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Arithmetic - Percentage, Profit & Loss

July 11, 2026
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Arithmetic - Chapter 5: Percentage, Profit & Loss

Percentage, Profit & Loss

Percentage, Profit & Loss
Figure 1: Percentage, profit and loss are essential concepts in business mathematics.

1. Percentage

A percentage is a fraction with denominator hundred. It is denoted by the symbol %.

Decrease% = (Decrease value / Original value) × 100
KEY FORMULAS

  • If price increases by r%, reduction in consumption to keep expenditure same = [r/(100+r) × 100]%
  • If price decreases by r%, increase in consumption to keep expenditure same = [r/(100-r) × 100]%
  • Population Formula: P(1 + r/100)ⁿ
  • Successive increase: (x + y + xy/100)%
  • Successive decrease: (-x - y - xy/100)%

2. Profit and Loss

BASIC TERMS

  • Cost Price (C.P.): Price at which an article is purchased
  • Selling Price (S.P.): Price at which an article is sold
  • Profit = S.P. - C.P. (when S.P. > C.P.)
  • Loss = C.P. - S.P. (when C.P. > S.P.)
IMPORTANT FORMULAS

  • Profit = (C.P. × Profit%) / 100
  • Loss = (C.P. × Loss%) / 100
  • S.P. = C.P. × (100 + Profit%) / 100
  • S.P. = C.P. × (100 - Loss%) / 100
  • C.P. = (100 × S.P.) / (100 + Profit%)
  • C.P. = (100 × S.P.) / (100 - Loss%)
REMEMBER

  • If a person sells two items at same S.P., one at gain x% and other at loss x%, then overall loss = (x²/100)%
  • Discount series: Equivalent discount = a + b - ab/100

3. Exercise (Questions 1-10)

Q1. 11% of which number is 33?

  • (a) 300
  • (b) 330
  • (c) 110
  • (d) None of these
Answer: (a) 11% of x = 33 ⇒ x = 300

Q2. If two numbers are respectively 20% and 50% of a third number, what is the percentage of the first number to the second?

  • (a) 10
  • (b) 20
  • (c) 30
  • (d) 40
Answer: (d) (20/50) × 100 = 40%

Q3. The digit at unit place of a two-digit number is increased by 100% and the digit at ten place is increased by 50%. The new number is 19 more than the original. What is the original number?

  • (a) 22
  • (b) 63
  • (c) 24
  • (d) 18
Answer: (d) Original number = 18

Q4. If 40% of customers decide to purchase items, and of those 15% purchase hats, then what per cent of the store's customers purchase hats?

  • (a) 4%
  • (b) 6%
  • (c) 15%
  • (d) 24%
Answer: (b) 40% × 15% = 6%

Q5. 10% of inhabitants died of cholera, then 25% of remaining left. The population is then reduced to 4050. Find the original inhabitants.

  • (a) 5000
  • (b) 6000
  • (c) 7000
  • (d) 8000
Answer: (b) 75% of 90% of x = 4050 ⇒ x = 6000

Q6. When price of a pressure cooker was increased by 15%, sale decreased by 15%. What was the net effect on sales?

  • (a) 15% decrease
  • (b) no effect
  • (c) 2.25% increase
  • (d) 2.25% decrease
Answer: (d) Net effect = -(15²/100) = -2.25%

Q7. In a housing society, 30% are men over 18 and 40% are women over 18. If there are 24 children, how many total residents?

  • (a) 32
  • (b) 80
  • (c) 94
  • (d) 112
Answer: (b) 30% are children. 30% of total = 24 ⇒ total = 80

Q8. A scooter costs ₹25,000 when brand new. At the end of each year, its value is only 80% of previous year. What is its value after 3 years?

  • (a) 10,000
  • (b) 12,500
  • (c) 12,800
  • (d) 12,000
Answer: (c) 25000 × 0.8 × 0.8 × 0.8 = ₹12,800

Q9. A student X passes with 515 marks, having scored 3% above minimum. If Y had obtained 710 marks, what % would have been above minimum?

  • (a) 40%
  • (b) 42%
  • (c) 50%
  • (d) Cannot be determined
Answer: (b) Minimum marks = 515/1.03 = 500. Y's % above = 710/500 × 100 = 142%

Q10. Ravi's salary is 150% of Amit's salary. Amit's salary is 80% of Ram's salary. What is the ratio of Ram's salary to Ravi's salary?

  • (a) 1 to 2
  • (b) 2 to 3
  • (c) 5 to 6
  • (d) 6 to 5
Answer: (c) Ram = 100, Amit = 80, Ravi = 120. Ratio = 100:120 = 5:6

4. Exercise (Questions 11-20)

Q11. The sum of two numbers is 28/25 of the first number. The second number is what percent of the first?

  • (a) 12%
  • (b) 14%
  • (c) 16%
  • (d) 18%
Answer: (a) y = 3x/25. y/x × 100 = 12%

Q12. An inspector rejects 0.08% of metres as defective. How many metres will he examine to reject 2 metres?

  • (a) 200m
  • (b) 250m
  • (c) 2500m
  • (d) 3000m
Answer: (c) 0.08% of x = 2 ⇒ x = 2500

Q13. A positive number is by mistake divided by 6 instead of being multiplied by 6. What is the % error on the basis of correct answer?

  • (a) 3
  • (b) 97
  • (c) 17
  • (d) 83
Answer: (b) Correct = 6x, Wrong = x/6. Error = (6x - x/6)/6x × 100 = 97.2%

Q14. If the radius of a circle is diminished by 10%, the area is diminished by

  • (a) 36%
  • (b) 20%
  • (c) 19%
  • (d) 10%
Answer: (c) 2r - r²/100 = 20 - 1 = 19%

Q15. A screw driver and a hammer currently have the same price. If screw driver rises by 5% and hammer goes up by 3%, then how much more will it cost to buy 3 screw drivers and 3 hammers?

  • (a) 3%
  • (b) 4%
  • (c) 5%
  • (d) 8%
Answer: (b) 24/600 × 100 = 4%

Q16. What is the total number of candidates at an examination, if 31% fail, and the number of those who pass exceeds the number of those who fail by 247?

  • (a) 605
  • (b) 560
  • (c) 650
  • (d) 1,650
Answer: (c) 38% of total = 247 ⇒ total = 650

Q17. 40% read newspaper X, 50% read Y and 10% read both. What percentage read neither?

  • (a) 10%
  • (b) 15%
  • (c) 20%
  • (d) 25%
Answer: (c) n(A∪B) = 40+50-10 = 80%. Neither = 100-80 = 20%

Q18. If manufacturer gains 10%, wholesale dealer 15% and retailer 25%, find the cost of production of a table whose retail price is ₹1265?

  • (a) 800
  • (b) 1000
  • (c) 900
  • (d) 600
Answer: (a) 125% of 115% of 110% of x = 1265 ⇒ x = 800

Q19. A man buys 50 pencils for ₹100 and sells 45 pencils for ₹90. Find his gain or loss?

  • (a) 20%
  • (b) 35%
  • (c) 25%
  • (d) No gain or loss
Answer: (d) C.P. of 45 = 90, S.P. of 45 = 90. No gain no loss.

Q20. Kaberi buys a radio three-fourths of its value and sells it for 20% more than its value. What will be its profit percentage?

  • (a) 50%
  • (b) 40%
  • (c) 60%
  • (d) 70%
Answer: (c) C.P. = 3x/4, S.P. = 6x/5. Profit = 60%

5. Exercise (Questions 21-34)

Q21. A person sells 36 oranges per rupee and suffers a loss of 4%. Find how many oranges per rupee to be sold to have a gain of 8%?

  • (a) 30
  • (b) 31
  • (c) 32
  • (d) 33
Answer: (c) 36 × 96/108 = 32

Q22. A shopkeeper's price is 50% above the cost price. If he allows a discount of 30%, what profit does he make?

  • (a) 5%
  • (b) 10%
  • (c) 15%
  • (d) 20%
Answer: (a) S.P. = 150 × 70% = 105. Profit = 5%

Q23. A man buys milk at ₹6 per litre and adds one third of water to it and sells mixture at ₹7.20 per litre. The gain is

  • (a) 40%
  • (b) 80%
  • (c) 60%
  • (d) 25%
Answer: (b) C.P. = 6, S.P. = 7.20 × 3/2 = 10.80. Profit = 80%

Q24. A dishonest fruit seller professes to sell at cost price but weighs 800 grams for a kg weight. Find his gain percent.

  • (a) 100%
  • (b) 150%
  • (c) 50%
  • (d) 200%
Answer: (a) Gain = 200/800 × 100 = 25%

Q25. A shopkeeper purchased 150 calculators at ₹250 each. He spent ₹2500 on transport. He fixed labelled price at ₹320 and allowed 5% discount. Find profit percentage.

  • (a) 14%
  • (b) 15%
  • (c) 16%
  • (d) 18%
Answer: (a) C.P. = 40000, S.P. = 45600. Profit = 14%

Q26. A man sold two steel chairs for ₹500 each. On one he gains 20% and on other loses 12%. How much does he gain or lose in the whole transaction?

  • (a) 1.5% gain
  • (b) 2% gain
  • (c) 1.5% loss
  • (d) 2% loss
Answer: (a) Gain = 1.5%

Q27. A man sells an article at a gain of 15%. If he had bought at 10% less and sold for ₹4 less, he would have gained 25%. Find the cost price.

  • (a) ₹150
  • (b) ₹160
  • (c) ₹170
  • (d) ₹180
Answer: (b) C.P. = ₹160

Q28. Five kg of butter was bought for ₹300. One kg becomes unsaleable. He sells remaining at a loss of 10%. At what price per kg was the butter sold?

  • (a) ₹67.50
  • (b) ₹52.50
  • (c) ₹60
  • (d) ₹72.50
Answer: (a) S.P. = 270/4 = ₹67.50

Q29. By selling 66 metres of cloth a man loses the selling price of 22 metres. Find the loss per cent.

  • (a) 20%
  • (b) 25%
  • (c) 30%
  • (d) 35%
Answer: (b) Loss = S.P. of 22m. C.P. = S.P. of 88m. Loss% = 22/88 × 100 = 25%

Q30. A single discount equal to a discount series of 10% and 20% is

  • (a) 25%
  • (b) 28%
  • (c) 30%
  • (d) 35%
Answer: (b) 10 + 20 - 200/100 = 28%

Q31. The list price of a watch is ₹160. Retailer bought it for ₹122.40. He got two successive discounts, one at 10% and other not legible. What is the second discount rate?

  • (a) 12%
  • (b) 14%
  • (c) 15%
  • (d) 18%
Answer: (c) 160 × 0.9 × (1-d/100) = 122.40 ⇒ d = 15%

Q32. For a certain article, if discount is 25%, profit is 25%. If discount is 10%, then profit is

  • (a) 10%
  • (b) 20%
  • (c) 35%
  • (d) 50%
Answer: (d) 75/90 = 125/(100+g) ⇒ g = 50%

Q33. A wholesaler sells 30 pens at the price of 27 pens to a retailer. The retailer sells the pens at market price. The profit for the retailer is

  • (a) 11%
  • (b) 10%
  • (c) 11 1/9%
  • (d) 9 1/11%
Answer: (c) Gain = (30-27)/27 × 100 = 11 1/9%

Q34. If a merchant estimates his profit as 20% of the selling price, what is his real profit per cent?

  • (a) 18%
  • (b) 20%
  • (c) 25%
  • (d) 28%
Answer: (c) 20/(100-20) × 100 = 25%

Complete Answer Key

1. (a)2. (d)3. (d)4. (b)5. (b)
6. (d)7. (b)8. (c)9. (b)10. (c)
11. (a)12. (c)13. (b)14. (c)15. (b)
16. (c)17. (c)18. (a)19. (d)20. (c)
21. (c)22. (a)23. (b)24. (a)25. (a)
26. (a)27. (b)28. (a)29. (b)30. (b)
31. (c)32. (d)33. (c)34. (c)
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