percentages

 

1. Definition and Basic Concepts

A percentage is a way of expressing a number as a fraction of 100. The symbol “%” denotes percentage.

• Formula:
  $\text{Percentage}=\frac{\text{Value}}{\text{Total Value}}\times 100$

Example:
If 45 out of 150 students passed, the pass percentage is:

$$\frac{45}{150}\times 100=30\mathrm{\%}$$

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2. Conversion Techniques

• Fraction to Percentage: Multiply by 100
  $\frac{3}{5}\times 100=60\mathrm{\%}$

• Decimal to Percentage: Multiply by 100
  $0.75\times 100=75\mathrm{\%}$

• Percentage to Fraction: Divide by 100
  $40\mathrm{\%}=\frac{40}{100}=\frac{2}{5}$

• Percentage to Decimal: Divide by 100
  $25\mathrm{\%}=0.25$

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3. Percentage Change

A. Percentage Increase/Decrease

$$\text{Percentage Change}=\frac{\text{New Value}-\text{Original Value}}{\text{Original Value}}\times 100$$

• Increase: If New Value > Original Value (result is positive)

• Decrease: If New Value < Original Value (result is negative)

Example:
A commodity’s price rises from ₹200 to ₹250:

$$\frac{250-200}{200}\times 100=25\mathrm{\%}\text{ increase}$$

B. Successive Percentage Change

If a value is increased by $x\mathrm{\%}$ and then by $y\mathrm{\%}$, the net change is:

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

• For increase and decrease, use + for increase and - for decrease.

Example:
Increase by 20%, then decrease by 10%:

$$20+(-10)+\frac{20\times -10}{100}=10-2=8\mathrm{\%}$$

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4. Application Areas

A. Business and Economics

• Profit and Loss:

  $$\text{Profit \%}=\frac{\text{Profit}}{\text{Cost Price}}\times 100$$$$\text{Loss \%}=\frac{\text{Loss}}{\text{Cost Price}}\times 100$$

• Discount:

  $$\text{Discount \%}=\frac{\text{Discount}}{\text{Marked Price}}\times 100$$

B. Population and Demographics

• Used to express growth/decline rates, literacy rates, etc.

C. Data Interpretation

• Percentages are crucial for interpreting pie charts, bar graphs, and tables in UPSC Mains1.

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5. Important Properties and Shortcuts

• If A is x% more than B, then B is less than A by:

  $$\frac{x}{100+x}\times 100\mathrm{\%}$$

• If A is x% less than B, then B is more than A by:

  $$\frac{x}{100-x}\times 100\mathrm{\%}$$

Example:
If A is 25% more than B, B is less than A by:

$$\frac{25}{125}\times 100=20\mathrm{\%}$$

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6. Reverse Percentage

To find the original value before a percentage increase or decrease:

• After Increase:

  $$\text{Original Value}=\frac{\text{Increased Value}}{1+\frac{x}{100}}$$

• After Decrease:

  $$\text{Original Value}=\frac{\text{Decreased Value}}{1-\frac{x}{100}}$$

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7. Alligation and Mixture Problems

Percentages are used to calculate the concentration of components in mixtures, especially in business and industry applications1.

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8. Practice Example (UPSC Level)

Q: The population of a town increased by 20% in the first year and decreased by 10% in the second year. What is the net percentage change?

Solution:
Let the initial population = 100
After 1st year: $100+20=120$
After 2nd year: $120-10\mathrm{\%}\text{ of }120=120-12=108$
Net change = $108-100=8$
Net percentage change = $8\mathrm{\%}$ increase

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9. Common Mistakes

• Confusing percentage points with percentage change.

• Applying percentage increase/decrease to the wrong base value.

• Ignoring compounding in successive changes.

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10. Tips for UPSC Mains

• Always write stepwise solutions and show conversions.

• Use percentages to compare data, analyze trends, and justify arguments in essays and GS papers.

• Practice data interpretation using percentages for graphs and charts.

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Summary:
Percentages are foundational in quantitative analysis for UPSC Mains, especially in economics, business, and data interpretation. Mastery of calculation techniques, reverse percentages, and successive changes is essential for accurate and efficient problem-solving12.

Basic Calculation and Conversion (1–10)

1. What is 25% of 240?

2. Express 0.45 as a percentage.

3. Convert 3/8 into a percentage.

4. What is 12.5% of 640?

5. Increase 200 by 15%.

6. Decrease 480 by 20%.

7. What percent of 80 is 20?

8. 75 is what percent more than 60?

9. If 30% of a number is 90, what is the number?

10. 120 is decreased to 96. What is the percentage decrease?

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Successive and Reverse Percentage (11–20)

11. A number is increased by 20% and then decreased by 10%. Find the net percentage change.

12. The price of an article is first increased by 10% and then again by 20%. What is the total percentage increase?

13. If the population of a town increases by 5% annually, what will be the population after 2 years if the present population is 20,000?

14. If the salary of a person is increased by 25% and then decreased by 20%, what is the net percentage change?

15. A number is decreased by 20% and then increased by 25%. What is the net percentage change?

16. If a number after being increased by 30% becomes 156, what was the original number?

17. If a number is reduced by 40%, by what percent must the reduced number be increased to restore the original number?

18. If the price of a commodity increases from ₹400 to ₹500, what is the percentage increase?

19. If 40% of a number is 80, what is 25% of the same number?

20. If a number is increased by 10% and then decreased by 10%, what is the net percentage change?

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Profit, Loss, Discount, and Data (21–30)

21. A shopkeeper marks his goods 20% above cost price and allows a discount of 10%. What is his profit percent?

22. An article is sold at a loss of 10%. If it had been sold for ₹40 more, there would have been a profit of 10%. Find the cost price.

23. A trader buys goods at a 15% discount and sells them at the marked price. What is his profit percent?

24. A man spends 80% of his income. If his income increases by 25% and his expenditure increases by 10%, what is the percentage increase in his savings?

25. The population of a city increases by 10% in the first year and by 20% in the next year. What is the total percentage increase in two years?

26. If the price of sugar increases by 25%, by what percent should a family reduce its consumption so that the expenditure does not increase?

27. A number is 20% more than another. By what percent is the second number less than the first?

28. The price of a commodity is increased by 20%. By what percent must consumption be reduced so that the expenditure remains the same?

29. If the cost price of an article is ₹500 and it is sold at a profit of 15%, what is the selling price?

30. If the marked price of an article is ₹1,000 and a discount of 20% is given, what is the selling price?

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Mixture, Alligation, and Compound Percentage (31–40)

31. In a mixture of 60 liters, milk and water are in the ratio 2:1. What is the percentage of water in the mixture?

32. A solution contains 30% alcohol. How much water should be added to 200 ml of this solution to make it 20% alcohol?

33. In an alloy, copper and zinc are in the ratio 5:3. What is the percentage of zinc in the alloy?

34. A sum of money increases by 20% in the first year, 30% in the second year, and 50% in the third year. What is the net percentage increase after three years?

35. If the price of petrol increases by 25% and a person wants to spend only 10% more, by what percent should he reduce his consumption?

36. If 40% of a number is equal to 60% of another number, what is the ratio of the first number to the second?

37. If the price of a book is reduced by 20%, a person can buy 5 more books for ₹400. What is the original price of the book?

38. In a mixture of 80 liters, the ratio of milk to water is 3:1. How many liters of water must be added to make the ratio 2:1?

39. A sum of money is increased by 10% and then by 20%. What is the effective percentage increase?

40. If the population of a town increases by 5% annually, what will be the population after 3 years if the present population is 10,000?

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Application and Data Interpretation (41–50)

41. In a class of 60 students, 45% are girls. How many boys are there?

42. Out of 500 students, 60% passed an exam. How many students failed?

43. If 25% of a number is 50, what is 40% of the same number?

44. The price of a commodity falls by 20%. By what percent must the consumption increase so that the expenditure remains the same?

45. In an election, a candidate gets 60% of the votes and wins by 2,400 votes. Find the total number of votes.

46. The population of a city increases by 5% every year. If the present population is 1,00,000, what will it be after 2 years?

47. The salary of a person is increased by 20% and then decreased by 10%. What is the net percentage change?

48. If a number is increased by 25% and then decreased by 20%, what is the net percentage change?

49. A person spends 80% of his income and saves ₹2,000. What is his income?

50. A reduction of 20% in the price of sugar enables a person to buy 2 kg more for ₹80. Find the original price per kg.