Percentages Concepts

1. Price-Consumption-Expenditure Paradox

• Formula: If price increases by (r%), required reduction in consumption to keep expenditure constant:

$$\frac{r}{100+r}\times100$$

If price decreases by (r%), required increase in consumption:

$$\frac{r}{100-r}\times100$$

• Example: Petrol price rises by 15%. By what percentage should consumption be reduced to keep expenditure constant? Solution:

$$\text{Reduction}=\frac{15}{115}\times100=13.04$$

Therefore, consumption must reduce by 13.04%.

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2. Successive Percentage Shifts

• Formula: For successive changes of (x%) and (y%):

$$\text{Net change}=x+y+\frac{xy}{100}$$

Take decreases as negative percentages.

• Example: A number is increased by 25% and then decreased by 20%. Find the net percentage change. Solution:

$$25+(-20)+\frac{25(-20)}{100}$$ $$=25-20-5=0$$

Therefore, there is no net change.

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3. Election & Population Puzzles

• Formula: After an (r%) increase:

$$\text{New value}=P\left(1+\frac r{100}\right)$$

After an (r%) decrease:

$$\text{New value}=P\left(1-\frac r{100}\right)$$

• Example: A town has a population of 50,000. It grows by 10% and then decreases by 5%. Find the final population. Solution: After 10% growth:

$$50000\times1.10=55000$$

After 5% decrease:

$$55000\times0.95=52250$$

Final population = 52,250.

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4. Salary & Expenditure Allocations

• Formula:

$$\text{Savings}=\text{Income}-\text{Expenditure}$$ \text{Savings %}=\frac{\text{Savings}}{\text{Income}}\times100

• Example: A person earns ₹30,000. He spends 60% on rent and 25% of the remaining amount on food. Find his savings. Solution: Rent:

$$60$$

Remaining:

$$30000-18000=12000$$

Food:

$$25$$

Savings:

$$12000-3000=\boxed{₹9000}$$

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5. Election Vote Percentage & Majority

• Formula:

$$\text{Majority}=\text{Winner's votes}-\text{Loser's votes}$$

If winner gets (x%) and loser gets (y%) of valid votes:

$$\text{Majority}=(x-y)$$

• Example: In an election, candidate A gets 55% of 20,000 valid votes. How many votes does A win by if B gets the remaining votes? Solution: A’s votes:

$$55$$

B’s votes:

$$20000-11000=9000$$

Majority:

$$11000-9000=\boxed{2000}$$

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6. Election with Invalid Votes

• Formula: If (r%) votes are invalid:

$$\text{Valid votes}=\text{Total votes}\times\frac{100-r}{100}$$

Percentages of candidates are generally calculated from valid votes.

• Example: 10% of 30,000 votes are invalid. Candidate A receives 60% of the valid votes. Find A’s votes. Solution: Valid votes:

$$30000\times\frac{90}{100}=27000$$

A’s votes:

$$27000\times\frac{60}{100}=\boxed{16200}$$

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7. Population Growth & Decline

• Formula: For annual growth of (r%) for (n) years:

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

For annual decline:

$$P_{\text{final}}=P\left(1-\frac r{100}\right)^n$$

• Example: A city’s population is 80,000 and grows by 5% annually. Find its population after 2 years. Solution:

$$80000(1.05)^2$$ $$=80000\times1.1025=\boxed{88200}$$

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8. Percentage of Percentage

• Formula: (x%) of (y%) of a number (P):

$$\frac{x}{100}\times\frac{y}{100}\times P$$

• Example: Find 20% of 30% of ₹5,000. Solution:

$$\frac{20}{100}\times\frac{30}{100}\times5000$$ $$=300$$

Answer = ₹300.

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9. Percentage Increase/Decrease from Original Value

• Formula:

$$$$

Always use the original value as the denominator.

• Example: A salary increases from ₹40,000 to ₹46,000. Find the percentage increase. Solution:

$$\text{Increase}=46000-40000=6000$$ $$$$

Answer = 15%.

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10. Reverse Percentage

• Formula: If the final value after an (r%) increase is (F):

$$\text{Original}=\frac{F\times100}{100+r}$$

If after an (r%) decrease:

$$\text{Original}=\frac{F\times100}{100-r}$$

• Example: After a 20% increase, a number becomes 360. Find the original number. Solution:

$$\text{Original}=\frac{360\times100}{120}$$ $$=\boxed{300}$$

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11. Expenditure Constant with Price Change

• Formula: Since

$$\text{Expenditure}=\text{Price}\times\text{Quantity}$$

for constant expenditure:

$$P_1Q_1=P_2Q_2$$

• Example: The price of rice increases from ₹40/kg to ₹50/kg. If a family previously bought 20 kg, how much can it buy for the same expenditure? Solution: Original expenditure:

$$40\times20=₹800$$

New quantity:

$$\frac{800}{50}=\boxed{16\text{ kg}}$$

Reduction:

$$20-16=4\text{ kg}=20$$

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12. Income-Expenditure-Savings Percentage Change

• Formula:

$$S=I-E$$

If income and expenditure change, calculate the new values first and then:

$$$$

• Example: A person’s income is ₹50,000 and expenditure is ₹40,000. Income increases by 10% while expenditure increases by 5%. Find the percentage increase in savings. Solution: Original savings:

$$50000-40000=10000$$

New income:

$$50000(1.10)=55000$$

New expenditure:

$$40000(1.05)=42000$$

New savings:

$$55000-42000=13000$$

Percentage increase:

\frac{3000}{10000}\times100=\boxed{30%}

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13. Required Percentage Increase to Restore Original Value

• Formula: If a value decreases by (r%), the percentage increase required to return to the original value is:

$$\frac{r}{100-r}\times100$$

If it increases by (r%), the required decrease is:

$$\frac{r}{100+r}\times100$$

• Example: A price falls by 20%. By what percentage must it increase to return to its original price? Solution:

$$\frac{20}{80}\times100=25$$

Answer = 25%.

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14. Percentage Comparison Between Two Values

• Formula: Percentage by which A is more than B:

$$\frac{A-B}{B}\times100$$

Percentage by which A is less than B:

$$\frac{B-A}{B}\times100$$

• Example: A’s salary is ₹48,000 and B’s salary is ₹40,000. A’s salary is what percentage more than B’s? Solution:

$$\frac{48000-40000}{40000}\times100$$ =\boxed{20%}

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Advanced Variants

15. Multiple Successive Price Changes with Constant Expenditure

• Formula: Apply each percentage change multiplicatively:

$$Q_{\text{new}}=Q\left(\frac{100}{100+r_1}\right)\left(\frac{100}{100+r_2}\right)$$

for successive price increases.

• Example: Petrol price increases by 20% and then by 25%. By what percentage must consumption decrease to keep expenditure unchanged? Solution: New price factor:

$$1.20\times1.25=1.50$$

Thus price becomes 150% of original. Required consumption:

$$\frac{100}{150}=66.67$$

Reduction:

100-66.67=\boxed{33.33%}

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16. Population with Different Annual Rates

• Formula: For different successive growth rates:

$$P_{\text{final}}=P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\cdots$$

• Example: A population of 40,000 increases by 10% in the first year and 20% in the second year. Find the final population. Solution:

$$40000(1.10)(1.20)$$ $$=40000(1.32)=\boxed{52800}$$

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17. Election: Winning Percentage from Majority

• Formula: If winner’s votes exceed loser’s votes by (M), and winner gets (x%) while loser gets (y%):

$$M=(x-y)$$

• Example: A candidate wins an election by 2,000 votes. The winner gets 55% of the valid votes. Find the total valid votes. Solution: Loser’s percentage:

$$100-55=45$$

Difference:

$$55-45=10$$

Therefore:

$$10$$ $$\text{Total votes}=\frac{2000\times100}{10}=\boxed{20000}$$

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18. Income, Expenditure and Savings with Percentage Changes

• Formula: When income changes by (x%) and expenditure by (y%):

$$I_2=I_1\left(1+\frac{x}{100}\right),\quad E_2=E_1\left(1+\frac{y}{100}\right)$$

Then (S_2=I_2-E_2).

• Example: A person’s income is ₹60,000 and expenditure is ₹48,000. Income rises by 20% and expenditure by 10%. Find the percentage increase in savings. Solution: Original savings:

$$60000-48000=12000$$

New income:

$$60000(1.2)=72000$$

New expenditure:

$$48000(1.1)=52800$$

New savings:

$$72000-52800=19200$$

Increase:

$$19200-12000=7200$$

Percentage increase:

\frac{7200}{12000}\times100=\boxed{60%}