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1 Chapter 12. Ratio and proportion Contents: A Ratio B Writing ratios as fractions C Equal ratios D Proportions E Using ratios to divide quantities F Scale diagrams G Gradient or slope 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black Y:\HAESE\IB_MYP2\IB_MYP2_12\ Tuesday, 12 August 2008 3:20:50 PM PETER. 234 RATIO AND PROPORTION (Chapter 12). A ratio is a comparison of quantities. We often hear statements about: a team's win-loss ratio the teacher-student ratio in a school the need to mix ingredients in a certain ratio or proportion: 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?

2 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 GERM. written as a ratio. 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. Also notice that the ratio is written without units such as mL or L. 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black y:\HAESE\IB_MYP2\IB_MYP2_12\ Thursday, 14 August 2008 1:40:57 PM PETER.

3 RATIO AND PROPORTION (Chapter 12) 235. It is important to mix the disinfectant and In both cases there is water in the correct ratio or proportion 1 part disinfectant to 4 parts water! 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. 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.

4 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. 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black Y:\HAESE\IB_MYP2\IB_MYP2_12\ Tuesday, 12 August 2008 12:25:33 PM PETER. 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.

5 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 In the figures alongside, find: a 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. a i shaded : unshaded = 1 : 3 b i shaded : unshaded = 4 : 1. ii shaded : total = 1 : 4 ii shaded : total = 4 : 5. iii Fraction of total shaded is 14.

6 Iii Fraction of total shaded is 45 . 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black Y:\HAESE\IB_MYP2\IB_MYP2_12\ Friday, 22 August 2008 2:07:37 PM PETER. 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. C. a Write as a ratio: 20%. A. 50%. i the sales of A to the sales of B. ii the sales of A to the total sales. 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.

7 12. a Find the total number of students 8. surveyed. 4. 0. b Write as a ratio: Car Bus Train Walk Bicycle i students arriving by car : students who walk ii students arriving by bus : total number of students surveyed. c What fraction of the students surveyed travel to school by bus? C EQUAL RATIOS. Consider the diagrams: A B C. The ratio of shaded area : unshaded area in each case is A 1 : 2 B 2 : 4 C 4 : 8. 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 95. 25. 75. 95. 25. 75. 95. 25. 75. 25. 75. 95. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black Y:\HAESE\IB_MYP2\IB_MYP2_12\ Friday, 22 August 2008 2:55:46 PM PETER. 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.

8 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 To convert a ratio to simplest form, divide Express the following ratios in simplest form: by the highest a 8 : 16 b 35 : 20 common factor. a 8 : 16 b 35 : 20. = 8 8 : 16 8 = 35 5 : 20 5. =1:2 =7:4. 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 b 10 : 5 c 6 : 14 d 6 : 18 e 20 : 50. f 15 : 25 g 9 : 15 h 21 : 49 i 56 : 14. Example 6 Self Tutor 2 3. Express the following ratios in simplest form: a 5 : 5 b 1 12 : 5. 2 3. a 5 : 5 b 1 12 : 5. 3. = 2. 5 5: 3. 5 5 = 2 :5 fconvert to an improper fractiong 3. =2:3 = 2 2 : 5 2. = 3 : 10. 100. 100.

9 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black Y:\HAESE\IB_MYP2\IB_MYP2_12\ Friday, 22 August 2008 2:11:25 PM PETER. RATIO AND PROPORTION (Chapter 12) 239. 2 Express the following ratios in simplest form: 1 4 5 1 2 1 1 3. a 5 : 5 b 4 : 4 c 3 : 3 d 1: 2 e 3: 4. 1 1. f 3 :2 g 3: 2 h 3 12 : 8 i 2 12 : 1 12 j 1. 2 : 1. 3. Example 7 Self Tutor We multiply or divide both parts of Express the following ratios in simplest form: a ratio by the same non-zero number to a 0:3 : 1:7 b 0:05 : 0:15 get an equal ratio. a 0:3 : 1:7 b 0:05 : 0:15. = 0:3 10 : 1:7 10 = 0:05 100 : 0:15 100. = 3 : 17 = 5 : 15. = 5 5 : 15 5. =1:3. 3 Express the following ratios in simplest form: a 0:4 : 0:5 b 1:3 : 1:8 c 0:1 : 0:9. d 0:2 : 0:6 e 0:4 : 0:2 f 1:4 : 0:7. g 0:03 : 0:15 h 0:02 : 0:12 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 210 cm 105 cm c the weight of the fish to the d the capacity of the small cola to breaking strain of the fishing line the capacity of the large cola 3 kg line 9 kg Cool Cola Cool Cola 2 Litre 500 mL.

10 E the number of squares to the f the number of circles to the number of stars number of dots 100. 100. 100. 100. 50. 50. 50. 50. 0. 0. 0. 0. 25. 75. 95. 25. 75. 95. 25. 75. 95. 25. 75. 95. 5. 5. 5. 5. IB MYP_2. cyan magenta yellow black y:\HAESE\IB_MYP2\IB_MYP2_12\ Thursday, 14 August 2008 1:49:43 PM PETER. 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 fHCF = 2g =1:3. 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: Remember to a 2 hours to 4 minutes b 45 cm to 3 m convert to the same units! a 2 hours to 4 minutes b 45 cm to 3 m = 2 60 min to 4 min = 45 cm to 300 cm = 120 min to 4 min = 45 : 300. = 120 : 4 = 45 15 : 300 15. = 120 4 : 4 4 fHCF = 4g = 3 : 20. = 30 : 1. 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 e 27 mm to 81 mm f 9 mL to 63 mL.