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Chapter 12 Ratio and proportion Contents: A B C D E cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\233IB_MYP2_12.CDR Tuesday, 12 August 2008 3:20:50 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 F G black Ratio Writing ratios as fractions Equal ratios Proportions Using ratios to divide quantities Scale diagrams Gradient or slope IB MYP_2 234 RATIO AND PROPORTION (Chapter 12) A ratio is a comparison of quantities. ² a team’s win-loss ratio ² the teacher-student ratio in a school ² the need to mix ingredients in a certain ratio or proportion: We often hear statements about: These are all examples of ratios. Ratios are also commonly used to indicate the scales on maps and scale diagrams. OPENING PROBLEMS Problem 1: Two-stroke fuel for a motor-bike is made by mixing 1 part oil with 7 parts petrol. How much oil and petrol is needed to fill a 20 litre tank? Problem 2: A map has a 1¡:¡1¡000¡000 scale. Jen measures the distance between two towns on the map to be 9:8 cm. How far are the towns actually apart? A RATIO A ratio is an ordered comparison of quantities of the same kind. Carol bought some industrial strength disinfectant for use in her hospital ward. The bottle instructs her to ‘mix one part disinfectant to four parts water’. Disinfectant and water are both liquids, so this statement can be written as a ratio. GERM KILL We say the ratio of disinfectant to water is 1 : 4 or “1 is to 4”. Note that the ratio of disinfectant to water is 1 : 4, but the ratio of water to disinfectant is 4 : 1. This is why order is important. cyan magenta yellow y:\HAESE\IB_MYP2\IB_MYP2_12\234IB_MYP2_12.CDR Thursday, 14 August 2008 1:40:57 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 Also notice that the ratio is written without units such as mL or L. black IB MYP_2 RATIO AND PROPORTION (Chapter 12) It is important to mix the disinfectant and water in the correct ratio or proportion so that the disinfectant will kill germs but without wasting chemicals unnecessarily. Carol may make a jug or a bucket of disinfectant. As long as she mixes it in the correct ratio, it will be effective. 235 In both cases there is 1 part disinfectant to 4 parts water! Although most ratios involve two quantities, they may involve more than two quantities. For example, To make a certain strength of concrete you need 1 bucket of cement, 2 buckets of sand, and 4 buckets of gravel. The ratio of cement to sand to gravel is 1 : 2 : 4, which we read as “1 is to 2 is to 4”. Example 1 Self Tutor Express as a ratio: 3 km is to 5 km “3 km is to 5 km” means 3 : 5 EXERCISE 12A 1 Express as a ratio: a $8 is to $5 b 7 mL is to 13 mL c 5 kg is to 2 kg d 2 tonne is to 7 tonne e E13:00 is to E1:00 f 8 mm is to 5 mm 2 Write a simple ratio for the following: a triangles to circles b cats to mice c teachers to students d trees to flowers Example 2 Self Tutor Express as a ratio: 7 minutes is to 2 hours We must express both quantities in the same units. cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\235IB_MYP2_12.CDR Tuesday, 12 August 2008 12:25:33 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 2 hours is the same as 120 minutes, so “7 minutes is to 2 hours” is really “7 minutes is to 120 minutes”, or 7 : 120. black IB MYP_2 236 RATIO AND PROPORTION (Chapter 12) 3 Express as a ratio: a 65 g is to 1 kg b 87 pence is to $1:00 c 5 months is to 2 years d 60 cents is to E2:40 e $2:00 is to 80 cents f 200 kg is to 1 tonne. Example 3 Self Tutor Write as a ratio: Kirsty spends two hours watching TV and three hours doing homework. We write TV : homework = 2 : 3. 4 Write as a ratio: a Peter has $11 and Jacki has $9. b In a theatre there are 3 girls for every boy. c The school spent E5 on volleyball equipment for every E1 on table tennis equipment. d There are 2 Japanese made cars for every 5 European made cars. e For every 15 km that you travel by car, I can travel 4 km by bicycle. f There are 2 blue-fin tuna for every 5 schnapper. g Mix 50 mL of liquid fertiliser with 2 litres of water. h Take 5 mg of medicine for every 10 kg of body weight. B WRITING RATIOS AS FRACTIONS When we compare the ratio of a part to its total, the ratio can be written as a fraction. For example, in the diagram alongside there are 3 red discs out of a total of 10 discs. The ratio of red discs to the total number of discs is 3 3 : 10, and the fraction of discs which are red is 10 . Example 4 Self Tutor a In the figures alongside, find: b i the ratio of the shaded area to the unshaded area ii the ratio of the shaded area to the total area iii the fraction of the total area which is shaded. cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\236IB_MYP2_12.CDR Friday, 22 August 2008 2:07:37 PM PETER 95 100 50 i shaded : unshaded = 4 : 1 ii shaded : total = 4 : 5 iii Fraction of total shaded is 45 . 75 25 0 b 5 95 100 50 75 25 0 5 95 100 50 75 25 0 i shaded : unshaded = 1 : 3 ii shaded : total = 1 : 4 iii Fraction of total shaded is 14 . 5 95 100 50 75 25 0 5 a black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 237 EXERCISE 12B 1 In each of the following diagrams: i Find the ratio of the shaded area to the unshaded area. ii Find the ratio of the shaded area to the total area. iii Find the fraction of the total area which is shaded. a b c d e f 2 The pie chart represents the sales of three different types of soap powders, A, B and C. a Write as a ratio: i the sales of A to the sales of B ii the sales of A to the total sales. C 20% 50% A 30% b What fraction of the total sales made is the sales of brand A? B 3 The column graph represents the results 20 of a survey to determine the method by 16 which students travel to school. 12 8 a Find the total number of students 4 surveyed. 0 Bus Train Car b Write as a ratio: i students arriving by car : students who walk ii students arriving by bus : total number of students surveyed. Walk Bicycle c What fraction of the students surveyed travel to school by bus? C EQUAL RATIOS Consider the diagrams: A B C cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\237IB_MYP2_12.CDR Friday, 22 August 2008 2:55:46 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 The ratio of shaded area : unshaded area in each case is A 1 : 2 B 2 : 4 C 4 : 8. black IB MYP_2 238 RATIO AND PROPORTION (Chapter 12) However, by looking carefully at the three diagrams we can see that the fraction of the total area which is shaded is the same in each case. The ratio of the shaded area to the unshaded area is also the same or equal in each case. We can write that 1 : 2 = 2 : 4 = 4 : 8: We can see whether ratios are equal in the same way we see if fractions are equal. 1 2 Just as 2 4 = = 48 , 1 : 2 = 2 : 4 = 4 : 8. If we multiply or divide both parts of a ratio by the same non-zero number, we obtain an equal ratio. A ratio is in simplest form when it is written in terms of whole numbers with no common factors. Example 5 Self Tutor Express the following ratios in simplest form: a 8 : 16 a b 35 : 20 8 : 16 = 8 ¥ 8 : 16 ¥ 8 =1:2 To convert a ratio to simplest form, divide by the highest common factor. 35 : 20 = 35 ¥ 5 : 20 ¥ 5 =7:4 b Two ratios are equal if they can be written in the same simplest form. EXERCISE 12C 1 Express the following ratios in simplest form: a 2:4 f 15 : 25 b 10 : 5 g 9 : 15 c 6 : 14 h 21 : 49 d 6 : 18 i 56 : 14 e 20 : 50 Example 6 Self Tutor a Express the following ratios in simplest form: a = 2 5 : 3 5 2 5 £5: 3 2 3 2 = £5 = =2:3 : 3 5 b 1 12 : 5 1 12 : 5 b 3 5 2 5 :5 fconvert to an improper fractiong £2 : 5£2 cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\238IB_MYP2_12.CDR Friday, 22 August 2008 2:11:25 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 = 3 : 10 black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 239 2 Express the following ratios in simplest form: a f 1 5 1 3 : 4 5 b :2 5 4 : g 3: 1 4 1 2 c 2 3 : 1 3 d 1: h 3 12 : 8 1 2 e 3: i 2 12 : 1 12 Example 7 j Self Tutor Express the following ratios in simplest form: a 0:3 : 1:7 a b 0:05 : 0:15 0:3 : 1:7 = 0:3 £ 10 : 1:7 £ 10 = 3 : 17 1 2 : 3 4 1 3 We multiply or divide both parts of a ratio by the same non-zero number to get an equal ratio. 0:05 : 0:15 = 0:05 £ 100 : 0:15 £ 100 = 5 : 15 = 5 ¥ 5 : 15 ¥ 5 =1:3 b 3 Express the following ratios in simplest form: a 0:4 : 0:5 d 0:2 : 0:6 g 0:03 : 0:15 b 1:3 : 1:8 e 0:4 : 0:2 h 0:02 : 0:12 c 0:1 : 0:9 f 1:4 : 0:7 i 0:18 : 0:06 4 Express as a ratio in simplest form: a the number of circles to squares b the height of the tall player to the height of the short player 105 cm 210 cm c the weight of the fish to the breaking strain of the fishing line d the capacity of the small cola to the capacity of the large cola 3 kg line Cool Cola 9 kg 2 Litre cyan magenta yellow y:\HAESE\IB_MYP2\IB_MYP2_12\239IB_MYP2_12.CDR Thursday, 14 August 2008 1:49:43 PM PETER 95 100 50 75 25 0 f the number of circles to the number of dots 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 e the number of squares to the number of stars Cool Cola 500 mL black IB MYP_2 240 RATIO AND PROPORTION (Chapter 12) Example 8 Self Tutor Express the ratio shaded area : unshaded area in simplest form. shaded area : unshaded area = 2 : 6 =2¥2 :6¥2 =1:3 fHCF = 2g 5 Express in simplest form the ratio of shaded area : unshaded area. a b c d e f Example 9 Self Tutor Express as a ratio in simplest form: a 2 hours to 4 minutes a b 45 cm to 3 m 2 hours to 4 minutes = 2 £ 60 min to 4 min = 120 min to 4 min = 120 : 4 = 120 ¥ 4 : 4 ¥ 4 fHCF = 4g = 30 : 1 Remember to convert to the same units! 45 cm to 3 m = 45 cm to 300 cm = 45 : 300 = 45 ¥ 15 : 300 ¥ 15 = 3 : 20 b 6 Express as a ratio in simplest form: a 20 cents to $1 b $3 to 60 pence c 15 kg to 30 kg d 13 cm to 26 cm g 800 g to 1 kg e 27 mm to 81 mm h 1 hour to 30 min f 9 mL to 63 mL i 40 mins to 1 12 hours k 2 m to 125 cm l 24 seconds to 1 12 min j 34 mm to 1 cm cyan magenta yellow y:\HAESE\IB_MYP2\IB_MYP2_12\240IB_MYP2_12.CDR Thursday, 14 August 2008 1:50:05 PM PETER 95 o 4 weeks to 1 year r 3 min to 45 seconds 100 50 75 25 0 5 95 100 50 75 25 0 5 95 n 1 week to 3 days q 1:5 kg to 125 g 100 50 75 25 0 5 95 100 50 75 25 0 5 m 2 L to 750 mL p 250 g to 1 kg black IB MYP_2 RATIO AND PROPORTION (Chapter 12) Example 10 241 Self Tutor Show that the ratio 4 : 6 is equal to 20 : 30. 4:6 and =4¥2:6¥2 =2:3 20 : 30 = 20 ¥ 10 : 30 ¥ 10 =2:3 ) 4 : 6 = 20 : 30 7 Which of the following pairs of ratios are equal? a 16 : 20, 4 : 5 d 5 : 6, 30 : 36 b 3 : 5, 21 : 35 e 12 : 16, 18 : 24 c 2 : 7, 8 : 21 f 15 : 35, 21 : 56 g 18 : 27, 6 : 4 h 2 : 2 12 , i 1 : 1 13 , INVESTIGATION 1 32 : 40 15 : 20 EQUAL RATIOS BY SPREADSHEET A painter needs to obtain a particular shade of green paint by mixing yellow paint with blue paint in the ratio 3 : 5. If he buys 45 L of yellow paint, how many litres of blue paint are required? SPREADSHEET What to do: 1 Open a new spreadsheet and enter the details shown alongside. 2 Highlight the formulae in row 3. Fill these down to row 5. Your spreadsheet should look like this: Notice that in each row, the ratio of yellow paint to blue paint is still 3 : 5. 3 Now fill down the formulae until your spreadsheet shows 45 L of yellow paint. How many litres of blue paint are required? 4 The painter also needs to obtain a particular shade of purple by mixing red paint with blue paint in the ratio 4 : 7. In A1 type ‘Red’. In A2 enter 4 and in B2 enter 7. In A3 enter =A2+4 and in B3 enter =B2+7. Fill the formulae down to find out how many litres of red paint are required to mix with 42 L of blue paint. cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\241IB_MYP2_12.CDR Tuesday, 12 August 2008 12:56:37 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 5 White paint and black paint are to be mixed in the ratio 2 : 9 to make a shade of grey paint. How much of each type of paint is required to make 55 L of grey paint? black IB MYP_2 242 RATIO AND PROPORTION (Chapter 12) D PROPORTIONS A proportion is a statement that two ratios are equal. For example, the statement that 4 : 6 = 20 : 30 is a proportion, as both ratios simplify to 2 : 3 . You should know the difference in meaning between ratio and proportion. Suppose we are given the proportion 2 : 5 = 6 : ¤: If we know that two ratios are equal and we know three of the numbers then we can always find the fourth number. Example 11 Self Tutor Find ¤ to complete the proportion: a 2:5=6:¤ b 16 : 20 = ¤ : 35 2:5=6:¤ a b We begin by reducing the LHS to simplest form. 16 : 20 = 16 ¥ 4 : 20 ¥ 4 =4:5 £3 2:5=6:¤ £7 £3 ) ¤ = 5 £ 3 = 15 ) 4 : 5 = ¤ : 35 £7 ) ¤ = 4 £ 7 = 28 Example 12 Self Tutor The student to leader ratio at a youth camp must be 9 : 2. How many leaders are required if there are 63 students enrolled? students : leaders = 9 : 2 ) 63 : ¤ = 9 : 2 or 9 parts is 63 ) ¥7 ) ) We can use a unitary method for ratios. 63 : ¤ = 9 : 2 ) ¥7 63 =7 9 2 parts is 7 £ 2 = 14 1 part is ¤¥7=2 ) ¤ = 14 cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\242IB_MYP2_12.CDR Tuesday, 12 August 2008 1:38:31 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 So, 14 leaders are needed. black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 243 EXERCISE 12D 1 Find the missing numbers in the following proportions: a d g j 3:4=6:¤ 5 : 8 = ¤ : 40 7 : 21 = ¤ : 33 5 : 100 = ¤ : 40 b e h k 3 : 6 = 12 : ¤ 1 : 3 = ¤ : 27 15 : 25 = 30 : ¤ 18 : 30 = 24 : ¤ c f i l 2 : 5 = ¤ : 15 4 : 1 = 24 : ¤ ¤ : 18 = 32 : 48 ¤ : 12 = 33 : 44 2 A disaster relief team consists of engineers and doctors in the ratio of 2 : 5. a If there are 18 engineers, find the number of doctors. b If there are 65 doctors, find the number of engineers. 3 The ratio of two angles in a triangle is 3 : 1. Find the: a larger angle if the smaller is 18o b smaller angle if the larger is 63o . 4 The ratio of teachers to students in a school is 1 : 15. If there are 675 students, how many teachers are there? 5 An MP3 player is bought for E240 and sold for E270. Find the ratio of the cost price to the selling price. 6 The maximum speeds of a boat and a car are in the ratio 2 : 7. If the maximum speed of the boat is 30 km per hour, find the maximum speed of the car. 7 A farmer has sheep and cattle in the ratio 8 : 3. a How many sheep has the farmer if he has 180 cattle? b Find the ratio of the number of sheep to the total number of animals. c Find the ratio of the total number of animals to the number of cattle. 8 The price of a table is reduced from $56 to $48. The set of chairs which go with the table was originally priced at $140. If the price of the chairs is reduced in the same ratio as that of the table, find the new price of the chairs. 9 Sue invested money in stocks, shares, and property in the ratio 6 : 4 : 5. If she invested E36 000 in property, how much did she invest in the other two areas? INVESTIGATION 2 THE GOLDEN RATIO AND THE HUMAN BODY Look at the four given rectangles and choose the one which you find most appealing. A D C cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\243IB_MYP2_12.CDR Tuesday, 12 August 2008 1:41:21 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 B black IB MYP_2 244 RATIO AND PROPORTION (Chapter 12) If you said Rectangle C you would agree with many artists and architects from the past. Rectangle C is called a Golden Rectangle and is said to be one of the most visually appealing geometric shapes. The ratio of length to width in a Golden Rectangle is the ratio 1:618 033 98::: : 1, but 1:6 : 1 is close enough for our purposes here. The ratio 1:6 : 1 is known as the Golden Ratio. The Greeks used the Golden Ratio extensively in paintings, architecture, sculpture, and designs on pottery. Leonardo Da Vinci suggested that the ratio of certain body measurements is close to the Golden Ratio. Keeping in mind that we are all different shapes and sizes, let us see if we can find the Golden Ratio using our body measurements. You will need: a tape measure and a partner. What to do: forearm 1 Ask your partner to carefully take your body measurements as indicated in the following diagrams. Record these measurements and calculate the required ratios. arm ² length of arm from fingertip to armpit = :::::: mm length of forearm from fingertip to elbow = ...... mm So, arm : forearm = :::::: : :::::: = :::::: : 1 ² hairline to chin = :::::: mm: hairline to bottom of nose = :::::: mm. hairline to chin : hairline to nose = :::::: : :::::: = :::::: : 1 nose chin 2 How do your ratios compare to the Golden Ratio? 3 Compile a class list of these ratios. Are they all close to the Golden Ratio? Check any ratios which are a long way away from the Golden Ratio to make sure they were measured accurately. 4 Can you find any other ratios of body measurements which are close to the Golden Ratio? Further Research: 5 Find out more about where the Golden Ratio occurs in nature. height 6 Research the dimensions of the Great Pyramids of Ancient Egypt. Calculate the ratio base length : height for each pyramid. cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\244IB_MYP2_12.CDR Friday, 22 August 2008 2:14:25 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 base black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 245 E USING RATIOS TO DIVIDE QUANTITIES In the diagram alongside we see that the ratio of the shaded area to the unshaded area is 3 : 2. 3 5 There are a total of 3 + 2 = 5 parts, so and 25 is unshaded. is shaded When a quantity is divided in the ratio 3 : 2, the larger part is 2 5 part is 3 5 of the total and the smaller of the total. Example 13 Self Tutor Line segment [AB] is divided into five equal intervals. a In what ratio does X divide line segment AB? A b What fraction of [AB] is [AX]? X B b [AX] has length 3 units [AB] has length 5 units ) [AX] is 35 of [AB]. a X divides [AB] in the ratio 3 : 2: EXERCISE 12E 1 [CD] is divided into equal intervals. i In what ratio does X divide [CD]? ii What fraction of [CD] is [CX]? a b C X C D c X D d C X D D e X C f D X C C X D 2 Estimate the ratio in which X divides [AB], then check your estimate by measuring: a b A X A B c X A B 3 The line segment [AM] is divided into equal intervals. Which point divides [AM] in the ratio: magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\245IB_MYP2_12.CDR Friday, 22 August 2008 2:21:17 PM PETER 95 100 50 75 25 0 X black B A B C D E F G H I J K L M c 4:8 g 1:1 5 95 100 50 75 25 0 5 95 b 11 : 1 f 6:6 100 50 75 25 0 5 95 100 50 75 25 0 a 5:7 e 2:1 5 B d A cyan X d 1:2 h 1 : 3? IB MYP_2 246 RATIO AND PROPORTION (Chapter 12) Example 14 Self Tutor I wish to divide $12 000 in the ratio 2 : 3 to give to my children Pam and Sam. How much does each one receive? There are 2 + 3 = 5 parts. 2 5 of $12 000 = 2 5 £ $12 000 = $24 000 5 ) Pam gets = 3 5 3 5 = $36 000 5 and Sam gets of $12 000 £ $12 000 = $7200 = $4800 or $12 000 ¡ $4800 = $7200 4 The recommended cordial to water ratio is 1 : 4. Calculate how many mL of cordial are needed to make: a a 300 mL glass of mixture b a 2L container of mixture. 5 The recommended disinfectant to water ratio is 1 : 20. How many mL of concentrated disinfectant are required to make a 9 L bucket of mixture? 6 Answer Problem 1 of the Opening Problem on page 234. A bag of 25 marbles is divided between Jill and John in the ratio 2 : 3. 7 a What fraction does Jill receive? b How many marbles does Jill receive? c How many marbles does John receive? 8 Divide: a $20 in the ratio 1 : 4 b E49 in the ratio 5 : 2. 9 $400 is divided in the ratio 3 : 2. What is the larger share? 10 U160 000 is divided in the ratio 3 : 5. What is the smaller share? Example 15 Self Tutor To make standard concrete, gravel, sand and cement are mixed in the ratio 5 : 3 : 1. I wish to make 18 tonnes of concrete. How much gravel, sand and cement must I purchase? There are 5 + 3 + 1 = 9 parts. cyan £ 18 tonnes = 10 tonnes of gravel £ 18 tonnes = 6 tonnes of sand magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\246IB_MYP2_12.CDR Tuesday, 12 August 2008 1:50:20 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 £ 18 tonnes = 2 tonnes of cement: 100 25 0 5 95 100 50 75 25 0 5 and 5 9 3 9 1 9 50 I need 75 ) black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 247 11 My fortune of $810 000 is to be divided in the ratio 4 : 3 : 2. How much does each person receive? 12 An alloy is made from copper, zinc and tin in the ratio 17 : 2 : 1. How much zinc is required to make 10 tonnes of the alloy? 13 When Michael makes pici pasta, he mixes semolina, “00” flour, and water in the ratio 6 : 3 : 2. If he uses 150 g of “00” flour, what mass does he require of: a semolina 14 Joe a b c d b water? and Bob share the cost of a video game in the ratio 3 : 7: What fraction does each pay? If the game costs $35, how much does each pay? If Joe pays $12, how much does Bob pay? If Bob pays $42, what is the price of the video game? F SCALE DIAGRAMS When designing a house it would be ridiculous for an architect to draw a full-size plan. BEDROOM 1 Instead the architect draws a smaller diagram in which all measurements have been divided by the same number or scale factor. BATHRM BEDROOM 2 LIVING ROOM KITCHEN For house plans a scale factor of 100 would be suitable. LAUNDRY P Similarly, a map of Brazil must preserve the shape of the country. All distances are therefore divided by the same scale factor. In this case the scale factor is 80¡000¡000. 0 1000 km These diagrams are called scale diagrams. In scale diagrams: cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\247IB_MYP2_12.CDR Friday, 22 August 2008 2:22:12 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 ² All lengths have been changed by the same scale factor. ² All angles are unaltered. black IB MYP_2 248 RATIO AND PROPORTION (Chapter 12) To properly use a scale diagram we need to know the scale used. Scales are commonly given in the following ways: ² Scale: 1 cm represents 50 m. This tells us that 1 cm on the scale diagram represents 50 m on the real thing. ² A divided bar can be used to show the scale. Scale This scale tells us that 1 cm on the scale 50 100 150 200 250 0 diagram represents 50 m on the real thing. metres ² Scale: 1 : 5000 This ratio tells us that 1 unit on the scale diagram represents 5000 of the same units on the real thing. For example, 1 cm would represent 5000 cm or 50 m, 1 mm would represent 5000 mm or 5 m. Scales are written in ratio form as drawn length : actual length. We usually simplify the scale to an equal ratio of the form 1 : the scale factor. Example 16 Self Tutor On a scale diagram 1 cm represents 20 m. a Write the scale as a ratio. b What is the scale factor? 1 cm to 20 m = 1 cm to (20 £ 100) cm = 1 cm to 2000 cm = 1 : 2000 a b The ratio simplifies to 1 : 2000 so the scale factor is 2000. Example 17 Self Tutor Interpret the ratio 1 : 5000 as a scale. 1 : 5000 means 1 cm represents 5000 cm ) 1 cm represents (5000 ¥ 100) m f100 cm = 1 mg ) 1 cm represents 50 m EXERCISE 12F.1 1 Write the following scales as ratios and state the corresponding scale factors: a 1 cm represents 10 m b 1 cm represents 50 km c 1 mm represents 2 m d 1 cm represents 250 km e 1 mm represents 5 m f 1 cm represents 200 km. 2 Interpret the following ratios as scales, explaining what 1 cm represents: cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\248IB_MYP2_12.CDR Friday, 22 August 2008 2:24:38 PM PETER 95 100 50 c 1 : 500 f 1 : 5 000 000 75 25 0 5 95 100 50 75 25 0 5 95 100 50 b 1 : 3000 e 1 : 100 000 75 25 0 5 95 100 50 75 25 0 5 a 1 : 200 d 1 : 20 000 black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 249 USING SCALES AND RATIOS Suppose we need a scale diagram of a rectangular area to give to a landscape gardener. The area is 12 m by 5 m and the landscaper wants us to use a scale of 1 : 100. What lengths will we draw on the scale diagram? They must be much less than the actual lengths so that we can fit them on paper. We divide by the scale factor, as division by larger numbers produces a smaller answer. drawn length = actual length ¥ scale factor Example 18 Self Tutor An object 12 m long is drawn with the scale 1 : 100. Find the drawn length of the object. 12 m = 1200 cm drawn length = actual length ¥ 100 fscale factor 100g drawn length = (1200 ¥ 100) cm drawn length = 12 cm ) ) Now suppose we have a map with a scale of 1 : 500 000 marked. We measure the distance between towns A and B to be 15 cm. We can calculate the actual distance between towns A and B using the formula: actual length = drawn length £ scale factor Example 19 Self Tutor For a scale of 1 : 500 000, find the actual length represented by a drawn length of 15 cm. ) ) actual length = drawn length £ 500 000 actual length = 15 cm £ 500 000 = 7 500 000 cm = (7 500 000 ¥ 100) m = 75 000 m = (75 000 ¥ 1000) km actual length = 75 km fscale factor 500 000g f1 m = 100 cmg f1 km = 1000 mg EXERCISE 12F.2 1 If the scale is 1 : 50, find the length drawn to represent an actual length of: a 20 m b 4:6 m c 340 cm d 7:2 m 2 If the scale is 1 : 1000, find the actual length represented by a drawn length of: cyan magenta yellow y:\HAESE\IB_MYP2\IB_MYP2_12\249IB_MYP2_12.CDR Thursday, 14 August 2008 1:56:02 PM PETER 95 100 50 75 25 0 c 4:6 cm 5 95 100 50 75 25 0 b 12 cm 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a 2:5 mm black d 7:4 mm IB MYP_2 250 RATIO AND PROPORTION (Chapter 12) 3 Make a scale drawing of: a a square with sides 200 m b a triangle with sides 16 m, 12 m and 8 m c a rectangle 8 km by 6 km d a circle of radius 16 km scale: scale: scale: scale: 1 cm represents 50 m 1 : 400 1 cm represents 2 km 1 : 1 000 000 4 Select an appropriate scale and make a scale diagram of: a a rectangular house block 42 m by 36 m b a kitchen door 2:2 m by 1:2 m c a triangular paddock with sides 84 m, 76 m and 60 m d the front of an A-framed house with base 12 m and height 9:6 m. 5 Consider the scale diagram of a rectangle. Use your ruler to find the actual dimensions given that the scale is 1 : 200. Which of the following could it represent: A a book B a postage stamp C a basketball court D a garage for a car? 6 A scale diagram of a building is shown with scale 1 : 1000. a If the height is 5 cm and width is 3 cm on the drawing, find the actual height and width of the building in metres. b If the height of the windows on the drawing are 2:5 mm, how high are the actual windows? c If the actual height of the entrance door is 3:2 m, what is its height on the scale drawing? Example 20 Self Tutor This is a scale diagram of a ship. Use your ruler and the given scale to determine: a the total length of the ship b the height of the taller mast c the distance between the masts. Scale 1 : 1000 total length cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\250IB_MYP2_12.CDR Tuesday, 12 August 2008 2:31:04 PM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a The measured length of the ship is 3:8 cm. So, the actual length is 3:8 cm £ 1000 = 3800 cm = 38 m. b The measured height of the taller mast is 2:5 cm So, the actual height is 2:5 cm £ 1000 = 2500 cm = 25 m. c The measured distance between the masts = 1:4 cm. So, the actual distance is 1:4 cm £ 1000 = 1400 cm = 14 m. black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 7 251 Consider the diagram alongside. Find: a the length of the vehicle b the diameter of a tyre c the height of the top of the vehicle above ground level d the width of the bottom of the door. Scale 1 : 80 8 For this part map of the USA, find the actual distance in kilometres in a straight line between: a New York and New Orleans b El Paso and Miami c Seattle and Denver. scale: 1:¡50¡000¡000 9 The actual length of the aeroplane shown in the scale drawing is 64 m. Find: a the scale used in the drawing b the actual wingspan of the aeroplane c the actual width of the fuselage. a cyan magenta yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\251IB_MYP2_12.CDR Friday, 15 August 2008 9:11:43 AM PETER 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 10 The diagram given is of an enlarged bee, drawn to a scale of 1 : 0:25. Find the actual length of the dimensions marked: a wing span, a b body length, b c total length, c. black c b IB MYP_2 RATIO AND PROPORTION (Chapter 12) 11 The floor plan of this house has been drawn to a scale 1 : 120. Find: BEDROOM 2 a the external dimensions of the house including verandah b the dimensions of the verandah c the cost of covering each of the bedroom floors with carpet tiles if carpet tiles cost $45:50 per square metre laid. KITCHEN WC BATHRM P LAUNDRY 252 VERANDAH WARDROBE LIVING ROOM 12 The diagram given shows a microscopic organism enlarged using the scale 1 : 0:001. Find the actual length of the dimensions marked: a width, a b height, b. BEDROOM 1 b a ACTIVITY 1 SCALE DIAGRAMS One way to make a scale drawing is to draw a grid over the picture to be enlarged or reduced. We then copy the picture onto corresponding positions on a larger or smaller grid. Grid paper is available on the worksheet. You could use a photocopier to further enlarge or reduce it. GRID PAPER ACTIVITY 2 DISTANCES IN EUROPE You will need: a ruler marked in mm, an atlas or map showing some major cities in Europe (available on the CD). MAP What to do: cyan magenta 95 yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\252IB_MYP2_12.CDR Wednesday, 20 August 2008 3:08:47 PM PETER 100 50 75 25 0 5 100 95 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 1 Use your ruler and the scale given on the map to estimate the distance in a straight line from: a Paris to Rome b Vienna to Madrid c London to Warsaw d Belgrade to Amsterdam. black IB MYP_2 253 RATIO AND PROPORTION (Chapter 12) G GRADIENT OR SLOPE Have you ever seen a sign like the one pictured here? Watch out for one next time you are travelling through mountains. It is used to warn drivers that the road has a steep slope or a steep gradient. Consider yourself walking up the two hills pictured below. A TRUCKS USE LOW GEAR B You can quite easily see that hill B is steeper than hill A, but in addition to comparing gradient or slope, we often need to measure it. One way to measure the gradient is to use ratios. Picture yourself walking up the sloping side of a right angled triangle. Each step you take, you are moving horizontally to the right, and also vertically upwards. rise run We can measure the gradient by using these vertical and horizontal distances. They are the height and base of the triangle. We sometimes call the vertical distance the rise and the horizontal distance the run. B 1 unit For example, a gradient of 1 in 4 means we move up 1 unit for every 4 units we move across. A 4 units So, the line segment [AB] shown has gradient 1 in 4 or 1 : 4. Gradient is the rise : run ratio. EXERCISE 12G Q 1 Judge by sight which line segment [PQ] is the steepest: A B C D Q Q Q P P P P cyan magenta 95 yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\253IB_MYP2_12.CDR Monday, 1 September 2008 11:17:30 AM PETER Line [PQ] rise run rise : run A B C D 1 3 1:3 1: 1: 1: 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 2 Copy and complete the following table for the triangles of question 1: black IB MYP_2 254 RATIO AND PROPORTION (Chapter 12) 3 By comparing the ratios in the last column of question 2, what happens to the rise : run ratio as the steepness of the line increases? 4 Write down the rise : run ratios of the following triangles in the form 1 : x. a b c d 5 Draw a triangle which has a rise : run ratio of: a 1 in 5 6 b 3:1 c 1 in 4 d 2 in 3 e 3:5 f 7 in 2 S a Express the rise : run ratio of the line segments [PQ] and [RS] in simplest form. b What do you notice about the slopes of [PQ] and [RS]? c Copy and complete the following statement: [PQ] and [RS] are ...... line segments. Q R P KEY WORDS USED IN THIS CHAPTER ² actual length ² gradient ² scale factor ² drawn length ² proportion ² simplest form ² equal ratio ² ratio ² Golden Ratio ² scale diagram REVIEW SET 12A a Express 72 cents to $1:80 as a ratio in simplest form. b Express 10 : 15 as a ratio in simplest form. c 5 : 12 = ¤ : 96. Find the missing number. 1 d Express 1 15 : 4 5 in simplest form. e Write “In a class there are 4 girls for every 3 boys” as a ratio. f Find the ratio of the shaded area to the unshaded area in the figure shown: 2 Divide $400 in the ratio 3 : 7. 3 A watch is bought for E120 and sold for a profit of E40. Find the ratio of the cost price to the selling price. 4 In a school the ratio of students playing football, basketball, and hockey is 12 : 9 : 8. If 144 students play football, how many students play hockey? cyan magenta 95 yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\254IB_MYP2_12.CDR Monday, 1 September 2008 11:17:57 AM PETER 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 5 The actual length of the bus shown alongside is 10 m. Find: a the scale used for this diagram b the actual height of the windows c the actual height of the bus. black IB MYP_2 RATIO AND PROPORTION (Chapter 12) 255 6 A fruit grower plants apricot trees and peach trees in the ratio 4 : 5. If he plants a total of 3600 trees, how many of each type did he plant? 7 [AB] is divided into equal intervals. In what ratio does P divide [AB]? A P B a Divide $200 in the ratio 2 : 3. b George specified that his bank balance be divided amongst his 3 daughters in the ratio 2 : 3 : 4. If his bank balance was E7200, how much did each daughter receive? 8 9 Make a scale drawing of a rectangular room 4 m by 6 m. Use a scale of 1 : 500. 10 A law firm has lawyers and secretaries in the ratio 5 : 2. If the law firm has 30 lawyers, how many secretaries are there? 11 Express the rise : run ratio of the following triangles in simplest form: a b REVIEW SET 12B Express 96 : 72 as a ratio in simplest form. Write 750 mL is to 2 litres as a ratio in simplest form. Write as a ratio: “I saw five Audis for every six BMWs”. Find the ratio of the shaded area : total area in the figure shown. e Express 0:4 : 0:9 as a ratio in simplest form. f Express 2 hours 20 minutes : 4 hours in simplest form. g Express the ratio 3 : 2 12 in simplest form. 1 a b c d 2 Find the missing numbers in the following proportions: a 2:3=8:¤ b 5 : 8 = ¤ : 24 3 A farm has 6000 animals. 2500 are sheep and the rest are cattle. Find the ratio of: a number of sheep : number of cattle b number of sheep : total number of animals. cyan magenta 95 yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\255IB_MYP2_12.CDR Monday, 1 September 2008 11:18:10 AM PETER 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 4 Two families share the cost of buying 75 kg of meat in the ratio 7 : 8. How much meat should each family receive? black IB MYP_2 256 RATIO AND PROPORTION (Chapter 12) 5 During the football season the school team’s win-loss ratio was 2 : 3. If the team lost twelve matches, how many did it win? 6 A man owning 7000 hectares of land divided it between his three sons in the ratio 6 : 9 : 5. How much land did each son receive? 7 Draw a scale diagram of a rectangular window which is 2 m long and 1 m high. Use the scale 1 : 250. 8 A B C D E F G H I The line segment [AI] is divided into equal intervals. Which point divides [AI] in the ratio: a 3:5 b 1:3 c 1:1? 9 A matchbox model Jaguar XJ6 is 75 mm long. The scale stamped on the bottom of the car is 1 : 64. What is the length of the real car? 10 Express the rise : run ratio of the following triangles in simplest form: a b 11 The ratio of girls to boys in a sports club is 4 : 5 . If there are 20 girls, what is the total number of people in the sports club? PUZZLE WORD PUZZLE Complete the missing words in the puzzle below to discover the shaded word. cyan magenta 95 yellow Y:\HAESE\IB_MYP2\IB_MYP2_12\256IB_MYP2_12.CDR Monday, 1 September 2008 11:18:24 AM PETER 100 50 0 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 V 75 H A L P B T U B U L T A N T X M M T 25 C L 5 A N C L black L Y IB MYP_2