1. 1A_Ch3(1)

2. 1A_Ch3(2) 3.1Simple Problems Involving Percentages A Using Percentage to Find a Number B Finding the Percentage C Finding the Original Number from a Given Percentage Index

3. 1A_Ch3(3) 3.2Percentage Change A Percentage Increase B Percentage Decrease C Percentage Change Index

4. 1A_Ch3(4) 3.3Profit and Loss A Profit B Loss Index

5. Using Percentage to Find a Number y% of a number A = A × y% Index A) 1A_Ch3(5) 3.1Simple Problems Involving Percentages + ExampleExample + Index 3.1Index 3.1

6. Find the value of each of the following. (a) 850 × 25% Index 1A_Ch3(6) 3.1Simple Problems Involving Percentages (a)25% of 850(b)10% of 12.8 = 850 × 0.25 = 212.5 (b) 12.8 × 10%= 12.8 × 0.1 = 1.28

7. Index 1A_Ch3(7) 3.1Simple Problems Involving Percentages There are 1 400 staff in a company. It is known that 88% of the staff in the company are university graduates. How many staff in that company are university graduates? Number of staff who are university graduates = 1 400 × 88% = 1 400 × = 1 232 Fulfill Exercise Objective  Use percentage to find a number.

8. Index 1A_Ch3(8) 3.1Simple Problems Involving Percentages In 2002, the population of Hong Kong was 6.8 million and 16% of them were aged under 15. How many people were 15 or above? Number of people aged under 15 = 6.8 × 16% million = 6.8 × 0.16 million = 1.088 million ∴ Number of people aged 15 or above = (6.8 – 1.088) million = 5.712 million Fulfill Exercise Objective  Use percentage to find a number. + Key Concept 3.1.1Key Concept 3.1.1

9. Finding the Percentage Index B) 1A_Ch3(9) 3.1Simple Problems Involving Percentages 1.To find out what percentage of a is b, we write. baba × 100% 2.To find out what percentage of b is a, we write. abab × 100% + ExampleExample + Index 3.1Index 3.1

10. (a)What percentage of 45 is 36? (b)What percentage of 36 is 45? Index 1A_Ch3(10) 3.1Simple Problems Involving Percentages (a)The required percentage = = 80% (b)The required percentage = = 125%

11. Index 1A_Ch3(11) 3.1Simple Problems Involving Percentages Among the 800 people in the election team, 256 of the team members vote for candidate A and the rest vote for candidate B. (a)What percentage of the votes has gone to candidate A? (b)What percentage of the number of votes for A is that for B?

12. Index 1A_Ch3(12) 3.1Simple Problems Involving Percentages (a)The required percentage = = 32% (b)The number of votes that candidate B gets = 800 – 256 = 544 The required percentage = = 212.5% Fulfill Exercise Objective  Find the required percentage. + Back to QuestionBack to Question + Key Concept 3.1.2Key Concept 3.1.2

13. Finding the Original Number from a Given Percentage Index C) 1A_Ch3(13) 3.1Simple Problems Involving Percentages Given : x% of the original number A = y Then yx%yx% A = + ExampleExample + Index 3.1Index 3.1

14. Find the unknown in each of the following. Index 1A_Ch3(14) 3.1Simple Problems Involving Percentages (a)$24 is 60% of $m.(b)180 kg is 125% of n kg. m × 0.6= 24 m= 24 ÷ 0.6 = 40 (a) $m × 60%= $24 n × 1.25= 180 n= 180 ÷ 1.25 = 144 (b) n kg × 125%= 180 kg

15. Index 1A_Ch3(15) 3.1Simple Problems Involving Percentages A football team won 85% of its games last year. If they won 34 games altogether, how many games did the team play? Let n be the total number of games the team played. Then 85% of n is 34. i.e. 85% × n= 34 n × 0.85= 34 n= 34 ÷ 0.85 = 40 ∴ The total number of games the team played was 40. Fulfill Exercise Objective  Find the original number.

16. Index 1A_Ch3(16) 3.1Simple Problems Involving Percentages In the academic year 1999 – 2000, 55% of the students in the University of Hong Kong were male. If the number of female students in that year was 6 300, then what was the total number of students? Percentage of female students = 100% – 55% = 45%

17. Index 1A_Ch3(17) 3.1Simple Problems Involving Percentages Let n be the total number of students. Then 45% of n is 6 300. i.e. 45% × n= 6 300 n × 0.45= 6 300 n= 6 300 ÷ 0.45 = 14 000 ∴ The total number of students was 14 000. Fulfill Exercise Objective  Find the original number. + Back to QuestionBack to Question + Key Concept 3.1.3Key Concept 3.1.3

18. Percentage Increase Index A) 1A_Ch3(18) 3.2Percentage Change + ExampleExample + Index 3.2Index 3.2 1.Increase = New value – Original value 2.Percentage increase = Increase Original value × 100% 3.Increase = Original value × Percentage increase 4.New value = Original value × (1 + Percentage increase)

19. Find the percentage increase in each of the following. Index 1A_Ch3(19) 3.2Percentage Change (a)Increase = 156 – 120 = 36 ∴ Percentage increase = = 30% (a)An increase from 120 to 156. (b)An increase of 10.5 mL from 25 mL. (b)Increase = 10.5 mL ∴ Percentage increase = = 42%

20. Increase each of the following quantities by the given percentage: Index 1A_Ch3(20) 3.2Percentage Change (a)New value = 82 × (1 + 15%) = 82 × 1.15 = 94.3 (a)82 by 15%(b)$144 by 35% (b)New value = $144 × (1 + 35%) = $144 × 1.35 = $194.4

21. Index 1A_Ch3(21) 3.2Percentage Change The price increase of a movie recorded on DVD and on video CD is both $5. If the original prices of a DVD and a video CD are $125 and $40 respectively, find the percentage increase in the price of (a) a DVD, (b) a video CD.

22. Index 1A_Ch3(22) 3.2Percentage Change (a)Percentage increase in the price of a DVD = = 4% (b)Percentage increase in the price of a video CD = = 12.5% Fulfill Exercise Objective  Find the percentage increase. + Back to QuestionBack to Question

23. Index 1A_Ch3(23) 3.2Percentage Change To celebrate the New Year, a certain brand of chocolate beans added 10% to the weight of each pack for free. If the original weight of each pack of the chocolate beans was 50 g, what was the weight of each pack after the increase? The new weight = 50 × (1 + 10%) g = 50 × 1.1 g = 55 g Fulfill Exercise Objective  Find the new value. + Key Concept 3.2.1Key Concept 3.2.1

24. Percentage Decrease Index B) 1A_Ch3(24) 3.2Percentage Change + ExampleExample + Index 3.2Index 3.2 1.Decrease = Original value – New value 2.Percentage decrease = Decrease Original value × 100% 3.Decrease = Original value × Percentage decrease 4.New value = Original value × (1 – Percentage decrease)

25. Find the percentage decrease in each of the following. Index 1A_Ch3(25) 3.2Percentage Change (a)Decrease = 300 – 48 = 252 ∴ Percentage decrease = = 84% (a)An decrease from 300 to 48. (b)An decrease of 3 g from 12 g. (b)Decrease = 3 g ∴ Percentage decrease = = 25%

26. Decrease each of the following quantities by the given percentage: Index 1A_Ch3(26) 3.2Percentage Change (a)New value = 150 × (1 – 27%) = 150 × 0.73 = 109.5 (a)150 by 27%(b)855 cm by 60% (b)New value = 855 × (1 – 60%) cm = 855 × 0.4 cm = 342 cm

27. Index 1A_Ch3(27) 3.2Percentage Change The worldwide population of the black rhinoceroes has dropped from 65 000 in the early 1970s to 2 405 in the late 1990s. What is the percentage decrease in the number of black rhinoceroes? The decrease = 65 000 – 2 405 = 62 595 The percentage decrease = = 96.3% Fulfill Exercise Objective  Find the percentage decrease.

28. Index 1A_Ch3(28) 3.2Percentage Change Kitty scored 80 marks in Mathematics last term. If her score is decreased by 5% this term, what is her score in this term? The score of this term = 80 × (1 – 5%) = 80 × 0.95 = 76 Fulfill Exercise Objective  Find the new value. + Key Concept 3.2.2Key Concept 3.2.2

29. Percentage Change Index C) 1A_Ch3(29) 3.2Percentage Change + ExampleExample + Index 3.2Index 3.2 ‧ Percentage change = New value – Original value Original value × 100% New value > Original value New value < Original value Change Sign of percentage change Increase Decrease + –

30. Index 1A_Ch3(30) 3.2Percentage Change (a)If 100 is changed to 110, then percentage change = = +10% (b)If 50 is changed to 40, then percentage change = = –20%

31. Index 1A_Ch3(31) 3.2Percentage Change The price of a digital camera was $4 000 last year. This year the price becomes $3 400. Find the percentage change in price. Percentage change Fulfill Exercise Objective  Find the percentage change. == = –15% + Key Concept 3.2.3Key Concept 3.2.3

32. Profit Index A) 1A_Ch3(32) 3.3Profit and Loss 1.When a merchant pays to buy goods, the amount he pays is called the cost price. 2.When the merchant sells goods at a price, the amount he receives is called the selling price. 3.If selling price > cost price, the merchant will make a profit. 4.If we compare the profit with the cost price of the goods and express the result as a percentage, the percentage is called the profit per cent (written as profit %).

33. Profit Index A) 1A_Ch3(33) 3.3Profit and Loss + ExampleExample + Index 3.3Index 3.3 5.If selling price > cost price, ii.Profit % = Profit Cost price × 100% iii.Profit = Cost price × Profit % iv.Selling price = Cost price × (1 + Profit %) i.Profit = Selling price – Cost price

34. The selling price of a toy car is $225, and the cost price is $180. Index 1A_Ch3(34) 3.3Profit and Loss (a)Find the profit of the toy car. (b)Find the profit per cent of the toy car. (a)Profit= $(225 – 180)= $45 (b)Profit % = = 25%

35. Index 1A_Ch3(35) Mr Wong bought a pair of jeans for $80 and sold them for $130. Mr Cheung bought a T-shirt for $40 and sold it for $90. Who made a greater profit %? Profit made by Mr Wong = 3.3Profit and Loss = $(130 – 80) = $50 ∴ Profit % = 62.5%

36. Index 1A_Ch3(36) Profit made by Mr Cheung = 3.3Profit and Loss = $(90 – 40) = $50 ∴ Profit % = 125% ∴ Mr Cheung made a greater profit %. Fulfill Exercise Objective  Find the profit %. + Back to QuestionBack to Question

37. Index 1A_Ch3(37) A car was bought for $350 000 and was sold at a profit of 20%. How much was it sold for? Selling price + Key Concept 3.3.1Key Concept 3.3.1 3.3Profit and Loss = $350 000 × (1 + 20%) = $420 000 = $350 000 × 1.2 Fulfill Exercise Objective  Find the selling price.

38. Loss Index B) 1A_Ch3(38) 3.3Profit and Loss 1.If selling price < cost price, there will be a loss. 2.We can express loss as a percentage of the cost price. The result is called the loss per cent (written as loss %).

39. Loss Index B) 1A_Ch3(39) 3.3Profit and Loss + ExampleExample 3.If selling price < cost price, ii.Loss % = Loss Cost price × 100% iii.Loss = Cost price × Loss % iv.Selling price = Cost price × (1 – Loss %) i.Loss = Cost price – Selling price + Index 3.3Index 3.3

40. Index 1A_Ch3(40) If a bicycle is bought at $1 000 and sold for $800, then loss = $(1 000 – 800) = $200 and loss % = = 20% 3.3Profit and Loss

41. Index 1A_Ch3(41) A box of 100 coloured pencils costs $250. In a sale, each pencil is sold at $2 a piece.What is the loss per cent after all of the pencils have been sold? Total cost price + Key Concept 3.3.2Key Concept 3.3.2 3.3Profit and Loss = $250 Total selling price = $2 × 100 = $200 Total loss= $(250 – 200)= $50 ∴ Loss % = = 20% Fulfill Exercise Objective  Find the loss %.

42. Discount Index 1A_Ch3(42) 3.4Discount 1.The difference between the marked price and the selling price is called the discount. 2.We can express the discount as a percentage of the marked price, which is called the discount per cent (written as discount %).

43. Discount Index 1A_Ch3(43) 3.4Discount + ExampleExample 3.i.Discount = Marked price – Selling price ii.Discount % = Discount Marked price × 100% iii.Discount = Marked price × Discount % iv.Selling price = Marked price × (1 – Discount %)

44. A sofa is marked at $2 800 and sold at $2 100 in a sale. What is the discount and discount %? Index 1A_Ch3(44) Discount = $(2 800 – 2 100) = $700 = 25% 3.4Discount Discount % =

45. Index 1A_Ch3(45) A pack of E-Power Battery marked at $24 is now sold for $21 only. A pack of D-cell Battery, on the other hand, marked at $20 is now sold for $17.9. Which pack of Battery is sold at a larger discount per cent? 3.4Discount

46. Index 1A_Ch3(46) For E-Power Battery, 3.4Discount = $(24 – 21) discount % = = 12.5% For D-cell Battery, = $(20 – 17.9) discount % = = 10.5% = $3 = $2.1 ∴ E-Power Battery is sold at a larger discount per cent. Fulfill Exercise Objective  Find the discount %. discount + Back to QuestionBack to Question

47. Index 1A_Ch3(47) After a 44% discount, a radio is now sold for $112. What was the original marked price of the radio? 3.4Discount 44% off Now $112 only Let $P be the original marked price of the radio. Then 112= P × (1 – 44%) 112= P × 0.56 P= 112 ÷ 0.56 = 200 ∴ The original marked price was $200. Fulfill Exercise Objective  Find the marked price.

48. Index 1A_Ch3(48) A dictionary marked at $260 is sold at a discount of 25%. 3.4Discount (a)Find the selling price. (b)If the cost price of the dictionary is $200, find the loss %. Dictionary 25% off (a)The selling price = $260 × (1 – 25%) = $260 × 75% = = $195

49. Index 1A_Ch3(49) + Key Concept 3.4.1Key Concept 3.4.1 3.4Discount (b)Loss = cost price – selling price = $(200 – 195) = $5 = = 2.5% Loss % Fulfill Exercise Objective  Miscellaneous problems. + Back to QuestionBack to Question

50. Simple Interest Index 1A_Ch3(50) 3.5Simple Interest 1.If the interest in each period is calculated on the same principal, then the interest obtained is called simple interest. 2.The calculation of interest is based on a percentage of the principal and that percentage is called the interest rate.

51. Simple Interest Index 1A_Ch3(51) 3.5Simple Interest 3.Let $I stand for the simple interest, $P stand for the principal, R% stand for the interest rate per annum, T years stand for the period of time, $A stand for the amount, then + ExampleExample i.ii. + ExampleExample

52. Index 1A_Ch3(52) If $1 000 is deposited in a bank at an interest rate of 6% p.a., find the interest received a year later. 3.5Simple Interest Interest received = $1 000 × 6% = $60

53. Index 1A_Ch3(53) Deposit $6 000 in a bank at an interest rate of 8% p.a. for 2 years. What is the simple interest received? 3.5Simple Interest Simple interest received = = $960

54. Index 1A_Ch3(54) Simon wants to deposit $30 000 in a bank for 2.5 years. Bank A offers an interest rate of 8% p.a. and Bank B offers 7.5% p.a. How much more simple interest will Bank A give to Simon than Bank B? Simple interest given by Bank A 3.5Simple Interest = = $6 000

55. Index 1A_Ch3(55) Simple interest given by Bank B 3.5Simple Interest = = $5 625 The difference between the simple interests ∴ Bank A will give $375 more simple interest to Simon than Bank B. Fulfill Exercise Objective  Find the simple interest. = $(6 000 – 5 625) = $375 + Back to QuestionBack to Question

56. Index 1A_Ch3(56) Kimmy borrows some money from a bank to start her own business. The interest rate is 5% p.a. Find the amount of money she borrows from the bank if she has to pay a total of $30 000 as simple interest after 2 years. 3.5Simple Interest

57. Index 1A_Ch3(57) Let $P be the amount of money Kimmy borrows, then 3.5Simple Interest 30 000 = P = 30 000 × 10 = 300 000 ∴ The amount of money Kimmy borrows is $300 000. Fulfill Exercise Objective  Find the principal. + Back to QuestionBack to Question + Key Concept 3.5.1Key Concept 3.5.1

58. Index 1A_Ch3(58) A bank pays simple interest for money deposited at an interest rate of 10% p.a. If $5 000 is deposited in the bank, find the amount received after 3 years. 3.5Simple Interest Amount received after 3 years = = $6 500

59. Index 1A_Ch3(59) A bank pays simple interest at an interest rate of 8% p.a. If $25 000 is deposited in the bank, find 3.5Simple Interest (a)the amount received after 10 months, correct to the nearest $100, (b)the time required to get the amount $27 500. (a)Amount received after 10 months = 10 12 = $26 700, cor. to the nearest 100

60. Index 1A_Ch3(60) 3.5Simple Interest (b)Let T years be the time required, then 27 500 = == 0.1 T = = 1.25 ∴ The time required is 1.25 years, i.e. 1 year and 3 months. Fulfill Exercise Objective  Find the amount.  Find the period of time. + Back to QuestionBack to Question

61. Index 1A_Ch3(61) John deposits $1 000 000 in a bank where simple interest is calculated at a rate of R% p.a. 3.5Simple Interest (a)If the amount received by John doubles the principal after 10 years, find the value of R. (b)Suppose R = 8. (i)Find the amount received by John after 20 years. (ii)Peter borrows a sum of money from John and the interest rate is 8% p.a. If the amount that Peter has to pay back John after 2.5 years is $120 000, how much does Peter borrow? + SolnSoln + SolnSoln

62. Index 1A_Ch3(62) 3.5Simple Interest (a)The principal is $1 000 000. The amount received by John after 10 years Hence 2 000 000 = 2 = 1 + 0.1R 1 = 0.1R ∴ R = 10 = $1 000 000 × 2 = $2 000 000 + Back to QuestionBack to Question

63. Index 1A_Ch3(63) 3.5Simple Interest (b)(i)Amount received after 20 years = = $2 600 000 (ii)If $P is the sum of money that Peter borrows from John, then 120 000 = 120 000 = P(1 + 0.2) P = = 100 000 ∴ Peter borrowed $100 000 from John. Fulfill Exercise Objective  Find the interest rate.  Find the principal. + Key Concept 3.5.1Key Concept 3.5.1 + Back to QuestionBack to Question