A percentage is a ratio with a base of 100. Its most important hidden question is “percent of what?” A change from 20 to 25 is an increase of 5 relative to 20, or 25%. The same change read backwards is a decrease of 5 relative to 25, or 20%. The amount changed is identical; the base changes.

1. Convert between forms

p% = p/100. Thus 12.5% = 1/8 and 37.5% = 3/8. Common fractions can shorten calculations: 25% is 1/4, 20% is 1/5, 16⅔% is 1/6, 33⅓% is 1/3 and 75% is 3/4. Use exact fractions when a recurring percentage is involved; rounding 16⅔% to 16.67% too early can distort an exact answer.

To find p% of N, calculate N×p/100. To recover the whole when a part equals p%, divide the part by p/100. An increase by p% multiplies the original by 1+p/100; a decrease multiplies it by 1−p/100.

Worked example 1. If 18% of a number is 72, the whole is 72/(18/100)=400. Check: 10% of 400 is 40 and 8% is 32, totalling 72.

2. Percentage change and changing bases

Percentage change = (new−old)/old × 100. A negative result indicates a decrease. For a decrease from 80 to 68, the fall is 12, and 12/80 = 15%. If the same quantity later returns from 68 to 80, the increase is 12/68 ≈ 17.65%. Equal absolute moves do not imply equal percentage moves.

When A is p% more than B, use B as the base: A=B(1+p/100). To find the percentage by which B is less than A, use A as the base. If A is 25% more than B, take B=100 and A=125; B is 25/125=20% less than A.

Worked example 2. A price rises from ₹800 to ₹920. The rise is ₹120; 120/800 ×100=15%. If it later falls back to ₹800, the fall is 120/920 ×100≈13.04%, not 15%.

3. Successive changes

Successive percentage changes multiply factors. A 20% rise followed by a 10% fall gives 1.20×0.90=1.08, a net 8% rise. Simply adding +20 and −10 gives 10%, which ignores the changed base. For two changes a% and b%, with decreases represented as negative numbers, net change is a+b+ab/100 percent.

For an increase and equal decrease p%, the product is (1+p/100)(1−p/100)=1−p²/10000. The result is a net decrease of p²/100 percent. A 10% rise and 10% fall leaves 99% of the starting amount, a 1% fall.

Worked example 3. A salary of ₹20,000 rises by 15% and then by 8%. The final salary is 20,000×1.15×1.08=₹24,840. The net rise is 24.2%, not 23%. The second rise applies to ₹23,000, not ₹20,000.

4. Reverse percentages

If a current amount is after a change, divide by the change factor to recover the original. A price after a 20% reduction is ₹960. Since the new price is 80% of the original, original=960/0.80=₹1,200. Adding 20% of ₹960 produces ₹1,152, which is wrong because ₹960 is the reduced base.

For repeated changes, reverse the factors in reverse order if reconstructing an intermediate stage. If a number grew by 10% and then 20%, the final multiplier is 1.32; divide final by 1.32 to recover the start. Do not subtract a combined 30% from the final.

5. Percentage points and weighted percentages

An increase from 40% to 45% is five percentage points. Relative to the old percentage, it is a 5/40=12.5% increase. The two descriptions answer different questions. For groups of unequal size, calculate the total numerator and denominator before finding the overall percentage. Averaging group percentages without weights is generally wrong.

Worked example 4. A class has 30 boys with 60% passing and 20 girls with 80% passing. Pass counts are 18 and 16. Overall, 34 of 50 pass, so the pass percentage is 68%. The simple average of 60% and 80% would be 70%, but the groups are not equal in size.

6. Application checks

Population, price, salary, votes, marks and discount questions all use the same base logic. Write the base beside each percentage. In a voting question, “60% of valid votes” and “60% of all votes” differ if some ballots are invalid. In a marks question, the denominator may be total possible marks, attempted marks or a section total. Read those words before computing.

For a final reasonableness check, 10% of N is N/10, 1% is N/100 and 0.1% is N/1000. These anchors catch misplaced decimal points. A 2% tax on ₹500 cannot be ₹100; 1% is ₹5, so 2% is ₹10.

Recall before practice

1. Explain the base used when A is 30% more than B. 2. Find the net effect of a 10% rise followed by a 10% fall. 3. Distinguish an increase of five percentage points from a 5% relative increase. 4. State how to recover an original price after a 25% discount.

Chapter 6 practice — 50 questions

Choose one option for each item. Keep a separate answer list; explanations follow this set.

1. In an examination Rahul scored 45% marks and passed by 150 marks. If the pass percentage is 30%, what is the maximum marks of the examination?

A. 300 B. 500 C. 1000 D. 1100

2. A's income is 150% more than B's income. By what percent is B's income less than A's income?

A. 14²⁄₇% B. 10% C. 28⁴⁄₇% D. 60%

3. In an examination Anita scored 50% marks and passed by 75 marks. If the pass percentage is 40%, what is the maximum marks of the examination?

A. 750 B. 300 C. 375 D. 150

4. In an examination Neha scored 40% marks and failed by 25 marks. If the pass percentage is 45%, what is the maximum marks of the examination?

A. 225 B. 500 C. 600 D. 250

5. In an examination Sanjay scored 30% marks and failed by 50 marks. If the pass percentage is 35%, what is the maximum marks of the examination?

A. 350 B. 1000 C. 1100 D. 500

6. A's income is 60% more than B's income. By what percent is B's income less than A's income?

A. 42⁶⁄₇% B. 10% C. 100% D. 37.5%

7. In an examination Rohit scored 27% marks and failed by 64 marks. If the pass percentage is 35%, what is the maximum marks of the examination?

A. 900 B. 800 C. 280 D. 400

8. A's income is 75% less than B's income. By what percent is B's income more than A's income?

A. 50% B. 300% C. 40% D. 20%

9. The population of a town is first decreased by 25% and then increased by 10%. What is the net percentage change?

A. decrease of 15% B. decrease of 14% C. increase of 17.5% D. decrease of 17.5%

10. The length of a rectangle is first decreased by 50% and then increased by 50%. What is the net percentage change?

A. increase of 25% B. decrease of 23% C. increase of 1% D. decrease of 25%

11. Express 3/5 as a percentage.

A. 6% B. 30% C. 60% D. 120%

12. What is 25% of 28?

A. 21 B. 70 C. 7 D. 32

13. What is 36% of 775?

A. 279 B. 496 C. 2790 D. 315

14. What is 35% of 3000?

A. 10500 B. 1050 C. 1950 D. 105

15. A's income is 25% more than B's income. By what percent is B's income less than A's income?

A. 11¹⁄₉% B. 50% C. 66⅔% D. 20%

16. In an examination Arjun scored 25% marks and failed by 48 marks. If the pass percentage is 33%, what is the maximum marks of the examination?

A. 600 B. 192 C. 300 D. 198

17. A's income is 11¹⁄₉% less than B's income. By what percent is B's income more than A's income?

A. 12.5% B. 80% C. 33⅓% D. 100%

18. In an examination Rahul scored 28% marks and failed by 48 marks. If the pass percentage is 40%, what is the maximum marks of the examination?

A. 160 B. 120 C. 400 D. 200

19. In an election between two candidates, the winner got 52% of the valid votes and won by 240 votes. What was the total number of votes polled?

A. 6,000 B. 480 C. 3,000 D. 7,200

20. The population of a town is first decreased by 15% and then increased by 20%. What is the net percentage change?

A. increase of 2% B. decrease of 2% C. increase of 6% D. increase of 5%

21. The population of a village is 2,60,000. If it decreases at the rate of 10% per annum, what will be its population after 2 years?

A. 3,14,600 B. 2,10,600 C. 2,08,000 D. 2,34,000

22. In an examination Priya scored 24% marks and failed by 72 marks. If the pass percentage is 36%, what is the maximum marks of the examination?

A. 600 B. 300 C. 200 D. 216

23. In an examination Suresh scored 18% marks and failed by 48 marks. If the pass percentage is 30%, what is the maximum marks of the examination?

A. 120 B. 160 C. 200 D. 400

24. In an examination Anita scored 56% marks and passed by 40 marks. If the pass percentage is 36%, what is the maximum marks of the examination?

A. 300 B. 100 C. 200 D. 72

25. In an examination Mohan scored 32% marks and failed by 64 marks. If the pass percentage is 40%, what is the maximum marks of the examination?

A. 800 B. 200 C. 320 D. 160

26. The population of a town is first increased by 12% and then decreased by 60%. What is the net percentage change?

A. decrease of 55.2% B. increase of 55.2% C. decrease of 48% D. decrease of 47%

27. The population of a village is 20,000. If it decreases at the rate of 10% per annum, what will be its population after 2 years?

A. 24,200 B. 18,000 C. 16,000 D. 16,200

28. Geeta spends 20% of his monthly income on rent and 25% of the remaining on food. If Geeta saves ₹14,400 every month, what is the monthly income?

A. ₹24,000 B. ₹26,000 C. ₹19,200 D. ₹28,800

29. A's income is 200% more than B's income. By what percent is B's income less than A's income?

A. 75% B. 66⅔% C. 37.5% D. 40%

30. The price of an article is first decreased by 12% and then decreased by 15%. What is the net percentage change?

A. decrease of 28.8% B. increase of 25.2% C. decrease of 27% D. decrease of 25.2%

31. The price of rice rises by 150%. By what percent must a family reduce its consumption of rice so that its expenditure on rice remains the same?

A. 60% B. 28⁴⁄₇% C. 16⅔% D. 11¹⁄₉%

32. The population of a village is 8. If it decreases at the rate of 50% per annum, what will be its population after 2 years?

A. 18 B. 4 C. 2 D. 8

33. In an examination Rekha scored 21% marks and failed by 36 marks. If the pass percentage is 33%, what is the maximum marks of the examination?

A. 400 B. 150 C. 300 D. 99

34. The population of a village is 27,000. If it decreases at the rate of 10% per annum, what will be its population after 2 years?

A. 32,670 B. 21,870 C. 21,600 D. 24,300

35. Mohan spends 30% of her monthly income on rent and 40% of the remaining on food. If Mohan saves ₹12,600 every month, what is the monthly income?

A. ₹30,000 B. ₹21,000 C. ₹25,200 D. ₹32,000

36. The price of milk rises by 50%. By what percent must a family reduce its consumption of milk so that its expenditure on milk remains the same?

A. 16⅔% B. 33⅓% C. 37.5% D. 11¹⁄₉%

37. The price of rice falls by 50%. By what percent must a family increase its consumption of rice so that its expenditure on rice remains the same?

A. 100% B. 12.5% C. 9¹⁄₁₁% D. 14²⁄₇%

38. A's income is 60% less than B's income. By what percent is B's income more than A's income?

A. 12.5% B. 33⅓% C. 150% D. 50%

39. Deepak spends 25% of her monthly income on rent and 30% of the remaining on food. If Deepak saves ₹23,625 every month, what is the monthly income?

A. ₹47,000 B. ₹47,250 C. ₹45,000 D. ₹33,750

40. A's income is 12.5% more than B's income. By what percent is B's income less than A's income?

A. 33⅓% B. 40% C. 28⁴⁄₇% D. 11¹⁄₉%

41. The price of milk rises by 12.5%. By what percent must a family reduce its consumption of milk so that its expenditure on milk remains the same?

A. 37.5% B. 14²⁄₇% C. 60% D. 11¹⁄₉%

42. The salary of a worker is first decreased by 60% and then decreased by 60%. What is the net percentage change?

A. increase of 84% B. decrease of 120% C. decrease of 84% D. decrease of 156%

43. The population of a village is 1,79,200. If it increases at the rate of 12.5% per annum, what will be its population after 2 years?

A. 2,24,000 B. 2,26,800 C. 1,37,200 D. 2,01,600

44. The population of a village is 2,048. If it decreases at the rate of 12.5% per annum, what will be its population after 3 years?

A. 1,568 B. 1,280 C. 1,372 D. 2,916

45. The salary of a worker is first decreased by 12% and then decreased by 20%. What is the net percentage change?

A. increase of 29.6% B. decrease of 32% C. decrease of 29.6% D. decrease of 34.4%

46. The price of wheat rises by 20%. By what percent must a family reduce its consumption of wheat so that its expenditure on wheat remains the same?

A. 50% B. 14²⁄₇% C. 16⅔% D. 60%

47. The length of a rectangle is first increased by 25% and then decreased by 50%. What is the net percentage change?

A. decrease of 25% B. decrease of 37.5% C. increase of 37.5% D. decrease of 24%

48. The length of a rectangle is first decreased by 15% and then decreased by 30%. What is the net percentage change?

A. increase of 40.5% B. decrease of 40.5% C. decrease of 49.5% D. decrease of 45%

49. The population of a village is 368. If it decreases at the rate of 25% per annum, what will be its population after 2 years?

A. 575 B. 207 C. 184 D. 276

50. The population of a village is 14,720. If it increases at the rate of 25% per annum, what will be its population after 3 years?

A. 28,750 B. 23,000 C. 25,760 D. 6,210

Chapter 6 — Answers and explanations

After checking the key, re-solve any miss without looking at the formula.

1. C. Difference between pass marks and marks obtained = (30 − 45)% of maximum = 15% of M = 150. So M = 150 × 100/15 = 1000.

APAR26-06-01 | Examination marks (pass / fail) | Easy

2. D. A is more than B by 3/2 of B. Take B = 2 units, then A = 5 units. B is less than A by 3 units out of 5, i.e. 3/5 = 60%. (Rule: more by a/b ⇒ less by a/(a+b).)

APAR26-06-06 | Percentage more / less comparison | Easy

3. A. Difference between pass marks and marks obtained = (40 − 50)% of maximum = 10% of M = 75. So M = 75 × 100/10 = 750.

APAR26-06-07 | Examination marks (pass / fail) | Easy

4. B. Difference between pass marks and marks obtained = (45 − 40)% of maximum = 5% of M = 25. So M = 25 × 100/5 = 500.

APAR26-06-04 | Examination marks (pass / fail) | Easy

5. B. Difference between pass marks and marks obtained = (35 − 30)% of maximum = 5% of M = 50. So M = 50 × 100/5 = 1000.

APAR26-06-14 | Examination marks (pass / fail) | Easy

6. D. A is more than B by 3/5 of B. Take B = 5 units, then A = 8 units. B is less than A by 3 units out of 8, i.e. 3/8 = 37.5%. (Rule: more by a/b ⇒ less by a/(a+b).)

APAR26-06-08 | Percentage more / less comparison | Easy

7. B. Difference between pass marks and marks obtained = (35 − 27)% of maximum = 8% of M = 64. So M = 64 × 100/8 = 800.

APAR26-06-10 | Examination marks (pass / fail) | Easy

8. B. A is less than B by 3/4 of B. Take B = 4 units, then A = 1 units. B is more than A by 3 units out of 1, i.e. 3/1 = 300%. (Rule: less by a/b ⇒ more by a/(b−a).)

APAR26-06-02 | Percentage more / less comparison | Easy

9. D. Net change = a + b + ab/100 with a = -25, b = 10: -25 + (10) + (-25 × 10)/100 = -17.5%. A 17.5% decrease. (Simply adding the two percentages, -15%, ignores the compounding term.)

APAR26-06-03 | Successive percentage change | Easy

10. D. Net change = a + b + ab/100 with a = -50, b = 50: -50 + (50) + (-50 × 50)/100 = -25%. A 25% decrease. (Simply adding the two percentages, 0%, ignores the compounding term.)

APAR26-06-15 | Successive percentage change | Easy

11. C. To convert a fraction to a percentage, multiply by 100: 3/5 × 100 = 300/5 = 60%. (Worth memorising: 1/5 = 20%.)

APAR26-06-09 | Fraction to percentage | Easy

12. C. Percent means 'per hundred', so 25% of 28 = (25/100) × 28 = 700/100 = 7.

APAR26-06-13 | Percentage of a number | Easy

13. A. Percent means 'per hundred', so 36% of 775 = (36/100) × 775 = 27900/100 = 279.

APAR26-06-05 | Percentage of a number | Easy

14. B. Percent means 'per hundred', so 35% of 3000 = (35/100) × 3000 = 105000/100 = 1050.

APAR26-06-12 | Percentage of a number | Easy

15. D. A is more than B by 1/4 of B. Take B = 4 units, then A = 5 units. B is less than A by 1 units out of 5, i.e. 1/5 = 20%. (Rule: more by a/b ⇒ less by a/(a+b).)

APAR26-06-11 | Percentage more / less comparison | Easy

16. A. Difference between pass marks and marks obtained = (33 − 25)% of maximum = 8% of M = 48. So M = 48 × 100/8 = 600.

APAR26-06-24 | Examination marks (pass / fail) | Medium

17. A. A is less than B by 1/9 of B. Take B = 9 units, then A = 8 units. B is more than A by 1 units out of 8, i.e. 1/8 = 12.5%. (Rule: less by a/b ⇒ more by a/(b−a).)

APAR26-06-29 | Percentage more / less comparison | Medium

18. C. Difference between pass marks and marks obtained = (40 − 28)% of maximum = 12% of M = 48. So M = 48 × 100/12 = 400.

APAR26-06-31 | Examination marks (pass / fail) | Medium

19. A. Winner 52%, loser 48% of valid votes; margin = 4% of valid votes = 240 ⇒ valid votes = 6,000. Winner's votes = 52% of 6,000 = 3,120.

APAR26-06-28 | Election / votes | Medium

20. A. Net change = a + b + ab/100 with a = -15, b = 20: -15 + (20) + (-15 × 20)/100 = 2%. A 2% increase. (Simply adding the two percentages, 5%, ignores the compounding term.)

APAR26-06-33 | Successive percentage change | Medium

21. B. Population after 2 years = P × (1 − 10/100)^2 = 2,60,000 × (9/10)^2 = 2,10,600.

APAR26-06-16 | Population growth / depreciation | Medium

22. A. Difference between pass marks and marks obtained = (36 − 24)% of maximum = 12% of M = 72. So M = 72 × 100/12 = 600.

APAR26-06-26 | Examination marks (pass / fail) | Medium

23. D. Difference between pass marks and marks obtained = (30 − 18)% of maximum = 12% of M = 48. So M = 48 × 100/12 = 400.

APAR26-06-19 | Examination marks (pass / fail) | Medium

24. C. Difference between pass marks and marks obtained = (36 − 56)% of maximum = 20% of M = 40. So M = 40 × 100/20 = 200.

APAR26-06-21 | Examination marks (pass / fail) | Medium

25. A. Difference between pass marks and marks obtained = (40 − 32)% of maximum = 8% of M = 64. So M = 64 × 100/8 = 800.

APAR26-06-35 | Examination marks (pass / fail) | Medium

26. A. Net change = a + b + ab/100 with a = 12, b = -60: 12 + (-60) + (12 × -60)/100 = -55.2%. A 55.2% decrease. (Simply adding the two percentages, -48%, ignores the compounding term.)

APAR26-06-23 | Successive percentage change | Medium

27. D. Population after 2 years = P × (1 − 10/100)^2 = 20,000 × (9/10)^2 = 16,200.

APAR26-06-20 | Population growth / depreciation | Medium

28. A. Let income = I. After rent: I × 80/100. After food: I × 80/100 × 75/100 = savings = ₹14,400. So I = ₹14,400 × 100/80 × 100/75 = ₹24,000. (Adding the percentages, 20% + 25% = 45%, is wrong because 25% applies to the remainder.)

APAR26-06-22 | Income, expenditure and savings | Medium

29. B. A is more than B by 2/1 of B. Take B = 1 units, then A = 3 units. B is less than A by 2 units out of 3, i.e. 2/3 = 66⅔%. (Rule: more by a/b ⇒ less by a/(a+b).)

APAR26-06-27 | Percentage more / less comparison | Medium

30. D. Net change = a + b + ab/100 with a = -12, b = -15: -12 + (-15) + (-12 × -15)/100 = -25.2%. A 25.2% decrease. (Simply adding the two percentages, -27%, ignores the compounding term.)

APAR26-06-38 | Successive percentage change | Medium

31. A. Price rises by 3/2: new price = 5/2 of old. To keep expenditure fixed, consumption must become 2/5 of old, a reduction of 3/5 = 3/5 = 60%.

APAR26-06-37 | Price change and consumption | Medium

32. C. Population after 2 years = P × (1 − 50/100)^2 = 8 × (1/2)^2 = 2.

APAR26-06-36 | Population growth / depreciation | Medium

33. C. Difference between pass marks and marks obtained = (33 − 21)% of maximum = 12% of M = 36. So M = 36 × 100/12 = 300.

APAR26-06-32 | Examination marks (pass / fail) | Medium

34. B. Population after 2 years = P × (1 − 10/100)^2 = 27,000 × (9/10)^2 = 21,870.

APAR26-06-25 | Population growth / depreciation | Medium

35. A. Let income = I. After rent: I × 70/100. After food: I × 70/100 × 60/100 = savings = ₹12,600. So I = ₹12,600 × 100/70 × 100/60 = ₹30,000. (Adding the percentages, 30% + 40% = 70%, is wrong because 40% applies to the remainder.)

APAR26-06-18 | Income, expenditure and savings | Medium

36. B. Price rises by 1/2: new price = 3/2 of old. To keep expenditure fixed, consumption must become 2/3 of old, a reduction of 1/3 = 1/3 = 33⅓%.

APAR26-06-39 | Price change and consumption | Medium

37. A. Price falls by 1/2: new price = 1/2 of old. To keep expenditure fixed, consumption must become 2/1 of old, an increase of 1/1 = 1/1 = 100%.

APAR26-06-30 | Price change and consumption | Medium

38. C. A is less than B by 3/5 of B. Take B = 5 units, then A = 2 units. B is more than A by 3 units out of 2, i.e. 3/2 = 150%. (Rule: less by a/b ⇒ more by a/(b−a).)

APAR26-06-17 | Percentage more / less comparison | Medium

39. C. Let income = I. After rent: I × 75/100. After food: I × 75/100 × 70/100 = savings = ₹23,625. So I = ₹23,625 × 100/75 × 100/70 = ₹45,000. (Adding the percentages, 25% + 30% = 55%, is wrong because 30% applies to the remainder.)

APAR26-06-40 | Income, expenditure and savings | Medium

40. D. A is more than B by 1/8 of B. Take B = 8 units, then A = 9 units. B is less than A by 1 units out of 9, i.e. 1/9 = 11¹⁄₉%. (Rule: more by a/b ⇒ less by a/(a+b).)

APAR26-06-34 | Percentage more / less comparison | Medium

41. D. Price rises by 1/8: new price = 9/8 of old. To keep expenditure fixed, consumption must become 8/9 of old, a reduction of 1/9 = 1/9 = 11¹⁄₉%.

APAR26-06-47 | Price change and consumption | Difficult

42. C. Net change = a + b + ab/100 with a = -60, b = -60: -60 + (-60) + (-60 × -60)/100 = -84%. A 84% decrease. (Simply adding the two percentages, -120%, ignores the compounding term.)

APAR26-06-44 | Successive percentage change | Difficult

43. B. Population after 2 years = P × (1 + 12.5/100)^2 = 1,79,200 × (9/8)^2 = 2,26,800.

APAR26-06-45 | Population growth / depreciation | Difficult

44. C. Population after 3 years = P × (1 − 12.5/100)^3 = 2,048 × (7/8)^3 = 1,372.

APAR26-06-43 | Population growth / depreciation | Difficult

45. C. Net change = a + b + ab/100 with a = -12, b = -20: -12 + (-20) + (-12 × -20)/100 = -29.6%. A 29.6% decrease. (Simply adding the two percentages, -32%, ignores the compounding term.)

APAR26-06-42 | Successive percentage change | Difficult

46. C. Price rises by 1/5: new price = 6/5 of old. To keep expenditure fixed, consumption must become 5/6 of old, a reduction of 1/6 = 1/6 = 16⅔%.

APAR26-06-50 | Price change and consumption | Difficult

47. B. Net change = a + b + ab/100 with a = 25, b = -50: 25 + (-50) + (25 × -50)/100 = -37.5%. A 37.5% decrease. (Simply adding the two percentages, -25%, ignores the compounding term.)

APAR26-06-46 | Successive percentage change | Difficult

48. B. Net change = a + b + ab/100 with a = -15, b = -30: -15 + (-30) + (-15 × -30)/100 = -40.5%. A 40.5% decrease. (Simply adding the two percentages, -45%, ignores the compounding term.)

APAR26-06-48 | Successive percentage change | Difficult

49. B. Population after 2 years = P × (1 − 25/100)^2 = 368 × (3/4)^2 = 207.

APAR26-06-49 | Population growth / depreciation | Difficult

50. A. Population after 3 years = P × (1 + 25/100)^3 = 14,720 × (5/4)^3 = 28,750.

APAR26-06-41 | Population growth / depreciation | Difficult