Percentages
Increase, decrease, successive percentage change.
Percentage to fraction conversions
Common percentages: 12.5% = 1/8, etc.
Percentage = per hundred. x% = x/100.
FRACTION ↔ PERCENT QUICK TABLE:
| Fraction | % | Fraction | % |
|---|---|---|---|
| 1/2 | 50 | 1/6 | 16.67 |
| 1/3 | 33.33 | 1/7 | 14.28 |
| 1/4 | 25 | 1/8 | 12.5 |
| 1/5 | 20 | 1/9 | 11.11 |
| 1/10 | 10 | 1/11 | 9.09 |
| 1/12 | 8.33 | 1/16 | 6.25 |
Memorize → instant calculation.
TYPES OF PROBLEMS:
1. Find x% of y: = (x/100) × y.
- 15% of 200 = 30.
2. What % is A of B? = (A/B) × 100.
- 60 is what % of 80? 60/80 × 100 = 75%.
3. Percentage increase: = ((new − old)/old) × 100.
- From 50 to 60: increase = 10/50 × 100 = 20%.
4. Percentage decrease: = ((old − new)/old) × 100.
- From 80 to 60: decrease = 20/80 × 100 = 25%.
SUCCESSIVE PERCENTAGES:
If a value increases by a%, then by b%:
Net change = a + b + ab/100 (signs as ± for increase/decrease).
Example: Price up 20%, then up 10%.
Net = 20 + 10 + 200/100 = 32%.
Example: Salary up 50%, then down 50%.
Net = 50 − 50 + (50×−50)/100 = −25% (so net DECREASE of 25%).
SHORTCUTS:
a% of b = b% of a. (Useful: 15% of 60 = 60% of 15 = 9. Either way.)
Increase by % then decrease by same % → always decrease:
- Loss % = (a²/100) where a is the common percentage.
- 20% up, 20% down → 4% loss.
Population growth:
- After n years at r% growth: P × (1 + r/100)ⁿ.
Income & expenditure:
- If income up x%, expenditure up y%, then saving change varies.
- Saving = Income − Expenditure.
EXAMPLE 1:
The price of an article is increased by 10% and then decreased by 10%. The net change is:
Net = 10 − 10 + (10×−10)/100 = −1%. So 1% decrease.
EXAMPLE 2:
A man's salary is increased by 20%. By what percent should the new salary be decreased to get back to the original?
Let original = 100. New = 120. To get 100 from 120, decrease by 20/120 × 100 = 16.67% (= 1/6 in fraction).
EXAMPLE 3:
In an election with two candidates, the winner got 60% of votes and won by 240 votes. Total votes?
Winner − Loser = 60% − 40% = 20% of total = 240. So total = 1200.
EXAMPLE 4 (population):
Population is 50000. Grows at 10% annually. Find population after 3 years.
P = 50000 × (1.1)³ = 50000 × 1.331 = 66550.
EXAM HOOKS:
- Use fractional shortcuts: 12.5% = 1/8; 33.33% = 1/3.
- Successive %: a + b + ab/100 (BOTH must be in same direction; if reverse, use signed values).
- "Decrease such that final = original after x% increase": decrease % = x/(100+x) × 100.
Successive percentage change
a + b + ab/100 formula.
For SSC and Banking exams, raw arithmetic is too slow. Memorize these conversions:
Common percentage ↔ fraction:
| % | Fraction |
|---|---|
| 6.25% | 1/16 |
| 8.33% | 1/12 |
| 9.09% | 1/11 |
| 10% | 1/10 |
| 11.11% | 1/9 |
| 12.5% | 1/8 |
| 14.28% | 1/7 |
| 16.67% | 1/6 |
| 20% | 1/5 |
| 25% | 1/4 |
| 33.33% | 1/3 |
| 50% | 1/2 |
Successive percentage change formula: if a quantity changes by a%, then by b%, the net change is:
Net % = a + b + (ab/100)
(Use negative signs for decreases.)
Example: price increases 20%, then decreases 10%. Net = 20 + (−10) + (20·−10/100) = 10 − 2 = +8%.
% increase ↔ % decrease relationship: if x increases by p% to y, then y must decrease by [p/(100+p)]·100% to return to x.
Example: price increased 25% (1.25×). To return: decrease by 25/125 × 100 = 20%.
Percentage and ratio: "A is x% more than B" means A/B = (100 + x)/100.
Example: salary of A is 25% more than B. A:B = 125:100 = 5:4. So if their total is ₹4500, A = 5/9 × 4500 = ₹2500.
Percentage = per hundred. x% = x/100.
FRACTION ↔ PERCENT QUICK TABLE:
| Fraction | % | Fraction | % |
|---|---|---|---|
| 1/2 | 50 | 1/6 | 16.67 |
| 1/3 | 33.33 | 1/7 | 14.28 |
| 1/4 | 25 | 1/8 | 12.5 |
| 1/5 | 20 | 1/9 | 11.11 |
| 1/10 | 10 | 1/11 | 9.09 |
| 1/12 | 8.33 | 1/16 | 6.25 |
Memorize → instant calculation.
TYPES OF PROBLEMS:
1. Find x% of y: = (x/100) × y.
- 15% of 200 = 30.
2. What % is A of B? = (A/B) × 100.
- 60 is what % of 80? 60/80 × 100 = 75%.
3. Percentage increase: = ((new − old)/old) × 100.
- From 50 to 60: increase = 10/50 × 100 = 20%.
4. Percentage decrease: = ((old − new)/old) × 100.
- From 80 to 60: decrease = 20/80 × 100 = 25%.
SUCCESSIVE PERCENTAGES:
If a value increases by a%, then by b%:
Net change = a + b + ab/100 (signs as ± for increase/decrease).
Example: Price up 20%, then up 10%.
Net = 20 + 10 + 200/100 = 32%.
Example: Salary up 50%, then down 50%.
Net = 50 − 50 + (50×−50)/100 = −25% (so net DECREASE of 25%).
SHORTCUTS:
a% of b = b% of a. (Useful: 15% of 60 = 60% of 15 = 9. Either way.)
Increase by % then decrease by same % → always decrease:
- Loss % = (a²/100) where a is the common percentage.
- 20% up, 20% down → 4% loss.
Population growth:
- After n years at r% growth: P × (1 + r/100)ⁿ.
Income & expenditure:
- If income up x%, expenditure up y%, then saving change varies.
- Saving = Income − Expenditure.
EXAMPLE 1:
The price of an article is increased by 10% and then decreased by 10%. The net change is:
Net = 10 − 10 + (10×−10)/100 = −1%. So 1% decrease.
EXAMPLE 2:
A man's salary is increased by 20%. By what percent should the new salary be decreased to get back to the original?
Let original = 100. New = 120. To get 100 from 120, decrease by 20/120 × 100 = 16.67% (= 1/6 in fraction).
EXAMPLE 3:
In an election with two candidates, the winner got 60% of votes and won by 240 votes. Total votes?
Winner − Loser = 60% − 40% = 20% of total = 240. So total = 1200.
EXAMPLE 4 (population):
Population is 50000. Grows at 10% annually. Find population after 3 years.
P = 50000 × (1.1)³ = 50000 × 1.331 = 66550.
EXAM HOOKS:
- Use fractional shortcuts: 12.5% = 1/8; 33.33% = 1/3.
- Successive %: a + b + ab/100 (BOTH must be in same direction; if reverse, use signed values).
- "Decrease such that final = original after x% increase": decrease % = x/(100+x) × 100.