S2 Mathematics Rate, Ratio and Proportion PDF

S2 Mathematics Rate, Ratio and Proportion 7 Notes Table of Contents Section 1: Rates.............................................................................................................

S2 Mathematics Rate, Ratio and Proportion 7 Notes Table of Contents Section 1: Rates....................................................................................................................... 2 Section 2: Ratios...................................................................................................................... 9 Section 3: Continued Ratios................................................................................................... 16 Section 4: Proportions............................................................................................................ 23 4.1 Concept of Proportion............................................................................................ 23 4.2 Scale Drawings...................................................................................................... 30 Section 5: Direct and Inverse Proportions............................................................................... 42 5.1 Direct Proportions.................................................................................................. 42 5.2 Inverse Proportions................................................................................................ 44 Section 6: Answers................................................................................................................. 46 Prepared by W. S. Yeung 2024/11/07 S2 Mathematics Rate, Ratio and Proportion Section 1: Rates Let’s look at the following examples. Car A takes 2 hours to travel 160 km while car B takes 3 hours to travel 270 km. Amy wants to know which car is faster. 160 (a) Distance that car A travelled in 1 hour = 2 = 80 km 270 (b) Distance that car B travelled in 1 hour = 3 = 90 km  Car B travels faster. In the above example, since the travelling times for both cars are different, we cannot determine which car is faster by simply comparing the distance travelled. It is more sensible to determine which car is faster by comparing their speeds. In fact, speed is an example of rate. The meaning of rate is stated as follows. A rate is a comparison of two quantities of different kinds by division. The symbol ‘/’ is used to denote ‘per’. For example, the speed of the car A in the above example can be expressed as 80 km/h. Example 1.1 A vehicle travels 360 km in 4 hours. Express its speed in each of the following units. (a) km/h (b) m/s Solution (a) The required speed (b) The required speed 360 km 90 km = = 4 hours 1 hour = 90 km/h (90)(1000) m = (60)(60) s = 25 m/s Practice 1.1 1. An aeroplane takes 6 hours to travel 6480 km. Express the speed of the aeroplane in each of the following units. (a) km/h (b) m/s 2. A ship sails 108 km in 3 hours. Express the speed of the ship in each of the following units. (a) km/h (b) m/s Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 2 S2 Mathematics Rate, Ratio and Proportion 3. A car travels 180 km in 4 hours. Express the speed of the car each of in the following units. (a) km/h (b) m/s 4. A car travels 405 km in 5 hours. Express the speed of the car in each of the following units. (a) km/h (b) m/s 5. A ship sails 96 km in 240 minutes. Express the speed of the ship in each of the following units. (a) km/h (b) m/min 6. Andy walks 7200 m in 4 hours. Express the walking speed of Andy in each of the following units. (a) km/h (b) m/min 7. A car completes a 900 km race in 5 hours. (a) Express the speed of the car in km/h. (b) Express the speed of the car in m/s. 8. A train travels 90 km in 45 minutes. (a) Express its speed in km/h. (b) Express the distance travelled in 1 hour and 20 minutes. (c) Express the time required to travel 300 km. Example 1.2 (a) Billy types 440 words in 11 minutes. Express the typing speed of Billy in words/min. (b) 3 kg of rice is sold at $96. Express the price rate of rice in $/kg. Solution 440 96 (a) The typing speed = (b) The price rate = 11 3 = 40 words/min = $32/kg Practice 1.2 1. A machine produces 520 sets of digit camera in 5 hours. Find the production rate in set/h. 2. 4 kg of meat is sold at $92. Find the price rate of meat in $/kg. 3. The selling price of 8 balls is $140. Find the price rate of ball in $/ball. 4. The rent of a house is $5400 for 3 months. Find the rate of rent in $/month. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 3 S2 Mathematics Rate, Ratio and Proportion 5. 15 bottles of milk weight 9 kg. Find the rate of milk in kg/bottle. 6. A car consumes 4 litres of petrol to cover 60 km. Find the rate of consumption in km/litre. 7. A motorcycle travels 120 km in 2.5 hours. Find the speed of the motorcycle in km/h. 8. A factory produces 2100 toys in 1 week. Find the rate of production in toys/day. 9. 1 dozen eggs cost $7.2. Find the price rate of egg in $/egg. 10. 4 litres of milk cost $28.8 and weight 6 kg. (a) Find the cost of milk in $/litre. (b) Find the cost of milk in $/kg. 11. 15 workers paint 420 chairs in 7 hours. (a) Find the rate of painting in chairs/h. (b) Find the rate of painting in chairs/worker. 12. Amy can read 60 pages of a book in 40 minutes. (a) Find her reading speed in pages/min. (b) Find her reading speed in pages/h. Example 1.3 Solution Car A travels at 36 km/h while car B travels at 36 km The speed of car A = 12 m/s. Somebody claims that car B travels 1h faster. Do you agree? Explain your answer. (36)(1000)m = (1)(60)(60)s = 10 m/s < 12 m/s = The speed of car B The claim is agreed. Practice 1.3 1. A cat runs at 1.5 km/hour while a dog runs at 40 m/min. Somebody claims that the cat runs faster. Do you agree? Explain your answer. 2. Sam ran 4.5 km in 18 minutes. Peter ran 7020 m in 0.5 hour. Somebody claims that Sam runs faster. Do you agree? Explain your answer. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 4 S2 Mathematics Rate, Ratio and Proportion Example 1.4 Solution The selling price of 150 cm of wire is $30. (a) The price of wire (a) Find the price of the wire in $/m. $30 = (b) Find the selling price of 13 m of the wire. 150cm (c) Find the length of the wire which is sold $30 = at $400. 1.5m = $20/m (b) The selling price = (13)(20) = $260 (c) The length of the wire 400 = 20 = 20 m Practice 1.4 1. A shop sells 2400 pairs of shoes in 15 days. (a) Express the rate of selling shoes in pairs/day. (b) Find the number of pairs of shoes can the shop sell in 26 days. (c) Find the number of days will the shop take to sell 3200 pairs of shoes. 2. The parking rate of a car park is $22 per hour. Find the parking fee for 5 hours. 3. The typing speed of Amy is 3120 words per hour. (a) Find the typing speed of Amy in words/min. (b) If Amy keeps typing at the same rate, find the time required to type 1560 words. 4. The selling price of 50 meter of copper wire is $1500. (a) Find the price rate of copper wire in $/m. (b) Find the selling price of 32 meter of copper wire. (c) Find the length of the copper wire which is sold at $2160. 5. The printing speed of a printer is 24 pages per minute. (a) If the printer takes 12 minutes to print a document, find the number of pages in the document. (b) Find the time required to print 144−page document. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 5 S2 Mathematics Rate, Ratio and Proportion Example 1.5 Solution Yeung’s family consumes 1925 g of rice in a (a) The rice consumption rate week. 1925 g = (a) Find the rice consumption rate of Yeung’s 7 day family in g/day. = 275 g/day (b) If the rice consumption rate of Yeung’s family remains unchanged, Mr Yeung claims that a (b) The rice consumption in November bag of 8 kg rice is enough for his family to = 275 (30) eat in the whole November. Do you agree? = 8250 g Explain your answer. = 8.25 kg > 8 kg  The claim is disagreed. Practice 1.5 1. A factory manufactures 18 000 sets of smart phones in 10 days. (a) Find the production rate of smart phone in sets/day. (b) The production rate is kept constant. Somebody claims that the factory can manufacturers 52000 sets of smart phones in 4 weeks. Do you agree? Explain your answer. 2. A photocopier can print 405 copies in 9 minutes. (a) Find the copying rate of the photocopier in copies/min. (b) Somebody claims that the photocopier can print 2800 copies in an hour. Do you agree? Explain your answer. 3. A baker makes 216 cakes in 9 hours. (a) Find the production rate of the baker in cakes/h. (b) Somebody claims that the baker can makes 140 cakes in 6 hours if he keeps the same production rate. Do you agree? Explain your answer. 4. The price of 4 dozen correction pens is $432. (a) Find the price of a correction pen. (b) Candy has $140. She claims that she can buy 15 correction pens. Do you agree? Explain your answer. 5. A factory manufacturer 1280 sets of LED TV in 4 days. (a) Find the production rate of LED TV in sets/day. (b) The production rate is kept constant. Somebody claims that the factory can manufacturers 2300 sets of LED TV in a week. Do you agree? Explain your answer. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 6 S2 Mathematics Rate, Ratio and Proportion 6. A coffee machine can make 75 cups of coffee in an hour. (a) Find the production rate of the machine in cups/min. (b) A café owner claims that the coffee machine can make at least 22 cups of coffee in 18 minutes. Do you agree? Explain your answer. 7. At a sport function, any participant who can run 12 km in 1.5 hour will receive a certificate. Suppose Candy runs 1.5 km in 12 minutes. (a) Find the running speed of Candy in km/min. (b) Candy claims that if she maintains the running speed described in (a), she will receive a certificate at the end of her run. Do you agree? Explain your answer. 8. A worker can make 100 T−shirts in a week. (a) If the worker works 5 days a week and 8 hours a day, find the production rate of the worker in T−shirts/h. (b) The manager of the factory claims that the worker can make 200 T−shirts in a week if he works overtime for 7 days a week and 12 hours day. Do you agree? Explain your answer. 9. A printer can print 270 pages of document in 30 minutes. (a) Find the printing speed of the printer in pages/min. (b) Daisy has to print a report of 110 pages with the printer for her boss. She claims that the report can be printed within 12 minutes. Do you agree? Explain your answer. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 7 S2 Mathematics Rate, Ratio and Proportion Example 1.6 Solution In a bank, CNY 800 can be exchanged for (a) The exchange rate HKD 1000. 800 = (a) Find the exchange rate in CNY/HKD. 1000 (b) How much CNY can we exchange for = 0.8 CNY/HKD HKD 1200? (c) How much HKD can we exchange for (b) The amount of CNY CNY 1000? = (1200)(0.8) (CNY: Renminbi; HKD: Hong Kong dollars) = CNY 960 (c) The amount of HKD 1000 = 0.8 = HKD 1250 Practice 1.6 1. In a currency exchange store, GBP 30 can be exchanged for HKD 315. (a) Find the exchange rate in HKD/GBP. (b) How many GBP can be exchanged for HKD 504? (c) How many HKD can be exchanged for GBP 100? 2. In a bank, HKD 390 can be exchanged for USD 50. (a) Find the exchange rate in HKD/USD. (b) How many HKD can be exchanged for USD 400? (c) How many USD can be exchanged for HKD 468? 3. In a currency exchange store, THB 32 can be exchanged for HKD 8. (a) Find the exchange rate in THB/HKD. (b) How many THB can be exchanged for HKD 2000? (c) How many HKD can be exchanged for THB 3600? 4. In a bank, CNY 156 can be changed for USD 25. (a) Find the exchange rate in CNY/USD. (a) How many CNY can be exchanged for USD 150? (b) How many USD can be exchanged for CNY 1872? 5. In a bank, CNY 100 can be exchanged for HKD 125. (a) How many CNY can be exchanged for HKD 1200? (b) How many HKD can be exchanged for CNY 1280? Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 8 S2 Mathematics Rate, Ratio and Proportion Section 2: Ratios A. The meaning of Ratio Let’s look at the following recipe of a fruit punch. Fruit punch recipe Mix 3 portions of orange juice with 4 portions of pineapple juice. 3 The recipe suggests that the volume of orange juice is of the volume of pineapple juice. To 4 describe this relation between these two volumes, the ratio of the volume of orange to that of 3 pineapple juice is 3 : 4 or. 4 a For two quantities a and b of the same kind, the ratio of a to b can be denoted as a : b or , b where a  0 and b  0. Notes: 1. Ratios have no units. 2. In general a : b  b : a. 3. a : b = 3 : 4 does not mean that a = 3 and b = 4. B. Simplification of Ratios Consider the ratio 6 : 8. 6 3 As = 8 4  6 : 8 and 3 : 4 are equal ratios. a a am a m In general, = and = , where m is any non−zero number, we have the following b bm b b m properties. If a, b and m are non−zero numbers, then 1. a : b = am: bm a b 2. a:b = : m m Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 9 S2 Mathematics Rate, Ratio and Proportion Example 2.1 Simplify each of the following ratios. 2 1 (a) 6 : 3 (b) : (c) 0.8 m :1.2 m (d) 0.5 kg : 200g 3 5 Solution (a) 6:3 (c) 0.8 m :1.2 m 6 3 = (0.8)(10) : (1.2)(10) = : 3 3 = 8 :12 = 2 :1 = 2: 3 2 1 (d) 0.5 kg : 200 g (b) : 3 5 = 500 g : 200 g 2 1 = 5:2 = (15) : (15) 3 5 = 10 : 3 Practice 2.1 Simplify each of the following ratios. 1 4 1. (a) 18 :12 (b) 0.6 : 2.8 (c) : (d) 1.25 m : 75 cm 2 3 12 8 1 3 2. (a) 35 : 49 (b) : (c) 0.3 : (d) : 2.5 7 7 3 8 3. (a) 18 cm :1m (b) 1.2 cm : 4 km (c) 2.5 : 6.5 (d) 20 min : 4 hours 3 5 4. (a) 30 : 8 (b) : (c) 4.2 L : 4800 mL (d) 480g : 1.2 kg 5 3 1 2 5. (a) 12:15 (b) : (c) 32 cm : 0.72 m (d) 25 min : 300 s 4 3 Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 10 S2 Mathematics Rate, Ratio and Proportion Example 2.2 Solution There are 1750 students at a school and 1000 of The required ratio them are boys. Find the ratio of the number of = 1000 : (1750 − 1000) boys to number of girls. = 1000 : 750 = 4:3 Practice 2.2 1. Andy and Billy have $2800 in total. If Andy has $800, find the ratio of the amount of money that Andy has to that of Billy has. 2. There are 30 balls in a bag, of which 12 are red and the rest are black. Find the ratio of the number of red balls to that of black balls. 3. There are 80 students in the Mathematics club. 48 of them are boys. Find the ratio of the number of boys to the number of girls. 4. The total number of oranges and banana bought is 120. If there are 55 oranges, find the ratio of the number of oranges to that of banana. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 11 S2 Mathematics Rate, Ratio and Proportion Example 2.3 Solution Andy and Billy share some cookies in the ratio of The number of cookies 5:3. If Billy gets 18 cookies, find the number 5 = 18   of cookies that Andy got. 3 = 30 Thinking 18 5k 3k Practice 2.3 1. The areas of two rooms are in the ratio of 9 : 7. If the area of the smaller room is 35 m2, find the area of the large room. 2. The ratio of Ann’s height to her father’s is 2 : 3. If her father’s height is 180 cm, find the height of Ann. 3. Mary wants to dilute a bottle of lemon juice by water in the ratio of 3 : 7. If there is 900 mL lemon juice, find the amount of water. 4. Amy and Betty share an amount of money in the ratio of 8 : 5. If Amy gets $2400, find the amount of money that Betty gets. 5. In a joint school Mathematics competition, the ratio of the number of male participants to the number of female participants is 7 : 5. If there are 2870 male participants, find the number of female participants in the joint school Mathematics competition. 6. The monthly salaries of Mr Wong and Mrs Wong are in the ratio of 8 : 5. If the monthly salaries of Mr Wong is $88000, find the monthly salary of Mrs Wong. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 12 S2 Mathematics Rate, Ratio and Proportion Example 2.4 Solution Ada and Bob share an amount of $72 in the ratio The amount that Ada has of 4 : 5. Find the amount that each of them  4  = 72   gets. 4+5 = $32 Thinking The amount that Bob has $72  5  = 72   4+5 = $40 $4k $5k Practice 2.4 1. Two numbers are in the ratio 4 : 7. If the sum of the numbers is 44, find the two numbers. 2. The ratio of the present ages of the son and his father is 1: 4. If the sum of their present ages is 40, find the present ages of the son and his father. 3. The total number of stamps owned by Andy and Billy is 1650. The ratio of the stamps owned by Andy and Billy is in the ratio 7 : 8. Find the number of stamps owned by Billy. 4. There are 875 students in a school. If the ratio of the number of boys to the number of girls is 4 : 3 , find the number of boys and the number of girls. 5. David and Elaine share an amount of $2800 in the ratio 2 : 5. Find the amount that each of them gets. 6. Tiffany and Vincent share an amount of $2500 in the ratio 7 : 3. If Tiffany gives $x to Vincent, they will have the same amount of money. Find x. 7. There are 425 staff members in a company. The number of male staff numbers and the number of female members are in the ratio 8 : 9. Find the number of male staff member and the number of female members in the company. 8. In a test, the ratio of the number of questions Peter answered incorrectly to the number of the questions he answered correctly is 18 : 25. Peter answered 72 questions incorrectly, find the number of the questions in the test. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 13 S2 Mathematics Rate, Ratio and Proportion Example 2.5 Solution A sum of money is divided between Candy and The sum of money Danny in the ratio 3 : 5. If Danny gets $18 5+3 = 18   more than Candy, find the sum of amount of 5−3 money. = $72 Thinking $3k $18 $5k Practice 2.5 1. The ratio of the number of boys to the number of girls is 7 : 6 in a class. If there are 3 boys more than girl, find the number of students in the class 2. The ratio of the number of teeth on gear A to the number of teeth on gear B is 5 : 8. If gear B has 36 teeth more than gear A, find the total number of teeth. 3. The ratio of the cost price to the selling price of a toy is 5 : 3. If the toy is sold at a loss of $72, find the cost price. 4. The ratio of the number of black marbles to the number of red marbles is 5 : 8. If the number of the red marble is greater than that of black marbles by 96, find the numbers of the red marbles. 5. The ratio of the number of male staff to that of female staff in a hospital is 5 : 3. If the number male staff is greater than that of female staff by 270, find the number of male staff in the hospital. 6. Amy and Billy share a jar of milk in the ratio of 7 : 5. If Amy gets 320 mL more milk than Billy, find the original volume of the jar of milk. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 14 S2 Mathematics Rate, Ratio and Proportion Example 2.6 There are 65 black marbles and some red marbles in a bag. The number of red marbles is less than that of the black marble by 30. (a) Find the ratio of the number of red marbles to the total number of marbles in the bag. (b) Andy puts an extra amount of red marbles into the bag so that the ratio of the number of red marbles to the total number of marbles becomes 1: 2. Find the number of extra red marbles. Solution (a) The required ratio = (65 − 30) : (65 − 30 + 65) = 35 :100 = 7 : 20 (b) Let the number of extra red marbles be x. 35 + x 1 = 100 + x 2 70 + 2x = 100 + x x = 30 The number of extra red marbles is 30. Practice 2.6 1. In a box, there are some apples and oranges. The weight of apples is 1.6 kg. The weight of orange is heavier than that of apples by 2.8 kg. (a) Find the ratio of the weight of apples to the total weight of fruits in the box. (b) Extra apples are put into the box so that the ratio of the weight of apples to the total weight of fruit becomes 1: 3. Find the weight of the extra apples. 2. Peter mixed some China tea and Indian tea. The weight of China Tea was 3.75 kg. The weight of India tea was 2.5 kg more than that of China tea. (a) Find the ratio of the weight of Indian tea to the total weight of tea. (b) Peter added an extra amount of China tea so that the ratio of the weight of Indian tea to the total weight of tea became 1: 3. Find the weight of the extra China tea. 3. The number of stamps owned by Andy is 420. The number of stamps owned by Billy is fewer than that of Andy by 140. (a) Find the ratio of the number of stamps owned by Andy to that of Billy. (b) Andy gives some stamps to Billy so that the ratio in (a) becomes 11: 9. Find the number of stamps that Andy gives to Billy. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 15 S2 Mathematics Rate, Ratio and Proportion Section 3: Continued Ratios 3.1. Basic Concept A ratio can be used to compare three or more quantities with the same unit. For example, comparing the quantities x, y and z can be denoted by x : y : z. Such a ratio is called a continued ratio. Although a continued ratio x : y : z cannot be expressed as a fraction, continued ratio can also be simplified by the following properties. For k  0 , (i) x : y : z = xk : yk : zk x y z (ii) x:y:z = : :. k k k Example 3.1 Simplify (a) 12 : 20 :16 , (b) 0.5 : 2 : 3.5 , 1 5 7 (c) : :. 3 6 15 Solution (a) 12 : 20 :16 1 5 7 (c) : : 12 20 16 3 6 15 = : : 4 4 4 1 5 7 = (30) : (30) : (30) = 3:5: 4 3 6 15 = 10 : 25 :14 (b) 0.5 : 2 : 3.5 = 5 : 20 : 35 = 1: 4: 7 Practice 3.1 Simplify the following ratios 2 3 4 1. (a) 24 : 8 :12 (b) : : (c) 0.9 : 6 :1.5 5 10 25 1 3 5 2. (a) 15 : 6 : 9 (b) 0.8 : 2.8 :1.2 (c) : : 4 8 12 1 2 3 3. (a) 24 : 30 : 54 (b) 1.8 : 0.6 : 4.5 (c) : : 5 15 20 Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 16 S2 Mathematics Rate, Ratio and Proportion ⚫ If a : b : c = 1: 2 : 5 , it is easy to obtain a : b = 1: 2 , a : c = 1: 5 and b : c = 2 : 5. ⚫ On the other hand, when a : b = 5 : 4 and b : c = 4 : 3 , how can we find the ratio a : b : c ? Since the common term b in both ratios corresponds to the same number 4, we can simply write a:b:c = 5: 4:3. i.e. a : b : c 5 : 4 4 : 3 5 : 4 : 3  a:b:c = 5:4:3 Example 3.2 (a) Let x, y and z be non−zero numbers such that x : y = 5 : 3 and x : z = 2 : 5. Find x : y : z. (b) Let x, y and z be non−zero numbers such that x : y = 3 : 5 and y : z = 10 : 7. Find x : y : z. (c) Let x, y and z be non−zero numbers such that x : z = 1: 2 and y : z = 5 : 8. Find x : y : z. Solution (a) x : y : z (b) x : y : z (b) x : y : z 5 : 3 3 : 5 1 : 2 2 : : 5 10 : 7 5 : 8 10 : 6 : 25 6 : 10 : 7 4 : 5 : 8  x : y : z = 10 : 6 : 25  x : y : z = 6 :10 : 7  x: y: z = 4:5:8 Practice 3.2 1. Let x, y and z be non−zero numbers such that x : y = 5 : 3 and y : z = 6 : 7. Find x : y : z. 2. Let x, y and z be non−zero numbers such that x : y = 4 : 3 and x : z = 6 : 5. Find x : y : z. 3. Let x, y and z be non−zero numbers such that x : z = 5 : 2 and y : z = 7 : 3. Find x : y : z. 4. Let a, b and c be non−zero numbers such that a : b = 3 : 5 and a : c = 2 : 3. Find a : b : c. 5. Let a, b and c be non−zero numbers such that a : b = 2 : 5 and b : c = 3 : 4. Find a : b : c. 6. Let p, q and r be non−zero numbers such that p : q = 5 : 4 and q : r = 3 : 2. Find p : q: r. 7. Let a, b and c be non−zero numbers such that a : b = 5 : 2 and b : c = 3 : 2. Find a : b : c. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 17 S2 Mathematics Rate, Ratio and Proportion Example 3.3 Let x, y and z be non−zero numbers such that x = 4y and x = 3z. Find x : y : z. Solution Step  Step  Step  x = 4y x = 3z x : y : z x 4 x 3 4 : 1 = = y 1 z 1 3 : : 1 x:y= 4 :1 x:z = 3 :1 12 : 3 : 4  x : y : z = 12 : 3 : 4 Practice 3.3 1. Let p, q and r be non−zero numbers such that p = 2q and q = 3r. Find p : q: r. 2. Let p, q and r be non−zero numbers such that p = 5r and 2q = 5r. Find p : q: r. 3. Let a, b and c be non−zero numbers such that a = 3b and b = 5c. Find a : b : c. 4. Let a, b and c be non−zero numbers such that 6a = 5b and 4b = 9c. Find a : b : c. Example 3.4 If 4x = 5y = 6z , find x : y : z. Solution 4x = 5y 5y = 6z x : y : z x 5 y 6 5 : 4 = = y 4 z 5 6 : 5 x:y= 5:4 y:z= 6:5 15 : 12 : 10  x : y : z = 15 :12 :10 Practice 3.4 1. Let x, y and z be non−zero numbers such that x = 2y = 3z. Find x : y : z. 2. Let x, y and z be non−zero numbers such that 2x = 3y = 5z. Find x : y : z. 3. Let a, b and c be non−zero numbers such that 3a = 4b = 5c. Find a : b : c. 4. Let a, b and c be non−zero numbers such that 3a = 8b = 12c. Find a : b : c. 5. Let a, b and c be non−zero numbers such that a = 2b = 6c. Find a : b : c. 6. Let a, b and c be non−zero numbers such that 5a = 8b = 30c. Find a : b : c. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 18 S2 Mathematics Rate, Ratio and Proportion Example 3.5 Solution The ratio of three integers is 4 : 9 : 5. If the The smallest integer sum of the three integers is 108, find the  4  = 108   smallest integer. 4+9+5 = 24 Practice 3.5 1. A wire of length 60 cm is cut into 3 parts in the ratio of 2 : 5 : 8. Find the length of the shortest part. 2. The number of $1, 50 cents and 20 cents coins in a bag is in the ratio of 4 : 7 : 3. If there are 420 coins in the bag, find the total value of the coins. 3. The ratio of the interior angles of a triangle is 2 : 3 : 7. Find the size of largest angle. 4. There are 128 marbles in a bag, including red, green and yellow marbles. The number of red marbles, green marble and yellow marbles are in the ratio 1: 3 : 4. Find the number of green marbles. Example 3.6 Solution A sum of money is divided among Andy, The total amount of money Billy and Candy in the ratio of 1: 2: 3. If  1+ 2 + 3  = 120   Candy gets $120, find the total amount of  3  money. = $240 Practice 3.6 1. Amy, Betty and Chris share a bag of marbles in the ratio of 4 : 7 : 5. Chris gets 15 marbles. Find the total number of marbles. 2. A piece of cloth is cut into 3 portions in the ratio of 2 : 3 : 2. If the length of longest portion is 9 m , find the original length of the cloth. 3. There are three kinds of tickets for express rail: first class, business class and economy class. The number of seats for first class, business class and economy class are in the ratio 5 :12 :18. If there are 270 economy class seats, find the total number of seats in the express rail. 4. A rope is cut into 3 piece and the ratio of their lengths is 2 : 3 : 7. If the length of the longest piece is 84 cm , find the original length of the rope. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 19 S2 Mathematics Rate, Ratio and Proportion Example 3.7 Solution An alloy made of gold, silver and copper G : S : C weights 210 g. The ratio of the weight of gold 1 : 4 to the weight of silver is 1: 4 and the ratio of 3 : 5 the weight of silver to the weight of copper is 3 : 12 : 20 3:5. Find the weight of gold in the alloy. The weight of gold  3  = 210    3 + 12 + 20  = 18 g Practice 3.7 1. The total weight of the Andy, Billy and Chris is 140 kg. The ratio of the weights of Andy to Billy is 5 : 4 and that of Billy to Chris is 3 : 2. Find the weight of Chris. 2. The total weight of three boxes of goods A, B and C is 115 kg. The ratio of the weight of A to that of B is 3 : 1 and the ratio of the weight of A to that of C is 2 : 5. (a) Find the ratio of the weight of good A to that of B to that of C. (b) Find the weight of the box of good A. Example 3.8 Solution A sum of amount of money is to be shared by (a) A : B : C Amy, Billy and Candy. The ratio of the amount 2 : 3 that Amy has to that of Billy is 2 : 3. The ratio 5 : 1 of the amount that Billy has to that of Candy is 10 : 15 : 3 5 : 1. Billy gets $90. The required ratio is 10 :15 : 3. (a) Find the ratio of the Amy’s share to Billy’s share to Candy’s share. (b) The sum of the amount (b) Find the sum of the amount of money.  10 + 15 + 3  = 90    15  = $168 Practice 3.8 1. The ratio of the number of white balls to that of blue balls is 5 : 6. The ratio of the number of blue balls to that of red balls is 4 :1. (a) Find the ratio of the number of white balls to that of blue ball to that of red ball. (b) If there are 60 blue balls, find the total number of balls. 2. In a music competition, the numbers of awards obtained by school X and school Y are in the ratio of 7 : 4. The numbers of awards obtained by school Y and school Z are in the ratio 6 : 5. (a) Find the ratio of awards obtained by school X to that of school Y to that of school Z. (b) It is known that school Z obtained 230 awards. Find the number of awards obtained by school X. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 20 S2 Mathematics Rate, Ratio and Proportion Example 3.9 Solution A piece of alloy is made up of gold, silver and (a) G : S : C copper. The weight of gold and the weight of 1 : 4 silver are in the ratio of 1: 4 , while the weight 3 : 5 of silver and the weight of copper are in the ratio 3 : 12 : 20 of 3 : 5. The required ratio is 3 :12 : 20. (a) Find the ratio of gold to silver to copper in the piece of alloy in weight. (b) The weight of the piece of alloy (b) If the weight of gold is 54 g less than that of  3 + 12 + 20  = 54   silver in the alloy, find the weight of the  12 − 3  piece of alloy. = 210 g Practice 3.9 1. Andy, Betty and Chris have some game cards. The number of Andy’s game cards and the number of Betty’s game cards are in the ratio of 5 : 2 while the number of Betty’s game cards and the number of Chris’s game cards are in the ratio of 3 : 1. (a) Find the ratio of the number of Andy’s game cards to that of Betty’s game cards to that of Chris’s game cards. (b) If Andy has 78 more game cards than Chris, find the total number of game cards that Andy, Betty and Chris has together. 2. Andy, Betty and Candy share some money. Andy’s share and Betty’s share were in the ratio of 3 : 5 while Andy’s share and Candy’s share were in the ratio of 2 : 3. (a) Find the ratio of Andy’s share to Betty’s share to Candy’s share. (b) Betty has just given $25 to Andy. Now Betty and Candy have the same amount of money. Find the total amount of money that Andy, Betty and Candy have. 5 3. The volume of the lemon juice is twice that of apple juice and the volume of a mango juice is 4 that of the lemon juice. (a) Find the ratio of the volume of the lemon juice to that of apple juice to that of mango juice. (b) If the total volume of the apple and lemon juice is 1500 mL, find the volume of the mango juice. 4. The weight of a girl is two−thirds that of a man and the weight of a boy is four−fifth that of the man. (a) Find the ratio of the girl to that of the boy to that of the man. (b) If the total weight of the girl and the boy is 88 kg, find the total weight of the girl, the boy and the man. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 21 S2 Mathematics Rate, Ratio and Proportion Example 3.10 Amy, Billy and Chris share some sweets in the ratio 6 : 8 : 9. Amy obtained 3000 sweets. (a) Find the total number of sweets and the sweets obtained by Chris. (b) If Chris give 500 sweets to Amy, find the ratio of Amy’s sweets to Billy sweets to Chris’s sweet. Solution (a) The total number of sweets (b) The number of sweet shared by Billy 6+8+9 = 11500 − 4500 − 3000 = 3000    6  = 4000 = 11500 The number of sweets shared by Chris The required ratio 9 = (3000 + 500) : 4000 : (4500 − 500) = 3000   6 = 3500 : 4000 : 4000 = 4500 = 7:8:8 Practice 3.10 1. $400 is shared among Andy, Billy and Cathy. The ratio of the shares of Andy to Billy is 3 : 5 and Andy to Cathy is 2 : 3. (a) Find the ratio of the Andy’s share to Billy’s share to Cathy’s share. (b) Find the amount that Andy, Billy and Cathy shared respectively. (c) If Cathy gives $4 to Andy, find the ratio of the amount of money that Andy has to that Billy has to that of Cathy has. 2. In a store, Andy, Billy and Candy shared the profits. The ratio of Andy’s share to Billy’s share is 2 :1. The ratio of Andy’s share to Candy’s share is 3 : 1. The amount of profit shared by Andy is $2400 (a) Find the total amount of profits and the profit shared by Billy. (b) If Andy give $800 to Billy, find the ratio of Andy’s share to Billy share to Candy’s share. 3. Andy, Betty and Candy share some marbles. Andy’s share and Betty’s share are in the ratio of 6 : 5 while Betty’s share and Candy’s share are in the ratio of 10 : 9. (a) Find the ratio of Andy’s share to Betty’s share to Candy’s share. (b) Betty has 30 marbles. Andy has just given 3 marbles to Candy. Find the ratio of Andy’s share to Billy’s share to Candy’s share. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 22 S2 Mathematics Rate, Ratio and Proportion Section 4: Proportions 4.1 Concept of Proportion A proportion is a statement that two ratios are equal.  a c If a : b and c : d are equal, then a : b = c : d  or =  is a proportion.  b d a 5 For example, 2 : 3 = 6 : 9 and = are proportions. b 7 A proportion can also be used to state the equality of two ratios of three or more quantities. For example, x : y : z = 1: 3 : 4 is a proportion. From this proportion, we know that x : y = 1: 3 , y : z = 3 : 4 and x : z = 1: 4. Example 4.1.1 (a) If x :18 = 5 : 3 , find the value x. (b) If x : (x + 3) = 5 : 6 , find the value of x. Solution x 5 x 5 (a) = (b) = 18 3 x+3 6 5 6x = 5x + 15 x = (18) 3 x = 15 = 30 Practice 4.1.1 1. If 5 : x = 15 : 6 , find the value of x. 2. If (x + 12) : 8 = 9 : 4 , find the values of x. 3. If 4 : 3 = 20 : x , find the value of x 4. If x : 8 = 7 : 4 , find the value of x. 1 5. If 5 : x = :10 , find the value of x. 2 6. If x : (x + 12) = 2 : 3 , find the value of x. 7. If (24 − b) : 3 = b : 5 , find the value of b. 8. If (b − 4) : (b + 4) = 1: 2 , find the value of b. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 23 S2 Mathematics Rate, Ratio and Proportion Example 4.1.2 (a) If x and y are non−zero numbers such that 5x : (x + y) = 3 : 2 , find x : y. (b) If x and y are non−zero numbers such that (3x + 5y) : (4x − y) = 2 :1, find x : y. Solution 5x 3 3x + 5y (a) = (b) = 2 x+y 2 4x − y 10x = 3x + 3y 3x + 5y = 8x − 2y 7x = 3y 5x = 7y x 3 x 7 = = y 7 y 5  x: y = 3:7  x:y = 7:5 Practice 4.1.2 1. If x and y are non−zero numbers such that 4x : (x + y) = 8 : 3 , find x : y. 2. If x and y are non−zero numbers such that (5x − 2y) : (x + y) = 3 :1, find x : y. 3. If x and y are non−zero numbers such that (x + y) : (x − y) = 6 :1, find x : y. 4. If x and y are non−zero numbers such that (2x + y) : (x − y) = 7 : 2 , find x : y. 5. If x and y are non−zero numbers such that (x + y) : (y − x) = 4 : 3 , find x : y. 6. If x and y are non−zero numbers such that (x + 5y) : (3y − x) = 5 : 2 , find x : y. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 24 S2 Mathematics Rate, Ratio and Proportion Example 4.1.3 If x and y are non−zero numbers such that (2x − y) : (x + y) = 4 : 3 , find (a) x : y , (b) (x + 2y) : (x − y). Solution 2x − y 4 (b) Let x = 7k , then y = 2k. (a) = x+y 3 (x + 2y) : (x − y) 6x − 3y = 4x + 4y = (7k + (2)(2k)) : (7k − 2k) 2x = 7y = 11k : 5k x 7 = 11: 5 = Common Mistake y 2 Let. x = 7k.and y = 2k  x:y = 7: 2 Practice 4.1.3 1. If a and b are non−zero numbers such that (2a + 3b) : (a − 2b) = 11: 2 , find (a) a : b , (b) (a + 2b) : (3a − b). 2. If a and b are non−zero numbers such that (a + 4b) : 3 = (3a + b) : 2 , find (a) a : b , (b) (2a − b) : (a + b). 3. If x and y are non−zero numbers such that (2x + y) : 3y = 3 : 5 , find (a) x : y , (b) (2y − x) : (x + 4y). 4. If a and b are non−zero numbers such that (a + 2b) : 3 = (a − b) : 2 , find (a) a : b , (b) (a − 3b) : (a + b). 5. If x and y are non−zero numbers such that (3x − 2y) : (2x + y) = 4 : 5 , find (a) x : y , (b) (x + 2y) : (4x − y). 6. If x and y are non−zero numbers such that (x + y) : ( −x + y) = 4 : 3 , find (a) x : y , (b) 3x : 2y , (c) (2x + y) : (x + 2y) , (d) (y − 2x) : (3x + y). Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 25 S2 Mathematics Rate, Ratio and Proportion Example 4.1.4 Solution Let a, b and c be non−zero numbers such that b−a 2 = (b − a) : (b + 2a) = 2 : 5 and a : c = 2 : 5. Find b + 2a 5 (2a + c) : (a + b). 5b − 5a = 2b + 4a 3b = 9a a:b = 1: 3  a : b : c 1 : 3 2 : 5 2 : 6 : 5 Let a = 2k , then b = 6k and c = 5k Common Mistake (2a + c) : (a + b) Let. a = 2k., b = 6k = (4k + 5k) : (2k + 6k) and c = 5k. = 9:8 Practice 4.1.4 1. Let a, b and c be non−zero numbers such that a : c = 2 : 3 and b : c = 2 :1. Find (a + 2b) : (3a − c). 2. Let x, y and z be non−zero numbers such that x : y = 5 : 3 and x : z = 2 : 3. Find (3x − z) : (2y + z). 3. Let x, y and z be non−zero numbers such that x : y = 3 : 4 and x : z = 2 : 5. Find (2x + z) : (x + 3y). 4. Let x, y and z be non−zero numbers such that x : z = 3 : 4 and y : z = 2 : 3. Find (x + 2y) : (y + z). 5. Let a, b and c be non−zero numbers such that 5a = 6b and a:c = 3: 2. Find (3a + 2b) : (a + b − c). 6. Let a, b and c be non−zero numbers such that (3a + b) : (a + b) = 5 : 2 and a : c = 6 : 5. Find (b + 2c) : (2a + b). 7. Let x, y and z be non−zero numbers such that (2x − z) : (x + z) = 3 : 2 and y : z = 4 : 3. Find (x − 2z) : (y + z). 8. 1983 CE Math Paper 1 Q.4 Let a, b and c be non−zero numbers such that a : b = 3 : 4 and a : c = 2 : 5. Find (a) a : b : c , ac (b) the value of. a + b2 2 Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 26 S2 Mathematics Rate, Ratio and Proportion 9. 1991 CE Math Paper 1 Q.4 Let a, b and c be non−zero numbers such that 2a = 3b = 5c. (a) Find the ratio a : b : c. (b) If a − b + c = 55 , find c. 10. 2020 DSE Math Paper 1 Q.4 a 6 b + 2c Let a, b and c be non−zero numbers such that = and 3a = 4c. Find. b 7 a + 2b Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 27 S2 Mathematics Rate, Ratio and Proportion Example 4.1.5 Solution 1 1 1 1 (a) If : = 4 : 3 , find x : y. (a) : = 4:3 x y x y 1 1 1 4 3 (b) If : : = 2 : 3 : 4 , find a : b : c. = : a b c 12 12 1 1 = : 3 4 x:y = 3: 4 (b) L.C.M. of 2, 3 and 4 is 12 1 1 1 2 3 4 : : = : : a b c 12 12 12 1 1 1 = : : 6 4 3  a:b:c = 6: 4:3 Practice 4.1.5 1 1 1. If : = 3 : 2 , find x : y x y 1 1 1 2. If : : = 4 : 5 : 6 , find x : y : z. x y z 1 1 3. If : = 5 : 2 , find x : y. x y 1 1 2 1 4. If : = : , find x : y. x y 3 4 1 1 1 5. If : : = 5 : 2 : 9 , find x : y : z. x y z 1 1 1 6. If : : = 4 : 6 : 9 , find x : y : z. x y z 7. If p : 20 :10 = 12 : q : 8 , find the values of p and q. 8. If (a + 2) : (a + 5) : b = 3 : 4 : 7 , find the values of a and b. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 28 S2 Mathematics Rate, Ratio and Proportion Example 4.1.6 The present ages of John and Mary are in the ratio 1: 2. Six years ago, the ratio of the ages of John and Mary was 2 : 5. Find the present age of Mary. Solution Let the present age of John be k. Common Mistake Then the present age of Mary be 2k. Let the present age of John and k−6 2 Mary be k and 2k respectively. = 2k − 6 5 5k − 30 = 4k − 12 k = 18 The present age of Mary is 36. Practice 4.1.6 1. 2011 CE Math Paper 1 Q.6 In a summer camp, the ratio of the number of boys to the number of girls is 7 : 6. If 17 boys and 4 girls leave the summer camp, then the number of boys and the number of girls are the same. Find the original number of girls in the summer camp. (4 marks) 2. 2012 DSE Practice Paper 1 Q.5 The ratio of the capacity of a bottle to that of a cup is 4 : 3. The total capacity of 7 bottles and 9 cups is 11 litres. Find the capacity of a bottle. (4 marks) 3. 2019 DSE Math Paper 1 Q.7 In a playground, the ratio of the number of adults to the number of children is 13 : 6. If 9 adults and 24 children enter the playground, then the ratio of the numbers of adults to the number of children is 8 : 7. Find the original number of adults in the playground. (4 marks) 4. Six years ago, the ratio of the age of Daniel to that of Fred was 5 : 2. Two years later, the ratio of the age of Daniel to that of Fred will be 7 : 4. Find the present age of Daniel. 5. Sharon has a piece of wire. She bends the piece of wire to form a rectangle in which its length to its width is in the ratio 5 : 3. If the length the rectangle is decreased by 5 cm and its width is increased by 5 cm , the rectangle will become a square. Find the area of the original rectangle. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 29 S2 Mathematics Rate, Ratio and Proportion Exercise 4.1.7 Type 1 1. 1991 CE Math Paper 2 Q.10 5. 1987 CE Math Paper 2 Q.34 1 1 If a:b = 3: 2 , b:c = 4:3 , then If : = 2:3 and a : c = 4 :1 , then a b (a + b) : (b + c) = a:b:c = A. 7 :10. A. 12: 8 : 3. B. 5:7. B. 8:3:2. C. 1:1. C. 4 : 6 :1. D. 7:5. D. 2:3:8. E. 10 : 7. E. 2:3: 4. 6. 1993 CE Math Paper 2 Q.35 2. 1994 CE Math Paper 2 Q.42 If a:b = 2: 3 and b : c = 5 : 3 , then If a:b = 2: 3 , a:c = 3: 4 and a+b+c = a : d = 4 : 5 , then b : c : d = a−b+c A. 2:3: 4. A. −2. B. 3: 4:5. 5 B.. C. 3 : 6 :10. 2 D. 18 :16 :15. C. 4. E. 40 : 45 : 48. 17 D.. 2 3. 1992 CE Math Paper 2 Q.10 E. 31. If a:b = 2: 3 , a:c = 3: 4 and b : d = 5 : 2 , find c : d. 7. 2018 DSE Math Paper 2 Q.10 A. 1: 5 Let a, b and c be non−zero numbers. If B. 16 : 45 a + 3b 3a = 4b and a : c = 2 : 5 , then = C. 10 : 3 b + 3c D. 20 : 9 5 A.. E. 5 :1 3 13 B.. 4. 1985 CE Math Paper 2 Q.11 33 If a : b = 1: 2 and b : c = 1: 3 , then 30 C.. a +b:b +c = 53 A. 1: 5. 75 D.. B. 2: 3. 38 C. 3: 4. D. 3:5. E. 3:8. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 30 S2 Mathematics Rate, Ratio and Proportion 8. 2006 CE Math Paper 2 Q.13 Type 2 Let x, y and z be non−zero numbers. If 1. 1983 CE Math Paper 2 Q.10 x : y = 1: 2 and y : z = 3 :1 , then If 2x = 3y = 5z , then x : y : z = (x + y) : (y + z) = A. 2:3:5. A. 3: 4. B. 5:3:2. B. 4:3. C. 6 :10 :15. C. 8:9. D. 15 :10 : 6. D. 9:8. E. 25 : 9 : 4. 9. 2009 DSE Sample Paper 2 Q.13 2. 1987 CE Math Paper 2 Q.42 If x, y and z are non−zero numbers such 1 1 1 If 3a = 2b = 5c , then : : = that 2x = 3y and x = 2z , then a b c (x + z) : (x + y) = A. 3: 2:5. A. 3:5. B. 5: 2:3. B. 6:7. 1 1 1 C. : :. C. 9:7. 3 2 5 D. 9 :10. 1 1 1 D. : :. 5 2 3 10. 2015 DSE Math Paper 2 Q.11 1 1 1 E. : :. Let a, b and c be non−zero numbers. If 2 3 5 a:c = 5:3 and b:c = 3: 2 , then (a + c) : (b + c) = 3. 2002 CE Math Paper 2 Q.13 A. 7:5. x+y−z If 2x = 3y = 4z , then = B. 8:5. x−y+z C. 16 :15. 1 A.. D. 19 :15. 5 1 B.. 3 5 C.. 3 7 D.. 5 4. 2014 DSE Math Paper 2 Q.12 4 5 7 It is given that = = , where a, b 5a 7b 9c and c are positive numbers. Which of the following is true? A. abc B. ac b C. bac D. bc a Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 31 S2 Mathematics Rate, Ratio and Proportion Type 3 4. 2011 CE Math Paper 2 Q.12 1. 1998 CE Math Paper 2 Q.15 Let m and n be non−zero numbers. If x + 2y 2m − n If = 5 , then x : y = = 3 , then m : n = 3x − 4y m − 2n A. 3:7. A. 1: 5. B. 7:3. B. 5 : 1. C. 7 :11. C. 5:7. D. 9:7. D. 7:5. E. 11: 7. 5. 2012 DSE Math Paper 2 Q.9 2. 2001 CE Math Paper 2 Q.28 If x and y are non−zero numbers such that x + 3y 2 x−y 6x + 5y If = , then = = 7 , then x : y = 2x − y 3 x+y 3y − 2x −5 A. 4:5. A.. 6 B. 4 :13. −3 C. 5: 4. B.. 5 D. 13 : 4. 3 C.. 5 6. 1984 CE Math Paper 2 Q.11 3 3x + 2y D.. If = 1 , then (x + y) : (x − y) = 4 x + 5y 5 A. 1: 5. E.. 6 B. 3:2. C. 5:6. 3. 2012 DSE Practice Paper 2 Q.12 D. 5 : 1. Let  and  be non−zero constants. If E. 7: 4. ( + ) : (3 − ) = 7 : 3 , then  :  = A. 5:9. 7. 2016 DSE Math Paper 2 Q.11 B. 9:5. If x and y are non−zero numbers such that C. 19 : 29. (3y − 4x) : (2x + y) = 5 : 6 , then x : y = D. 29 :19. A. 7:8. B. 8 : 29. C. 9 : 32. D. 13 : 34. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 32 S2 Mathematics Rate, Ratio and Proportion 8. 2021 DSE Math Paper 2 Q.11 Let  and  are non−zero numbers such 2 + 3 7 2 +  that = , then = 3 + 2 10  + 2 A. 1. 3 B.. 2 11 C.. 6 13 D.. 8 9. 2022 DSE Math Paper 2 Q.12 Let x, y and z be non−zero numbers. If x : y = 8 : 5 and 2x = 4z − 3y , then y : z = A. 16 :17. B. 17 :16. C. 20 : 31. D. 31: 20. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 33 S2 Mathematics Rate, Ratio and Proportion 4.2 Scale Drawings Scale drawing is a two−dimensional drawing of a place or a real object in a specified ratio relative to the actual size of the place or the object. The specified ratio in the scale drawing is called the scale which is defined as length on the scale drawing Scale = actual length A scale is expressed in the form 1: n or n :1, where n is a number greater than 1. The following are examples of scale drawing. Reduced Drawing Enlarged Drawing Scale: 2 :1 Figure B Scale: 1:120000 Figure A 1. Figure A is a reduced drawing of Hong Kong. The scale 1:120000 means that the length of 1 cm on the drawing represents an actual length of 120 000 cm (i.e. 1.2 km) 2. Figure B is an enlarged drawing of a bee. The scale 2 :1 means that the length of 2 cm on the drawing represents an actual length of 1 cm. Note: (a) When the scale is expressed in the form 1: n , where n  1, the object on the drawing is reduced. (b) When the scale is expressed in the form n :1, where n  1, the object on the drawing is enlarged. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 34 S2 Mathematics Rate, Ratio and Proportion Example 4.2.1 Solution If a distance of 2 cm on a map represents an The scale actual distance of 10 km, express the scale of the = 2 cm : 10 km map in the form 1: n. = 2 cm : 10(1000)(100) cm = 1 : 500000 Practice 4.2.1 1. If the length of a bus on a scale drawing is 15 cm and its actual length is 7.5 m, express the scale of the drawing in the form of 1: n. 2. The actual length of a butterfly is 9.6 cm. If the length of the butterfly is 3.2 cm on a drawing, express the scale of drawing in the form of 1: n. 3. The actual length of a real caterpillar is 40 mm. If the length of the caterpillar is 1 cm on a drawing, express the scale of the drawing in the form of 1: n. 4. The length of a dog is 5 cm in a photo. If the actual length of the dog is 75 cm, express the scale of the photo in the form 1: n. Example 4.2.2 Solution The length of an ant in the scale drawing is 4 cm. The scale The actual length of the ant is 8 mm. Express = 4 cm : 8 mm the scale of the drawing in the form n :1. = 40 mm : 8 mm = 5 :1 Practice 4.2.2 1. The length of a fly in the scale drawing is 8 cm. The actual length of the fly is 4 mm. Express the scale of the drawing in the form n :1. 2. The diameter of the red blood cell in a scale drawing is 1.6 cm. The actual diameter of the red blood cell is 0.008 mm. Express the scale of the drawing in the form n :1. 3. The length of an ant in a scale drawing is 3 cm. The actual length of the ant is 6 mm. Express the scale of the drawing in the form n :1. 4. The diameter of the white blood cell in a scale drawing is 1.2 cm. The actual diameter of the white blood cell is 0.012 mm. Express the scale of the drawing in the form n :1. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 35 S2 Mathematics Rate, Ratio and Proportion Example 4.2.3 Solution The length of a road on a map is 6 cm. If the The scale actual length of the road is 15 km, find the scale = 6 cm :15 km of the map in the form 1: n. = 6 cm :1500000 cm = 1: 250000 Practice 4.2.3 1. The length of a river on a map is 8 cm. If the actual length of the river is 24 km, find the scale of the map in the form 1: n. 2. The actual length of the river is 500 m. If the length of the river on a map is 2 cm, find the scale of the map in the form 1: n. 3. The actual length of the road is 1 km. If the length of the road on a map is 8 cm, find the scale of the map in the form 1: n. 4. The actual length of the path is 1 km. If the length of the road on a map is 5 mm, find the scale of the map in the form 1: n. 5. The length of a road is 12 mm on a map. The actual distance of a road is 24 km. Find the scale of the map in the form 1: n. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 36 S2 Mathematics Rate, Ratio and Proportion Example 4.2.4 The scale of a map is 1: 40000. (a) If the distance of a path on the map is 6 cm, find the actual distance of the path in m. (b) If the actual distance of a path is 4.8 km, find the distance of the path on the map. Solution 6 cm 1 dis tance on the map 1 (a) = (b) = Actual distance 40000 4.8km 40000 Actual distance = (40000)(6 cm) 4.8km distance on the map = = 240 000 cm 40000 = 2400 m 480000cm = 40000 = 12 cm Practice 4.2.4 1. The scale of a map is 1: 50000. (a) If the distance of a path on the map is 5 cm, find the actual distance of the path in km. (b) If the actual distance of a path is 12 km, find the distance of the path on the map. 2. The scale of a map is 1:1000000. (a) If two cities are 50 km apart, find the distance in cm between the cities on the map. (b) If the distance between two cities on the map is 7 cm, find the actual distance between the two cities in km. 3. The scale of a map is 1:1250000. (a) The actual distance between towns A and B is 50 km. Find the distance between towns A and B on the map. (b) If the length of a railway on the map is 18 cm, find the actual length of the railway. 4. The scale of a map is 1: 500000. (a) If the distance between towns X and Y is 3.2 cm on the map, find the actual distance between these two towns in km. (b) If the actual distance between towns A and B is 3.5 km, find the distance between these two towns on the map. 5. The scale of a map is 1: 800000. (a) If the distance between towns P and Q is 5.5 cm on the map, find the actual distance between these two towns in km. (b) If the actual distance between towns A and B is 32 km, find the distance between these two towns on the map. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 37 S2 Mathematics Rate, Ratio and Proportion Example 4.2.5 The scale of a floor plan of a flat is 1: 250. The length and the width of the flat on the floor plan are 5 cm and 4 cm respectively. (a) Find the actual length and the width of the flat in m. (b) Find the actual area of the flat in m2. Solution (a) The actual length (b) The actual area = 5(250) cm = (12.5)(10) = 1250 cm = 125 m2 = 12.5 m The actual width = 4 (250) cm = 1000 cm = 10 m From this example, we can observe that The actual area = (5)(250)  (4)(250) = (5  4)(250)2 = Area on the floor plan (250)2 Therefore, we can conclude that The actual area = Area on the drawing  (scale)2 Practice 4.2.5 1. The scale of a floor plan of a bedroom is 1:120. The length and the width of the bedroom on the floor plan are 2.5 cm and 3 cm respectively. (a) Find the actual length and the width of the bedroom in m. (b) Find the actual area of the bedroom in m2. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 38 S2 Mathematics Rate, Ratio and Proportion Example 4.2.6 3. 2006 CE Math Paper 2 Q.15 2000 CE Math Paper 1 Q.8 The scale of a map is 1: 8000. If the area The scale of a map is 1: 5000. If the area of of a park on the map is 2 cm2 , then the the passenger terminal of the Hong Kong actual area of the park is International Airport on a map is 220 cm. 2 A. 4 000 m2. find the actual area of the terminal on the B. 6 400 m2. ground. C. 12 800 m2 Solution D. 16 000 m2. The actual area  5000  2 4. 2009 CE Math Paper 2 Q.13 = 220    100  The scale of a map is 1: 5000. If the area = 550 000 m2 of a garden on the map is 4 cm2 , then the actual area of the garden is Practice 4.2.6 A. 100 m2. 1. 1997 CE Math Paper 2 Q.11 B. 200 m2. In a map of scale 1: 500 , the length and C. 10 000 m2. breadth of a rectangular field are 2 cm D. 20 000 m2. and 3 cm respectively. Find the actual area of this field. 5. 2017 DSE Math Paper 2 Q.11 A. 30 m 2 The scale of a map is 1: 20000. If the B. 150 m 2 area of a zoo on the map is 4 cm , then 2 C. 1500 m2 the actual area of the zoo is D. 3 000 m 2 A. 8  104 m2. E. 15 000 m2 B. 1.6  105 m2. C. 3.2  105 m2. 2. 1983 CE Math Paper 2 Q.40 D. 1 106 m2. The scale of a map is 1: 20000. On the map, the area of a farm is 2 cm2. The actual area of the farm is A. 400 m2. B. 800 m2. C. 40 000 m2. D. 80 000 m2. E. 8 000 000 m2. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 39 S2 Mathematics Rate, Ratio and Proportion Example 4.2.7 3. The scale of a map is 1: 2000. If the The scale of a map is 1: 300. If the actual actual area of a university is 1200 m2 , find area of a garden is 54 m2, then the area of the its area on the map. garden on the map is A. 2.4 cm2 A. 6 cm2 B. 3 cm2 B. 9 cm2 C. 4.8 cm2 C. 18 cm2 D. 6 cm2 D. 24 cm2 Solution 4. The scale of a map is 1: 50000. If the The area on the map actual area of a farm is 6  105 m2 , find its  100  2 area on the map. = 54    300  A. 1.2 cm2 = 6 cm2 B. 1.5 cm2 C. 2.4 cm2 Practice 4.2.7 D. 3 cm2 1. 2003 CE Math Paper 2 Q.15 The scale of a map is 1: 4000. If the actual area of a sports field is 8000 m2 , find its area on the map. A. 0.02 cm2 B. 0.05 cm2 C. 2 cm2 D. 5 cm2 2. The scale of a map is 1: 500. If the actual area of a park is 200 m , find its 2 area on the map. A. 4 cm2 B. 5 cm2 C. 8 cm2 D. 10 cm2 Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 40 S2 Mathematics Rate, Ratio and Proportion Example 4.2.8 3. The actual area of a country park is The actual area of a park is 24 m2. If the area 2.7  105 m2. If the area of the country of the park on the map is 6 cm , then the scale 2 park on the map is 3 cm2 , then the scale of of the map is the map is A. 1: 200 A. 1: 30000. B. 1: 400 B. 1: 90000. C. 1: 500 C. 1: 300000. D. 1:1000 D. 1: 900000. Solution 6 cm2 : 24 m2 = 1cm2 : 4 m2 Scale = 1cm : 2 m = 1cm : 200 cm = 1: 200 Practice 4.2.8 1. 2013 DSE Math Paper 2 Q.12 The actual area of a playground is 900 m2. If the area of the playground on a map is 36 cm2 , then the scale of the map is A. 1: 25. B. 1: 50. C. 1: 500. D. 1: 250000. 2. 2020 DSE Math Paper 2 Q.10 The actual area of a golf course is 0.75 km2. If the area of the course on a map is 300 cm2 , then the scale of the map is A. 1: 250. B. 1: 5000. C. 1: 62500. D. 1: 25000000. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 41 S2 Mathematics Rate, Ratio and Proportion Section 5: Direct and Inverse Proportions 5.1 Direct Proportions The table below shows some corresponding values of the number of oranges sold (x) and the total income ($y) of a supermarket. x 1 5 10 20 30 y 6 30 60 120 180 The corresponding pairs of x and y are always in the same ratio, i.e. 1: 6 = 5 : 30 = 10 : 60 = 20 :120 = 30 :180. We note that when x increases by a certain factor, y also increases by the same factor. In addition, the ratio of the values of x is equal to the ratio of corresponding values of y, i.e. 1: 5 = 6 : 30 1:10 = 6 : 60 20 : 30 = 120 :180 We say that x and y are in direct proportion, or x is directly proportional to y. If x and y are in direct proportion, then x1 x 2 = ( or x1 : y1 = x 2 : y 2 ) y1 y 2 where x 1 and x 2 are any two values of x; y 1 and y 2 are the corresponding values of y. x Note: In fact, when x and y are in direct proportion, is a non-zero constant. y Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 42 S2 Mathematics Rate, Ratio and Proportion Example 5.1.1 Solution x and y are in direct proportion. When x = 8 , 8 12 = y = 12. When x = 10 , find the value of y. 10 y y = 15 Practice 5.1.1 1. x and y are in direct proportion. When x = 15 , y = 9. When y = 6 , find the value of x. 2. x and y are in direct proportion. When x = 12 , y = 8. When x = 30 , find the value of y. 3. x and y are in direct proportion. When x = 5 , y = 7. When x = 35 , find the value of y. 4. x and y are in direct proportion. When x = 20 , y = 8. When y = 12 , find the value of x. 5. x and y are in direct proportion. When x = 4 , y = 6. When y = 15 , find the value of x. Example 5.1.2 Solution If 4 kg of peanuts cost $140, find the cost of Let $x be the cost of 10 kg of peanuts. 10 kg of peanuts. 10 x = 4 140 x = 350 The cost of 10 kg of peanuts is $350. Practice 5.1.2 1. If 6 L of paint is needed to paint 100 m2 of wall, find the amount of paint needed to paint 85 m2 of wall. 2. If a factory produces 750 headphones in 21 days, find the number of headphones that can be produced in 56 days. 3. If 2000 m2 of farmland needs 156 kg of fertilizer, find the amount of fertilizer needed for a farmland of 850 m2. 4. If a machine can fill 124 bottles in 64 minutes, find the number of bottles that can be filled in 80 minutes. 5. If 2.5 m of ribbon is sold at $16, find the length of ribbon that $240 can buy. 6. If a car travels a distant of 405 km on 27 L of petrol, find the amount of petrol needed to travel 435 km. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 43 S2 Mathematics Rate, Ratio and Proportion 5.2 Inverse Proportions There are $72 to buy some oranges from a market. The table below shows some corresponding values of the unit price of oranges ($x) and the number of oranges bought (y). x 2 3 4 6 8 y 36 24 18 12 9 1 We note that when x increases by a factor of k, y decreases by a factor of. In addition, the k ratio of the values of x is equal to the reciprocal of the ratio of corresponding values of y, i.e. 2: 3 = 2: 3 but 36 : 24 = 3 : 2 2: 4 = 1: 2 but 36 :18 = 2 :1 6:8 = 3: 4 but 12 : 9 = 4 : 3 We say that x and y are in inverse proportion, or x is inversely proportional to y. If x and y are in inverse proportion, then x1y1 = x 2 y 2 ( or x1 : x 2 = y 2 : y1 ) where x 1 and x 2 are any two values of x; y 1 and y 2 are the corresponding values of y. Note: In fact, when x and y are in inverse proportion, xy is a non-zero constant. Example 5.2.1 Solution x and y are in inverse proportion. When x = 7 , 7(12) = 3y y = 12. When x = 3 , find the value of y. y = 28 Practice 5.2.1 1. x and y are in inverse proportion. When x = 9 , y = 20. When y = 15 , find the value of x. 2. x and y are in inverse proportion. When x = 20 , y = 10. When x = 16 , find the value of y. 3. x and y are in inverse proportion. When x = 5 , y = 32. When x = 16 , find the value of y. 4. x and y are in inverse proportion. When x = 10 , y = 15. When y = 3 , find the value of x. 5. x and y are in inverse proportion. When x = 2 , y = 24. When x = 6 , find the value of y. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 44 S2 Mathematics Rate, Ratio and Proportion Example 5.2.2 Solution A car travels at a constant speed of 42 km/h. If Let t hours be the time taken. the car takes 6 hours to complete a journey, find 6(42) = 72t the time taken for the car to complete the same t = 3.5 journey if it travels at a constant speed of The time taken for the car to complete the same 72 km/h. journey is 3.5 hours. Practice 5.2.2 1. An empty tank can be filled up by 5 pipes in 12.6 hours. If the rate of water flowing out of each pipe is the same, find the time taken for the tank to be filled up by 9 pipes. 2. 15 boy scouts can clean up a beach in 6 hours. If all boy scouts work at the same rate, find the time taken for 9 boy scouts to clean up the beach. 3. There is enough food for 5 men to survive 12 days. If all men consume food at the same rate, find the number of days that 3 men can survive with the food. 4. An empty tank can be filled up by 3 pipes in 40 minutes. If the rate of water flowing out of each pipe is the same, find the number of pipes needed to fill up the tank in 12 minutes. 5. 50 workers take 42 days to build a house. If all workers work at the same rate, find the number of days needed for 60 workers to build the same house. 6. 15 workers take 20 days to complete a job. If all workers work at the same rate, find the number of extra days needed for 12 workers to complete the same job. Example 5.2.3 Solution A job can be finished by 15 workers in a certain Let n be the original number of days needed. number of days. If there are 3 more workers, 15n = (15 + 3)(n − 4) the job could be finished 4 days earlier. Find the 15n = 18n − 72 original number of days needed. n = 24 The original number of days needed is 24. Practice 5.2.3 1. A project can be done by some students in 24 days. If there are 2 more students, the project could be done 8 days earlier. Find the original number of students. 2. The money that Alice has is just enough to buy 20 oranges. If the price of an orange is increased by $1, Alice can buy 16 oranges with no money left. Find the original price of an orange. 3. It takes 5 workers 27 days to build a brick wall. If 4 more workers are employed, find the number of days needed to build the same wall. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 45 S2 Mathematics Rate, Ratio and Proportion Section 6: Answers Practice 1.1 Practice 1.3 1. (a) 1080 km/h 1. The claim is disagreed. (b) 300 m/s 2. The claim is agreed 2. (a) 36 km / h (b) 10 m / s Practice 1.4 3. (a) 45 km / h 1. (a) 160 pairs/day (b) 12.5 m / s (b) 4160 4. (a) 81km / h (c) 20 (b) 22.5 m / s 2. $110 5. (a) 24 km/h 3. (a) 52 words / min (b) 400 m/min (b) 30 min 6. (a) 1.8 km/h 4. (a) $30 / m (b) 30 m/min (b) $960 7. (a) 180 km/h (c) 72 m (b) 50 m/s 5. (a) 288 8. (a) 120 km/h (b) 6 minutes (b) 160 km (c) 2.5 h Practice 1.5 1. (a) 1800 sets/day Practice 1.2 (b) The claim is disagreed. 1. 104 set/h 2. (a) 45 copies/min 2. $23/kg (b) The claim is disagreed. 3. 17.5 dollars/ball 3. (a) 24 cakes/h 4. 1 800 dollars / month (b) The claim is agreed. 5. 0.6 kg/bottle 4. (a) $9 (b) The claim is agreed. 6. 15 km/ 5. (a) 320 sets/day 7. 48 km/h (b) The claim is disagreed. 8. 300 toys/day 6. (a) 1.25 cups/min 9. 0.6 dollars/egg (b) The claim is agreed. 10. (a) 7.2 dollars/liter 7. (a) 0.125 km/min. (b) 4.8 dollars/kg (b) The claim is disagreed. 11. (a) 60 chairs/h 8. (a) 2.5 T−shirts/h (b) 28 chairs/worker (b) The claim is agreed. 12. (a) 1.5 pages/min 9. (a) 9 pages/min (b) 90 pages/h (b) The claim is disagreed. Prepared by W. S. Yeung S2_07_Rate_Ratio_and_Proportion_N − 46 S2 Mathematics Rate, Ratio and Proportion Practice 1.6 Practice 2.2 1. (a) 10.5 HKD/GBP 1. 2: 5 (b) GBP 48 2. 2: 3 (c) HKD 1050 3. 3:2 2. (a) 7.8 HKD/USD 4. 11:13 (b) HKD 3120 (c) USD 60 Practice 2.3 3. (a) 4 THB/HKD 1. 45 m2. (b) THB 8000 2. 120