Percentage – Complete Short Notes for Competitive Exams

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1. Basic Concept

• “Per cent” means per hundred.

• So, Percentage = (Value × 100) / Total value

• Symbol: %

$$\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100$$

Example:
If a student scored 40 marks out of 80:

$$(40\mathrm{/}80)\times 100=50\mathrm{\%}$$

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2. Conversion Rules

| Type                                   | Formula                          | Example    || --------------------------------| ----------------- ------------| ------------- || Fraction → %                    | (Fraction × 100)%          | 3/4 = 75%  || % → Fraction                    | % ÷ 100                          | 25% = 1/4  || Decimal → %                    | Decimal × 100               | 0.45 = 45% || % → Decimal                    | % ÷ 100                          | 35% = 0.35 |

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3. Common Fraction–Percentage Equivalents

Fraction	Percentage	Fraction	Percentage
1/2	50%	1/6	16.66%
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%	3/4	75%

Tip: Remember key fractions for faster mental calculations.

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4. Important Formulas

1. Percentage Value:

   $$\text{Value}=\frac{\text{Percentage}\times \text{Total}}{\mathrm{100}}$$

2. Finding Total from Percentage:

   $$\text{Total}=\frac{\text{Value}\times 100}{\text{Percentage}}$$
3. Increase or Decrease in Percentage:$$\text{Change \%}=\frac{\text{Difference}}{\text{Original value}}\times 100$$

4. Resultant Change (Successive % Change):
   If a quantity increases by x% and then by y%,

   $$\text{Net Change \%}=x+y+\frac{xy}{\mathrm{100}}$$

   If one is increase and other is decrease,

   $$\text{Net Change \%}=x-y-\frac{xy}{100}$$

Example:
If a price increases by 20% and again by 10%,
Net = 20 + 10 + (20×10)/100 = 32% increase

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5. Percentage Increase and Decrease

Situation	Formula	Example
Increase from A to B	$\frac{B - A}{A} × 100$	40→50 = 25% ↑
Decrease from A to B	$\frac{A - B}{A} × 100$	50→40 = 20% ↓

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6. Reverse Percentage (Back Calculation)

Used when we are given increased or decreased value.

If a value increases by x%,

$$\text{Original}=\frac{\text{New value}\times 100}{100+\mathrm{x}}$$

If a value decreases by x%,

$$\text{Original} = \frac{\text{New value} × 100}{100 - x}$$

Example:
After a 20% rise, price = ₹240.
Original = (240×100)/120 = ₹200

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7. Percentage Comparison

To find how many percent one number is of another:

$$\text{Required \%} = \frac{\text{First}}{\text{Second}} × 100$$

Example:
What % is 20 of 80? → (20/80)×100 = 25%

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8. Percentage Change in Product

If one factor increases by x% and another by y%,

$$\text{Total Change \%}=x+y+\frac{xy}{\mathrm{100}}$$

Example:
If length ↑ 20% and breadth ↑ 10%,
Area ↑ = 20 + 10 + (20×10)/100 = 32%

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9. Population Formula

If population grows at rate r% per year, after n years:

$$P_n = P \left(1 + \frac{r}{100}\right)^n$$

If it decreases at rate r% per year:

$$P_n=P{(1-\frac{r}{100})}^n$$

Example:
Current population = 10000, growth = 5% per year, 2 years later = 10000×(1.05)² = 11025

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10. Profit, Loss, Discount (Linked Topics)

Concept	Formula
Profit %	(Profit / Cost Price) × 100
Loss %	(Loss / Cost Price) × 100
Discount %	(Discount / Marked Price) × 100

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11. Salary / Income Problems

If salary increases by x% and expenses by y%, and income changes accordingly,
use successive percentage formula to find net change in savings.

Example:
Income ↑ 20%, Expenses ↑ 10% ⇒ Net saving ↑ 8% approx.
(use formula for combined % change).

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12. Examination Marks Problems

Let total marks = T, obtained = O, percentage = P%

$$O=\frac{P\times T}{\mathrm{100}}$$

If a student scores 30% and fails by 12 marks,
then 40% marks = passing marks →
10% = 12 ⇒ Total = 12 × 10 = 120 marks.

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13. Relation Between Fraction and Percentage Change

Change	Equivalent Fraction
1%	1/100
5%	1/20
10%	1/10
20%	1/5
25%	1/4
50%	1/2

Useful for fast mental math.

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14. Tricks for Quick Solving

✅ To find “x% of y”, compute directly:

$$x\mathrm{\%}\text{ of }y=\frac{x\times y}{100}$$ (Interchangeable: x% of y = y% of x)

✅ For 10%, divide by 10
✅ For 5%, divide by 20
✅ For 1%, divide by 100
✅ For 2%, divide by 50

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15. Important Examples

1. What % is 45 of 150?
   → (45/150)×100 = 30%

2. If price ↑ 25% then ↓ 20%, net % change?
   = 25 - 20 - (25×20)/100 = -15% → 15% decrease

3. A’s income is 25% more than B’s. Find how much % less is B’s income than A’s?
   Let B = 100 ⇒ A = 125
   Difference = 25 → (25/125)×100 = 20% less

4. If number increases from 80 to 100, find % increase.
   = ((100−80)/80)×100 = 25% increase

5. A’s marks = 80% of B. Find ratio A:B.
   = 80:100 = 4:5

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16. Key Concept Relations

Concept	Relation
% → Fraction → Ratio	50% = 1/2 = 1:2
Fraction → %	(1/n)×100
Increase x% then decrease x%	Net = $x^2\mathrm{/100\%\; decrease}$
Equal % increase/decrease	Always results in decrease

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17. Quick Practice Formula Table

Type	Formula
% value	(Value/Total)×100
Increase %	((New−Old)/Old)×100
Decrease %	((Old−New)/Old)×100
Successive %	x + y + (xy/100)
Reverse %	(New×100)/(100±x)
Population	P(1±r/100)^n
% of %	(x×y)/100

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18. One-Line Summary

“Percentage means out of 100.
Convert easily between fractions, decimals & ratios.
Remember key formulas for change, comparison, and reverse calculation.”

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19. Exam Revision Checklist

☑ Definition and conversion rules
☑ Formula for increase/decrease
☑ Successive % formula
☑ Reverse % formula
☑ Population growth/decay
☑ Fraction–% table
☑ Shortcut % calculations

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✅ In One Line:

“% compares part to whole — 100 base concept; master reverse %, successive %, and x% of y tricks.”

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Focus Keywords:

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