(100 + r) × 100 %. If the price falls by r%, consumption can rise by r/(100 − r) × 100 %.
Population and depreciation
Population after n years = P × (1 + r/100)ⁿ
Value after n years of depreciation = P × (1 − r/100)ⁿ
Value after n years of depreciation = P × (1 − r/100)ⁿ
Solved examples
- A number is increased by 20% and then decreased by 20%. What is the net change?
Net = −(20²/100) = −4%. The number falls by 4%. - The price of sugar rises by 25%. By what percentage must a family cut consumption to keep its spending the same?
25/125 × 100 = 20%. - In an election, a candidate got 55% of the votes and won by 2,400 votes. How many votes were cast?
The margin is 55% − 45% = 10% of the votes. 10% = 2,400, so 100% = 24,000 votes. - A town’s population is 50,000 and grows 10% a year. What will it be in 2 years?
50,000 × 1.1 × 1.1 = 60,500.
Common traps
- Using the new value as the base for the percentage change.
- Adding successive percentages directly (20% + 10% is not 30%).
- Forgetting that “percentage points” is a difference of percentages, not a percentage change.
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