maths notes and questions chapter 6.docx

6A Introducing ratios Learning intentions To understand that ratios show a relationship between related quantities To understand that the order in which values are written in a ratio is important To be able to write a ratio from a situation that has been described To be able to write eq...

6A Introducing ratios Learning intentions To understand that ratios show a relationship between related quantities To understand that the order in which values are written in a ratio is important To be able to write a ratio from a situation that has been described To be able to write equivalent ratios to a given ratio by multiplying or dividing each quantity by the same number Ratios are regularly used in everyday life. They are used to show the relationship between two (or more) related quantities. Here are five common uses of ratios: Ingredients -- the ratio of different ingredients in a recipe (cooking, industrial) Maps -- most maps include a scale which is written as a ratio Sporting success -- showing a team's win to loss ratio or the ratio of kicking goals to points Comparing size -- the ratio of length, area or volume of different shapes Legal requirements -- minimum standards of supervision, staff to student ratios When dealing with ratios, the order in which the ratio is written is very important. For example, a team's win : loss ratio of 5 : 2 is very different to a team's win : loss ratio of 2 : 5. Ratios compare quantities of the same type and are given in the same unit. Therefore a ratio is not generally written with a unit of measurement. A compounding chemist or pharmacist is qualified to mix and prepare personalised medications. Quantities are measured precisely and mixed in the ratio prescribed by a patient's doctor. To be able to pull so much weight a tractor's gears need to be set using particular ratios. 6A Introducing ratios 363 LESSON STARTER Estimating ratios Estimate the following ratios. Ratio of boys to girls in your school Ratio of Year 8 students to Year 7 students in your school Ratio of your teacher's age to your age Ratio of hours you spend outside to hours you spend inside Ratio of hours you are awake to hours you are asleep Ratio of parents to children in an average Australian family Ratio of the length to the width of an A4 sheet of paper Discuss your answers as a class. KEY IDEAS Ratios show the relationship between two (or more) related quantities. For example, a drink was made with the ratio of cordial to water of 1 : 3. The colon : is the mathematical symbol used to represent ratios. The written ratio of a : b is read as the ratio of 'a to b' or 'a is to b'. The order in which the quantities are written in a ratio is important. Before ratios can be written and simplified, the quantities must be expressed in the same unit. Ifeachvalueinaratioismultipliedordividedbythesamenumberanequivalentratioisformed. For example, 1 : 3 and 2 : 6 and 10 : 30 are equivalent ratios. BUILDING UNDERSTANDING 1 State the ratio of shaded parts to unshaded parts for each grid. abc 2 Express each of the following pairs of quantities as a ratio. a 9 goals to 4 goals b 52 litres to 17 litres c 7 potatoes to 12 carrots d 3 blue marbles to 5 green marbles 364 Chapter 6 Ratios and rates Example 1 Writing ratios A sample of mixed nuts contains 5 cashews, 12 peanuts and 2 macadamia nuts. Write down: a the ratio of cashews to peanuts to macadamias b the ratio of cashews to the total number of nuts c the ratio of peanuts to other nuts SOLUTION a 5 : 12 : 2 b 5 : 19 c 12 : 7 EXPLANATION cashews : peanuts : macadamias 5 cashews, compared to the 19 total nuts 12 peanuts, compared to the 7 other nuts Now you try In a bag there are 3 green marbles, 7 blue marbles and 6 red marbles. Write down: a the ratio of green to blue to red marbles b the ratio of green marbles to the total number of marbles c the ratio of blue marbles to other marbles Example 2 Producing equivalent ratios Complete each pair of equivalent ratios. a 4:9=16: b 30:15= :5 SOLUTION a ×4 4 : 9 ×4 16 : 36 b ÷3 30 :15 ÷3 10 : 5 EXPLANATION Both numbers are multiplied by 4. Both numbers are divided by 3. Now you try Complete each pair of equivalent ratios. a 3:10=9: b 30:20= :4 6A Introducing ratios 365 FLUENCY 1--3, 4--5(1/2) 2, 3, 4--6(1/2) 3, 4(1/3), 5--6(1/2) Example 1 Example 1 Example 2 PROBLEM-SOLVING 7, 8 8--10 9--11 Exercise 6A 1 A bag contains 5 green marbles, 7 blue marbles and 3 yellow marbles. Write down the ratio of: a green marbles to yellow marbles b blue marbles to total marbles c yellow to blue to green marbles d green to yellow to blue marbles. 2 Over the past fortnight, it has rained on eight days and it has snowed on three days. Write down the ratio of: a rainy days to snowy days b snowy days to total days c fine days to rainy and snowy days d rainy days to non-rainy days. 3 In a box of 40 flavoured icy poles there were 13 lime, 9 lemonade, 11 raspberry and 7 orange icy poles. Write down the ratio of: a lime icy poles to orange icy poles b raspberry icy poles to lemonade icy poles c the four different flavours of icy poles in the order listed above d lime and orange icy poles to raspberry and lemonade icy poles. 4 Complete each pair of equivalent ratios. a 1:3=4: c 2:5= :10 e 5:10 = 1: g 12:18= :3 i 4:7 = 44: k 27:6= :2 5 Complete each pair of equivalent ratios. a 2:3:5=4: : c 1:7:9= : b 1:7=2: d 3:7= :21 f 12:16 = 3: h 20:50= :25 j 4:17 = 12: l 121:66= :6 b 4:12:16= :6: d 22:110:66 = 11: : :63 6 Write three equivalent ratios for each of the following ratios. a1:2 b2:5 c8:6 d9:3 7 Group the following ratios into three pairs of equivalent ratios. 2:5, 6:12, 7:4, 1:2, 4:10, 70:40 8 During a recent dry spell, it rained on only 3 days during the month of September. a What was the ratio of wet days to total days for the month of September? b Write equivalent ratios for a total of 10 days and 100 days. 366 Chapter 6 Ratios and rates 9 On their way to work, Andrew passes 15 sets of traffic lights and Pauline passes 10 sets of traffic lights. One morning Andrew was stopped by 12 red traffic lights. How many green traffic lights would Pauline need to pass through to have the equivalent ratio of red to green traffic lights as Andrew? 10 Write the ratio of vowels to consonants for each of the following place names. a Queensland b Canberra c Wagga Wagga d Australia 11 Name all Australian states that have a vowel to consonant ratio of 1 : 1. 12 Can 10 people be divided into two groups with a ratio of 1 : 2? Explain. 13 Bertrand wins one-third of his games of tennis this season. Write down his win to loss ratio. REASONING 12 12, 13 13--15 14 a b What is the ratio of black keys to white keys for the entire 88 keys of the piano? c Are these equivalent ratios? 15 Write the missing expression. a 2:x=6: b 2:x=5: c y:2x= :8x d 12xy:6y= :1 16 Using the dimensions provided, find the ratio of the shaded area to the unshaded area for each of the following diagrams. What is the ratio of black keys to white keys in one octave of the piano? ENRICHMENT: Area ratios -- -- 16--18 a 10 cm b 20 cm 4 cm 2 cm cd 4 cm 8 cm 30° 50° 5 cm 15 cm Triangles base = 10 cm height = 9 cm 17 Estimate the ratio of the shaded floor to the unshaded floor in this photo. 18 Design your own diagram for which the ratio of shaded area to unshaded area is: a 1:3 b 1:7 6B Simplifying ratios 367 6B Simplifying ratios Learning intentions To understand that simplifying a ratio involves finding an equivalent ratio with no common factors To be able to simplify a ratio involving whole numbers by dividing by the highest common factor To be able to simplify ratios involving fractions To be able to write simplified ratios involving quantities by converting units if necessary In a similar way to fractions, ratios are simplified by dividing each term by a common factor. A ratio is said to be in its simplest form when it contains whole numbers only and the highest common factor (HCF) between the terms in the ratio is 1. If a ratio contains fractions or decimals, it can be simplified by multiplying rather than dividing or cancelling. LESSON STARTER Class ratios Look around your classroom and write down the following ratios. a Ratio of girls to boys b Ratio of teachers to students c Ratio of wearing a watch to not wearing a watch d Ratio of white socks to black socks e Ratio of textbooks open to textbooks closed f Ratio of not having a pencil case to having a pencil case g Ratio of blonde hair to brown hair to black hair h Ratio of blue eyes to brown eyes to other colour eyes Design your own ratio question for your class or classroom. Can any of your ratio answers be simplified? The first outboard motor using fuel was designed by an American engineering student in 1903. Outboard motors require fuel to oil mixed in a certain ratio, such as 5 litres of fuel to 100 ml of oil, which simplifies to the ratio 50 : 1. KEY IDEAS Simplifying ratios A ratio is simplified by dividing both numbers in the ratio by their highest common factor (HCF). For example, the ratio 15 : 25 can be simplified to 3 : 5. ÷ 5 15 : 25 ÷ 5 3 : 5 Simplest form 368 Chapter 6 Ratios and rates It is conventional to express ratios in their simplest form. Ratios in simplest form use whole numbers only. If a ratio is expressed with fractions, it is simplified by converting the quantities to whole numbers. This is generally done by multiplying by the lowest common denominator (LCD). Before ratios are simplified the quantities must be expressed in the same unit. BUILDING UNDERSTANDING 1 State the ratio of shaded parts to unshaded parts for each of the following in simplest form. ab 2 To express the ratio 4 : 16 in simplest form, you would: A multiply both quantities by 2 B subtract 4 from both quantities C divide both quantities by 2 D divide both quantities by 4 3 Decide which one of the following ratios is not written in simplest form. A 1:5 B 3:9 C 2:5 D 11:17 4 Decide which one of the following ratios is written in simplest form. A 2:28 B 15:75 C 14:45 D 13:39 Example 3 Simplifying ratios Simplify the following ratios. a 7:21 b 450:200 SOLUTION EXPLANATION HCF of 7 and 21 is 7. Divide both numbers by 7. HCF of 450 and 200 is 50. Divide both numbers by 50. a b ÷ 7 7 : 21 1 : 3 ÷ 7 450 : 200 ÷50 ÷50 9: 4 Now you try Simplify the following ratios. a 8:20 b 210:240 6B Simplifying ratios 369 Example 4 Simplifying ratios involving fractions Simplify the following ratios. a 3:1 b 21:11 52 34 SOLUTION EXPLANATION LCD of 5 and 2 is 10. Alternatively: 35 : 12 ×10 ×10 6 :5 Convert mixed numerals to improper fractions. LCD of 3 and 4 is 12. Multiply both numbers by 12. a b 6 10 5 10 × 10 7 : 5 3 4 : 6 : 5 × 10 28 : 15 ×12 12 12 ×12 28 : 15 Now you try Simplify the following ratios. a 3:1 b 22:31 56 32 Example 5 Changing quantities to the same units First change the quantities to the same unit, and then express each pair of quantities as a ratio in simplest form. a 4 mm to 2 cm b 25 minutes to 2 hours SOLUTION a 4mmto2cm = 4mmto20mm ÷4 4 : 20 ÷4 1 : 5 b 25 minutes to 2 hours = 25 minutes to 120 minutes ÷5 25 : 120 ÷ 5 5 : 24 EXPLANATION 2cm = 20mm Once in same unit, write as a ratio. Simplify ratio by dividing by the HCF of 4. 2 hours = 120 minutes Once in same unit, write as a ratio. Simplify ratio by dividing by the HCF of 5. Now you try First change the quantities to the same unit, and then express each pair of quantities as a ratio in simplest form. a 3 m to 50 cm b 40 seconds to 2 minutes 370 Chapter 6 Ratios and rates FLUENCY 1, 2(1/2), 4(1/2), 6(1/2) 2--6(1/2) 2--6(1/4) Example 3a Example 3b Example 3 Example 4a Example 4b Example 5 PROBLEM-SOLVING 7, 8 7--9 7, 9, 10 Exercise 6B 1 Simplify the following ratios. ai2:8 b i 40:180 2 Simplify the following ratios. ii 4:24 ii 150 : 100 c 8:10 d g 18:14 h k 51:17 l o 200 : 125 p a 10:50 e 21:28 i 45:35 m 1200:100 b 6:18 f 24:80 j 81:27 n 70:420 25:40 26:13 300 : 550 90:75 85:35:15 24:48:84 27 : 15 9:7 812 3 Simplify the following ratios. (Note: You can divide all three numbers by the highest common factor.) a 2:4:6 b 12:21:33 e 12:24:36 f 100:300:250 4 Simplify the following ratios. a 13:12 b14:15 e 3:1 f 7:4 84 105 5 Simplify the following ratios. a 11:3 b 21:3 c 42:60:12 d g 270:420:60 h c 25 : 34 d g 11:2 h c 31:12 d 41:33 24553534 6 First change the quantities to the same unit, and then express each pair of quantities as a ratio in simplest form. a 12mmto3cm d 60mmto2.1m g 2 kg to 440 g j 20 minutes to 2 hours m 180 minutes to 2 days p 8 weeks to 12 days b 7cmto5mm e 3kgto450g h 1.25Lto250mL k 3 hours to 15 minutes n 8 months to 3 years q 50 cents to \$4 c 120 m to 1 km f 200 g to 2.5 kg i 400 mL to 1 L l 3 days to 8 hours o 4 days to 4 weeks r \$7.50 to 25 cents D 1 : 1 10 15 7 Which of these ratios is the simplest form of the ratio 12 : 2? A 2 : 12 B 1 : 4 C 14 : 1 8 Kwok was absent from school on 8 days in Term 1. There were 44 school days in Term 1. Write the following ratios in simplest form. a Ratio of days absent to total days b Ratio of days present to total days c Ratio of days absent to days present 6B Simplifying ratios 371 9 Over the past four weeks, the Paske family had eaten takeaway food for dinner on eight occasions. They had hamburgers twice, fish and chips three times and pizza three times. Every other night they had home-cooked dinners. Write the following ratios in simplest form. a Ratio of hamburgers to fish and chips to pizza b Ratio of fish and chips to pizza c Ratio of takeaway dinners to home-cooked dinners d Ratio of home-cooked dinners to total dinners 10 When Lisa makes fruit salad for her family, she uses 5 bananas, 5 apples, 2 passionfruit, 4 oranges, 3 pears, 1 lemon (for juice) and 20 strawberries. a Write the ratio of the fruits in Lisa's fruit salad. b Lisa wanted to make four times the amount of fruit salad to take to a party. Write an equivalent ratio that shows how many of each fruit Lisa would need. c Write these ratios in simplest form: i bananas to strawberries ii strawberries to other fruits. 11 Andrew incorrectly simplified 12 cm to 3 mm as a ratio of 4 : 1. What was Andrew's mistake and what is the correct simplified ratio? 12 Mariah has \$4 and Rogan has 50 cents. To write a ratio in simplest form, values must be written in the same unit. a First convert both units to dollars, and then express the ratio of Mariah's money to Rogan's money in simplest form. b As an alternative, convert both units to cents and express the ratio in simplest form. Do you arrive at the same simplified ratio? REASONING 11 11, 12 12--14 13 a b Write two quantities of distance, in different units, which have a ratio of 4 : 3. Write two quantities of time, in different units, which have a ratio of 2 : 5. 14 Simplify the following ratios. a 2a:4b b 15x:3y c a:a2 d 5f:24f e hk:3k f 24xyz:60yz 15 Aspect ratio is the relationship between the width and height of the image as displayed on a screen. a An old analogue television has an aspect ratio of 1.3. : 1. Write this ratio in simplest form. b A high definition digital television has an aspect ratio of 1.7. : 1. Write this ratio in simplest form. c Although these two ratios are clearly not equivalent, there is a connection between the two. What is it? d The digital television aspect ratio of 1.7. : 1 was chosen as the best compromise to show widescreen movies on television. Many major films are presented in Panavision, which has an aspect ratio of 2.35 : 1. Write this ratio in simplest form. e Investigate the history of aspect ratio in films and television. f Research how formats of unequal ratios are converted through the techniques of cropping, ENRICHMENT: Aspect ratios -- -- 15 zooming, letterboxing, pillarboxing or distorting. g Investigate aspect ratios of other common media. i Find the aspect ratio for several different newspapers. ii Find the aspect ratio for several common sizes of photographs. iii Find the aspect ratio for a piece of A4 paper (a surd is involved!) 6C Dividing a quantity in a given ratio Ratios are closely connected to fractions. To help solve problems involving ratios, we can think of each quantity in terms of a 'number of parts'. A drink made in the ratio of cordial to water of 1 : 4 means that there is '1 part' of cordial and '4 parts' of water. There is a total of '5 parts' in the drink. In terms of ratios, it does not matter whether the drink is a 250 mL glass, a 2 L bottle or a 50 L urn. Changing the size of the drink will not change the ratio of 'parts'; it will simply change the size of each part. The fraction of cordial in the drink is number of cordial parts = 1. total number of parts 5 The fraction of water in the drink is number of water parts = 4. total number of parts 5 Thinking about ratios in terms of 'parts' helps us to divide quantities into particular ratios. LESSON STARTER Sharing \$120 For each situation below you have \$120 to share out in the given ratios. With a partner, work out how much each person receives. Situation 1: Two people, Jack and Pippa, ratio of 1 : 1 Situation 2: Two people, Andrew and Alex, ratio of 2 : 2 Situation 3: Two people, Jess and Teresa, ratio of 2 : 3 Situation 4: Two people, Kyle and Kara, ratio of 3 : 5 Situation 5: Three people, Matt, Leos and Djarrin, ratio of 1:1:1 Situation 6: Three people, Christine, Prue and Carolyn, ratio of 3:5:7 Discuss the strategy that you used to share the \$120. Does your strategy work for every situation? Learning intentions To understand that a quantity can be divided into a ratio To be able to divide a quantity in a particular ratio To be able to find the total quantity given a ratio and the actual size of one component A jeweller mixes other metal alloys with gold to increase its durability. One yellow gold alloy has a ratio of: gold : silver : copper = 15 : 3 : 2. Hence a 40 g yellow gold bracelet contains 15 × 40 = 30 g of pure gold. 20 KEY IDEAS Think of a ratio in terms of 'parts'. A ratio of 2 : 3 has 2 parts of one quantity for every 3 parts of another quantity and a total of 5 parts. Using the unitary method to divide a quantity in a given ratio: Find the total number of parts in the ratio. Find the value of one part. Find the value of the number of parts required in the ratio. 6C Dividing a quantity in a given ratio 373 For example: Share \$20 in ratio of 2 : 3. Think of sharing \$20 into 2 parts and 3 parts. Total number of parts = 2 + 3 = 5. Value of one part = \$20 ÷ 5 = \$4. Therefore 2 parts = \$8, and 3 parts = \$12. Using fractions to divide a quantity in a given ratio: Fraction of the amount required = number in ratio ÷5 × 2 \$20 = 5 parts \$ 4 = 1 part \$ 8 = 2 parts ÷5 × 2 total number of parts Calculate the fraction of the amount for each share of the ratio. For example: Share \$20 in ratio of 2 : 3. Fractions of the amount required are 25 and 35. Therefore 25 of \$20 = \$8 and 35 of \$20 = \$12. To find a total quantity from a given ratio: Use the concept of 'parts' and the unitary method to find the value of one part and therefore the value of the total parts can be calculated. Or Use equivalent ratios to find the value of each quantity in the ratio and then add the numbers together to find the total. BUILDING UNDERSTANDING 1 Find the total number of parts in the following ratios. a 3:7 b 1:5 c 11:3 d 2:3:4 2 The ratio of girls to boys in a class is 3:5. a What fraction of the class is girls? b What fraction of the class is boys? 3 The diagram shows four glasses that contain different amounts of cordial. Water is then added to fill each glass. For each drink shown, what is the ratio of cordial to water? abcd 4 What fraction of each of the drinks above is cordial? 374 Chapter 6 Ratios and rates Example 6 Dividing a quantity in a particular ratio Divide 54 m in a ratio of 4:5. SOLUTION Unitary method: Total number of parts = 9 EXPLANATION Total number of parts = 4 + 5 = 9 Value of 1 part = 54 m ÷ 9 = 6 m Check numbers add to total: 24 + 30 = 54 Fraction = number in ratio total number of parts Check numbers add to total: 24 + 30 = 54 ÷ 9 9 parts = 54 m × 4 1 part = 6 m 4 parts = 24 m ÷ 9 × 4 × 5 1 part = 6 m 5 parts = 30 m × 5 54 m divided in a ratio of 4 : 5 is 24 m and 30 m. Fractions method: 4 of 54 = 4 × 54 = 24 9 9 1 5 of 54 = 5 × 54 = 30 991 54 m divided in a ratio of 4 : 5 is 24 m and 30 m. Now you try Divide 30 m in a ratio of 2:3. Example 7 Dividing a quantity in a ratio with three terms Divide \$300 in the ratio of 2:1:3. SOLUTION Unitary method: Total number of parts = 6 EXPLANATION Total number of parts = 2 + 1 + 3 = 6 Value of 1 part = \$300 ÷ 6 = \$50 Check numbers add to total: \$100 + \$50 + \$150 = \$300 ÷ 6 6 parts = \$300 × 2 1 part = \$50 2 parts = \$100 ÷ 6 × 2 × 3 1 part = \$50 3 parts = \$150 × 3 \$300 divided in a ratio of 2 : 1 : 3 is \$100, \$50 and \$150. 6C Dividing a quantity in a given ratio 375 Fractions method: 2 of 300 = 2 × 300 = 100 6 6 1 1 of 300 = 1 × 300 = 50 661 3 of 300 = 3 × 300 = 150 661 \$300 divided in a ratio of 2 : 1 : 3 is \$100, \$50 and \$150. Fraction = number in ratio total number of parts Check numbers add to total: \$100 + \$50 + \$150 = \$300 Now you try Divide \$200 in the ratio of 5:2:3. Example 8 Finding a total quantity from a given ratio The ratio of boys to girls at Birdsville College is 2 : 3. If there are 246 boys at the school, how many students attend Birdsville College? SOLUTION Unitary method: EXPLANATION Ratio of boys : girls is 2:3 Boys have '2 parts' = 246 Value of 1 part = 246 ÷ 2 = 123 Total parts = 2 + 3 = 5parts 5 parts = 5 × 123 = 615 ÷ 2 × 5 2 parts = 246 1 part = 123 5 parts = 615 ÷ 2 × 5 615 students attend Birdsville College. Equivalent ratios method: boys : girls Use equivalent ratios. × 123 = 2 : 3 × 123 = 246 : 369 615 students attend Birdsville College. Multiply each quantity by 123. Total number of students = 246 boys + 369 girls = 615 Now you try The ratio of boys to girls at a school is 4 : 5. If there are 240 boys, how many students are there in total? 376 Chapter 6 Ratios and rates FLUENCY 1, 2--4(1⁄2) 2--5(1⁄2) 2--5(1⁄2) Example 6 Example 6 Example 6 Example 7 Example 7 PROBLEM-SOLVING 6, 7 7--10 9--12 Example 8 Example 8 Exercise 6C 1 Divide: a 40 m in the ratio of 3 : 5 b 2 Divide: a \$60 in the ratio of 2 : 3 b d 360 kg in the ratio of 5 : 7 e 3 Share \$400 in the ratio: a 1:3 b 2:3 4 Divide: a \$200 in the ratio of 1:2:2 c 12 kg in the ratio of 1:2:3 e 320 kg in the ratio of 12:13:15 5 Share 600 lollies in the ratio: a 1:9 b 2:1:3 14 kg in the ratio of 4 : 3 \$1000 in the ratio of 3 : 17 72 m in the ratio of 1 : 2 c 3:5 c c f \$110 in the ratio of 7 : 4 48 kg in the ratio of 1 : 5 155 m in the ratio of 4 : 1 d 9:11 b \$400 in the ratio of 1:3:4 d 88 kg in the ratio of 2:1:5 f \$50000 in the ratio of 1:2:3:4 c 2:5:5 d 12:7:8:3 6 Evergreen Fertiliser is made up of the three vital nutrients nitrogen, potassium and phosphorus in a mass ratio of 4:5:3. How much of each nutrient is in a 1.5 kg bag? 7 The angles of a triangle are in the ratio of 2 : 3 : 4. Find the size of each angle. 8 Three friends, Cam, Molly and Seb, share a prize of \$750 in a ratio of 3 : 4 : 8. How much more money does Seb receive than Cam? 9 Trudy and Bella made 80 biscuits. If Trudy made 3 biscuits in the time that Bella made 2 biscuits, how many biscuits did Trudy make? 10 In Year 8, the ratio of boys to girls is 5 : 7. If there are 140 girls in Year 8, what is the total number of students in Year 8? 11 A light rye bread requires a ratio of wholemeal flour to plain flour of 4 : 3. A baker making a large quantity of loaves uses 126 cups of plain flour. What is the total amount of flour used by the baker? 12 A textbook contains three chapters and the ratio of pages in these chapters is 3 : 2 : 5. If there are 24 pages in the smallest chapter, how many pages are in the textbook? 6C Dividing a quantity in a given ratio 377 REASONING 13 13, 14 14 13 The ratio of the cost of a shirt to the cost of a jacket is 2 : 5. If the jacket cost \$240 more than the shirt, find the cost of the shirt and the cost of the jacket. 14 In a class of 24 students the ratio of girls to boys is 1 : 2. a How many more girls are needed to make the ratio 1 : 1? b If 4 more girls and 4 more boys joined the class, what would be the new ratio? c On one day, the ratio was 3 : 7. How many boys and how many girls were absent? 15 Three students Ramshid, Tony and Maria entered a group Geography competition. Tony spent 3 hours researching the topic. Maria spent 2 12 hours designing the poster. Ramshid spent 5 hours preparing the final presentation. Their group won second prize in the competition and received a prize of \$250. Ramshid, Tony and Maria decide to share the prize in a ratio of 3 : 2 : 1. a How much money did each student receive? Round the answer to nearest cent. b If the prize was divided up according to the time spent on the project, what would be the new ratio? Write the ratio in whole numbers. c How much money did each student receive with the new ratio? Round the answer to the nearest cent. d Although she spent the least time on the project, Maria would prefer to divide the prize according to time spent. How much more money did Maria receive with the new ratio? e Tony preferred that the original ratio remained. How much was Tony better off using the original ratio? f Which ratio would Ramshid prefer and why? g The group ended up going with a ratio based on time spent but then rounded amounts to the nearest \$10. How much prize money did each student receive? ENRICHMENT: A fair share -- -- 15 6E Introducing rates 385 6E Introducing rates Learning intentions To understand that rates compare two quantities measured in different units To be able to simplify rates To be able to find average rates To understand that a rate like \$12/h (or \$12 per hour) means \$12 for 1 hour If you were to monitor what you said each day, you might well find that you speak about rates many times per day! A ratio shows the relationship between the same type of quantities with the same units, but a rate shows the relationship between two different types of quantities with different units. The following are all examples of rates: Cost of petrol was \$1.45 per litre. Rump steak was on special for \$18/kg. Dad drove to school at an average speed of 52 km/h. After the match, your heart rate was 140 beats/minute. Unlike ratios, a rate compares different types of quantities, and so units must be shown. For example: The ratio of boys to girls in a school was 4 : 5. The average rate of growth of an adolescent boy is 12 cm/year. LESSON STARTER State the rate For each of the following statements, write down a corresponding rate. The Lodges travelled 400 km in 5 hours. Gary was paid \$98 for a 4-hour shift at work. Felicity spent \$600 on a two-day shopping spree. Max had grown 9 cm in the last three months. Vuong paid \$37 for half a cubic metre of crushed rock. Paul cycled a total distance of 350 km for the week. What was the rate (in questions/minute) at which you answered these questions? Irrigation rates for watering crops at various growth stages is essential knowledge for efficient farm management. Rates such as: pump flow rates, L /s; drip irrigation rates, L /h; travelling irrigator rates, acres/h; and irrigation frequency rates, n /week. 386 Chapter 6 Ratios and rates KEY IDEAS Rates compare quantities measured in different units. All rates must include units for each quantity. The two different units are separated by a slash '/', which is the mathematical symbol for 'per'. For example: 20 km/h = 20 km per hour = 20 km for each hour It is convention to write rates in their simplest form. This involves writing the rate for only one unit of the second quantity. For example: If \$45 is spent in 3 hours, \$45 in 3 hours Non simplified rate ÷3 ÷3 \$15 in 1 hour Simplified rate (\$15/h) The average rate is calculated by dividing the total change in one quantity by the total change in the second quantity. For example: If a 400 page book is read in 4 days, 400 pages in 4 days ÷4 ÷4 100 pages in 1 day Therefore average reading rate = 100 pages/day. BUILDING UNDERSTANDING 1 Which of the following are examples of rates? A \$5.50 B 180mL/min C \$60/h D 5 23 E 4.2 runs/over F 0.6 g/L G 200 cm2 H 84 c/L 2 Match each rate in the first column with its most likely rate in the second column. Employee's wage Speed of a car Cost of building a new home Population growth Resting heart rate 90 people/day \$2100/m2 68 km/h 64 beats/min \$15/h 3 State typical units for each of the following rates. a Price of sausages c Typing speed e Energy nutrition information g Pain relief medication b Petrol costs d Goal conversion rate f Water usage in the shower h Cricket team's run rate 6E Introducing rates 387 Example 12 Writing simplified rates Express each of the following as a simplified rate. a 12 students for two teachers b \$28 for 4 kilograms SOLUTION EXPLANATION Write the given rate in a sentence. Divide both sides by 2 to find the number of students per 1 teacher. Write the given rate in a sentence. Divide both sides by 4 to find the amount per 1 kg. a b 12 students for 2 teachers ÷2 ÷2 6 students for 1 teacher ∴ 6 students/teacher \$28 for 4 kg ÷4 ÷4 \$7 for 1 kg ∴ \$7/kg Now you try Express each of the following as a simplified rate. a 21 students for three teachers b \$30 for 5 kilograms Example 13 Finding average rates Tom was 120 cm tall when he turned 10 years old. He was 185 cm tall when he turned 20 years old. Find Tom's average rate of growth per year between 10 and 20 years of age. SOLUTION EXPLANATION Growth = 185 − 120 = 65 cm Divide both numbers by 10. ÷ 10 65 cm in 10 years ÷ 10 Average rate of growth = 6.5 cm/year. 6.5 cm in 1 year Now you try Michelle grew from 90 cm to 130 cm over 5 years. Find her average rate of growth per year. 388 Chapter 6 Ratios and rates FLUENCY 1, 2(1/2) 2--3(1/2) 2--3(1/3) Example 12a Example 12b Example 12 Example 13 PROBLEM-SOLVING 4--6 6--8 7--9 Exercise 6E 1 Express each of the following as a simplified rate. a i 15 students for 3 teachers b i \$40for10kg 2 Express each of the following as a simplified rate. a 12 days in 4 years c \$180 in 6 hours e \$126000 to purchase 9 acres g 12000 revolutions in 10 minutes i 60 minutes to run 15 kilometres 3 Find the average rate of change for each situation. a Relma drove 6000 kilometres in 20 days. c A cricket team scored 78 runs in 12 overs. e Russell gained 6 kilograms in 4 years. ii 20 loads for 5 workers ii \$7for2g b 15 goals in 3 games d \$17.50 for 5 kilograms f 36000 cans in 8 hours h 80mm rainfall in 5 days j 15 kilometres run in 60 minutes b Holly saved \$420 over three years. d Saskia grew 120 centimetres in 16 years. f The temperature dropped 5°C in 2 hours. 4 A dripping tap filled a 9 litre bucket in 3 hours. a What was the dripping rate of the tap in litres/hour? b How long would it take the tap to fill a 21 litre bucket? 5 Martine grew at an average rate of 6 cm/year for the first 18 years of her life. If Martine was 50 cm long when she was born, how tall was Martine when she turned 18? 6 If 30 salad rolls were bought to feed 20 people at a picnic, and the total cost was \$120, find the following rates. a Salad rolls/person b Cost/person c Cost/roll 7 The number of hours of sunshine was recorded each day for one week in April. The results are listed. Monday 6 hours, Tuesday 8 hours, Wednesday 3 hours, Thursday 5 hours, Friday 7 hours, Saturday 6 hours, Sunday 7 hours a Find the average number of hours of sunshine: i per weekday ii per weekend day iii per week iv per day. b Given the above rates, how many hours of sunshine would you expect in April? 8 Harvey finished a 10 kilometre race in 37 minutes and 30 seconds. Jacques finished a 16 kilometre race in 53 minutes and 20 seconds. Calculate the running rate of each runner. Which runner had a faster running pace? 6E Introducing rates 389 9 The Tungamah Football Club had 12 000 members. After five successful years and two premierships, they now have 18000 members. a What has been the average rate of membership growth per year for the past 5 years? b If this membership growth rate continues, how many more years will it take for the club to have 32 000 members? REASONING 10 10, 11 11 10 a b How can you convert km/L to L/km? A car uses 24 L of petrol to travel 216 km. Express these quantities as a simplified rate in: i km/L ii L/km. 11 The Teleconnect satellite telecommunications company has a variable call charge rate for phone calls of up to 30 minutes. The charges are 50 c/min for first 10 minutes, 75 c/min for the second 10 minutes and \$1/min for the third 10 minutes. a Find the cost of phone calls of these given lengths. i 8 minutes ii 13 minutes iii 24 minutes iv 30 minutes b What is the average charge rate per minute for a 30 minute call? Connectplus, a rival telecommunications company, charges a constant call rate of 60 c/minute. c If you normally made calls that were 15 minutes long, which company has the better deal for you? d If you normally made calls that were 25 minutes long, which company has the better deal for you? e What is the length of phone call for which both companies would charge the same amount? 12 In Victoria, due to drought conditions, the state government in 2008 urged all residents to save water. The goal was set for each Victorian to use no more than 155 litres of water per day. a How many people live in your household? b According to the Victorian government, how many litres of water can your household use per day? c Perform some experiments and calculate the following rates of water flow. ENRICHMENT: Target 155 -- -- 12 i Shower rate (L/min) iii Hose (L/min) v Running tap (L/min) vii Dishwasher (L/wash) ii Washing machine (L/load) iv Toilet (L/flush, L/half flush) vi Drinking water (L/day) viii Water for food preparation (L/day) d Estimate the average daily rate of water usage for your household. e Ask your parents for a recent water bill and find out what your family household water usage rate was for the past three months. Before the initiative, Victorians were using an average of 164 litres/day/person. Twelve months after the initiative, Victorians were using 151 litres/day/person. f How much water per year for the state of Victoria does this saving of 13 litres/day/person represent? g What is the rate at which your family is charged for its water? 390 Chapter 6 Ratios and rates 6A 6A 6B 6C 6C 6D 6D 6D 6E 6E 6E 6E 1 An excursion to Sea World has 6 teachers and 144 students. Sixty-nine of the students are boys; the rest are girls. Write down, in simplest form, the ratio of: a teachers to students b boys to girls c girls to the total of people on the excursion. 2 Complete each pair of equivalent ratios. a 2:5 = \_\_\_\_\_:15 b c 24:8 = 12:\_\_\_\_ d 3 Simplify the following ratios. a 24:36 b 24:36:40 \_\_\_\_:3 = 12:9 7:8:3 = 21:\_\_\_\_:\_\_\_\_ 3 c 4:9 d 0.9:12 4 Divide: a \$800 in the ratio of 5:3 c 1 km in the ratio of 2 : 3 e 40 cents to \$1.20 b \$7500 in the ratio of 8:3:1 5 Tony and Vi share their company's profits in the ratio of 15 : 7. Tony's share totals \$1200. a How much profit did the company make? b What is Vi's share? 6 A map uses a scale of 1 : 40 000. Find the actual distance, in metres, if the distance on the map measures 8 mm. 7 A model car has a scale of 1 : 100. Find the scale length, in centimetres, of a Lexus that is 4.32 metres long. 8 Determine the scale ratio for 4 mm on a drawing representing an actual distance of 8 km. 9 Express each of the following as a simplified rate. a 180 students on 3 buses b \$5.60 for 4 kg c 186kmin212hours 10 Find the average rate for each situation. a Thelma drove 8000 km in 50 days b Callum saved \$1250 in 6 months c Ainslie grew 20 cm in 2 12 years 11 Who earns the most, and by how much, if Kelly is paid \$96570 a year and Todd earns \$7985 each month? 12 Which is faster 70 km/h or 21 m/s? Progress quiz ISBN Photo 6F Solving rate problems 391 6F Solving rate problems 'One thing that is certain in life is that change is inevitable.' We are constantly interested in rates of change and how things change over a period of time. We are also regularly faced with problems involving specific rates. Strong arithmetic skills and knowing whether to multiply or divide by the given rate allows many rate problems to be solved quickly and accurately. Over the next few days, keep a record of any rates you observe or experience. Look out for the slash '/' sign and listen for the 'per' word. LESSON STARTER Estimate the rate For each of the following statements, estimate a corresponding rate. Commercial rate: The number of commercials on TV per hour Typing rate: Your typing speed in words per minute Laughing rate: The number of times a teenager laughs per hour Growth rate: The average growth rate of a child from 0 to 15 years of age Running rate: Your running pace in metres per second Homework rate: The number of subjects with homework per night Clapping rate: The standard rate of audience clapping in claps per minute Thankyou rate: The number of opportunities to say thank you per day Compare your rates. Which rate is the 'highest'? Which rate is the 'lowest'? Discuss your answers. Learning intentions To understand that rates can be used to model many situations To be able to solve problems involving rates The rate that food energy is burned depends on a person's weight. For example, playing soccer for 30 minutes uses around 4.3 calories/kg. Hence, a 60 kg teenager playing soccer burns energy at a rate of 260 calories per half hour. KEY IDEAS When a rate is provided, a change in one quantity implies that an equivalent change must occur in the other quantity. For example: Patrick earns \$20/hour. How much will he earn in 6 hours? \$20 for 1 hour ×6 ×6 \$120 for 6 hours For example: Patrick earns \$20/hour. How long will it take him to earn \$70? \$20 for 1 hour × 312 \$70 for 3 1 hours × 312 2 Carefully consider the units involved in each question and answer. 392 Chapter 6 Ratios and rates BUILDING UNDERSTANDING 1 State the missing components in each rate problem. a × 3 60 km in 1 hour × 3 180 km in \_\_\_ c \_\_\_ 7 questions in 3 minutes 70 questions in \_\_\_ 2 State the missing components in each rate problem. a ÷3 ×5 b \_\_\_ \_\_\_ \$36 for 3 hours ÷3 \_\_\_ for 1 hour \_\_\_ \_\_\_ for 5 hours 150 rotations in 5 minutes \_\_\_ \_\_\_\_\_\_\_\_\_\_\_ in 1 minute \_\_\_ \_\_\_\_\_\_\_\_\_\_\_ in 7 minutes b × 5 \_\_\_ d \_\_\_ \$25 in 1 hour \_\_\_ \$125 in \_\_\_ 120 litres in 1 minute \_\_\_ \_\_\_ in 6 minutes Example 14 Solving rate problems a Rachael can type at 74 words/minute. How many words can she type in 15 minutes? b Leanne works in a doughnut van and sells on average 60 doughnuts every 15 minutes. How long is it likely to take her to sell 800 doughnuts? a b ×15 74 words in 1 minute 1110 words in 15 minutes SOLUTION EXPLANATION 74 × 15 370 740 1110 Selling rate = 60 doughnuts/15 minutes Divide both quantities by HCF of 15. Multiply both quantities by 200. Convert answer to hours and minutes. ×15 Rachael can type 1110 words in 15 minutes. ÷ 15 60 doughnuts in 15 minutes × 200 4 doughnuts in 1 minute 800 doughnuts in 200 minutes ÷ 15 × 200 Leanne is likely to take 3 hours and 20 minutes to sell 800 doughnuts. Now you try a Joan can type of 80 words/minute. How many words can she type in 25 minutes? b A hotdog vendor sells on average 60 hotdogs every 20 minutes. How long is it likely to take to sell 210 hotdogs? Three water hoses from three different taps are used to fill a large swimming pool. The first hose alone takes 200 hours to fill the pool. The second hose alone takes 120 hours to fill the pool and the third hose alone takes only 50 hours to fill the pool. How long would it take to fill the pool if all three hoses were used? SOLUTION In 600 hours: hose1wouldfill3pools hose2wouldfill5pools hose3wouldfill12pools Therefore in 600 hours the three hoses together would fill 20 pools. 600 hours for 20 pools ÷20 ÷20 h 30 hours for 1 pool It would take 30 hours to fill the pool if all three hoses were used. EXPLANATION LCM of 200, 120 and 50 is 600. Hose1 = 600h ÷ 200h/pool = 3pools Hose2 = 600h ÷ 120h/pool = 5pools Hose3 = 600h ÷ 50h/pool = 12pools Together = 3 + 5 + 12 = 20 pools filled. Simplify rate by dividing by HCF. Now you try Two hoses are used to fill a large pool. The first hose alone takes 100 hours to fill the pool and the second hose takes only 150 hours to fill the pool. How long would it take to fill the pool if both hoses were used? FLUENCY 1--6 2--7 3--7 Example 14a Example 14b Example 14a Example 14b 6F Solving rate problems 393 Example 15 Solving combination rate problems Exercise 6F 1 a Geoff can type 58 words/minute. How many words can be type in 10 minutes? b Philip works at a juice bar and sells on average 40 juices every 10 minutes. How long is it likely to take him to sell 200 juices? 2 A factory produces 40 plastic bottles/minute. a How many bottles can the factory produce in 60 minutes? b How many bottles can the factory produce in an 8 hour day of operation? 3 Mario is a professional home painter. When painting a new home he uses an average of 2.5 litres of paint per hour. How many litres of paint would Mario use in a week if he paints for 40 hours? 394 Chapter 6 Ratios and rates 4 A truck travels 7 km per litre of fuel. How many litres are needed for the truck to travel 280 km? 5 Daniel practises his guitar for 40 minutes every day. How many days will it take him to log up 100 hours of practice? 6 A flywheel rotates at a rate of 1500 revolutions per minute. a How many revolutions does the flywheel make in 15 minutes? b How many revolutions does the flywheel make in 15 seconds? c How long does it take for the flywheel to complete 15000 revolutions? d How long does it take for the flywheel to complete 150 revolutions? 7 Putra is an elite rower. When training, he has a steady working heart rate of 125 beats per minute (bpm). Putra's resting heart rate is 46 bpm. a How many times does Putra's heart beat during a 30 minute workout? b How many times does Putra's heart beat during 30 minutes of 'rest'? c If his coach says that he can stop his workout once his heart has beaten 10000 times, for how long would Putra need to train? 8 What is the cost of paving a driveway that is 18 m long and 4 m wide, if the paving costs \$35 per square metre? 9 A saltwater swimming pool requires 2 kg of salt to be added for every 10000 litres of water. Joan's swimming pool is 1.5 metres deep, 5 metres wide and 15 metres long. How much salt will she need to add to her pool? 10 TheBionicWomangivesBatmana12secondstartina2kilometrerace.IftheBionicWomanrunsat 5 km/min, and Batman runs at 3 km/min, who will win the race and by how many seconds? PROBLEM-SOLVING 8--9 8--10 9--11 11 At a school camp there is enough food for 150 students for 5 days. a How long would the food last if there were only 100 students? b If the food ran out after only 4 days, how many students 6F Solving rate problems 395 REASONING 12 12--14 13--16 Example 15 ENRICHMENT: Value for money -- -- 17 12 Michelle can complete a landscaping job in 6 days and Danielle can complete the same job in 4 days. Working together, in how many days could they complete the job? 13 Three bricklayers Maric, Hugh and Ethan are cladding a new home. If Maric were to work alone, the job would take him 8 days to complete. If Hugh were to work alone, the job would take him 6 days to complete, and if Ethan were to work by himself, the job would take him 12 days to complete. a If the three men work together, how long will it take them to complete the job? b What fraction of the house will each bricklayer complete? 14 Four cans of dog food will feed 3 dogs for 1 day. a How many cans are needed to feed 10 dogs for 6 days? b How many dogs can be fed for 9 days from 60 cans? c For how many days will 40 cans feed 2 dogs? 15 If it takes 4 workers, 4 hours to dig 4 holes, how long would it take 2 workers to dig 6 holes? 16 State the units required for the answer to each of the following rate calculations. a \$205/kg × 48kg b 62s × 12m/s c 500 beats ÷ 65 beats/min (bpm) d 4000 revolutions ÷ 120 revs/min 17 Soft drink can be purchased from supermarkets in a variety of sizes. Below are the costs for four different sizes of a certain brand of soft drink. 600 mL 'buddies' 1.25 L bottles 2 L bottles 10 × 375 mL cans \$2.70 each \$1.60 each \$2.20 each \$6.00 per pack a Find the economy rate (in \$/L) for each size of soft drink. b Find and compare the cost of 30 litres of soft drink purchased entirely in each of the four different sizes. c If you only have \$60 to spend on soft drink for a party, what is the difference between the greatest amount and the least amount of soft drink you could buy? Assume you have less than \$1.60 left. Most supermarkets now include the economy rate of each item underneath the price tag to allow customers to compare value for money. d Carry out some research at your local supermarket on the economy rates of a particular food item with a range of available sizes (such as drinks, breakfast cereals, sugar, flour). Write a report on your findings. 396 Chapter 6 Ratios and rates 6G Speed A rate that we come across almost every day is speed. Speed is the rate of distance travelled per unit of time. On most occasions, speed is not constant and therefore we are interested in the average speed of an object. Average speed is calculated by dividing the distance travelled by the time taken. Average speed = Distance travelled Time taken Given the average speed formula, we can tell that all units of speed must have a unit of length in the numerator, followed by a unit of time in the denominator. Therefore 'mm/h' is a unit of speed and could be an appropriate unit for the speed of a snail! Two common units of speed are m/s and km/h. Average speed cameras use automatic number plate recognition technology to record the exact time a car passes under a camera. Average speed is calculated using the time and distance between cameras, and fines are issued for speeding. LESSON STARTER Which is faster? With a partner, determine the faster of the two listed alternatives. a Car A travelling at 10 m/s b Walker C travelling at 4 km/h c Jogger E running at 1450 m/h d Plane G flying at 700 km/h Learning intentions To understand that speed is a rate relating distance and time To be able to find an average speed (given a distance and the time taken) To be able to find the distance travelled (given an average speed and the time taken) To be able to find the time taken (given an average speed and the distance) To be able to convert between different units of speed Car B travelling at 40 km/h Walker D travelling at 100 m/min Jogger F running at 3 m/s Plane H flying at 11 km/min 6G Speed 397 KEY IDEAS Speed is a measure of how fast an object is travelling. If the speed of an object does not change over time, the object is travelling at a constant speed. 'Cruise control' helps a car travel at a constant speed. When speed is not constant, due to acceleration or deceleration, we are often interested to know the average speed of the object. Average speed is calculated by the formula: Average speed = Distance travelled or s = dt Time taken Depending on the unknown value, the above formula can be rearranged to make d or t the subject. The three formulas involving s, d, and t are: s = dt d = s × t t = ds Care must be taken with units for speed, and on occasions units will need to be converted. The most common units of speed are m/s and km/h. d st d st d st BUILDING UNDERSTANDING 1 Which of the following is not a unit of speed? A m/s B km/h C cm/h D L/kg 2 If Average speed = Distance travelled then the Distance travelled must equal: Time taken A Average speed × Time taken B Average speed C Time taken Average speed 3 If Average speed = Distance travelled then Time taken must equal: Time taken A Distance travelled × Average speed B Distance travelled C Average speed Distance travelled 4 If an object travels 800 metres in 10 seconds, the average speed of the object is: A 8000m/s B 800km/h C 80km/h E m/min Time taken Average speed D 80m/s 398 Chapter 6 Ratios and rates Example 16 Finding average speed Find the average speed in km/h of: a a cyclist who travels 140 km in 5 hours b a runner who travels 3 km in 15 minutes. SOLUTION a s = dt = 140 km 5 h = 28 km/h Alternative method: ÷ 5 140 km in 5 hours 28 km in 1 hour Average speed = 28 km/h b s= dt = 3km 15 min = 3 ÷ 41 = 3 × 4 = 12 km/h Alternative method: × 4 3 km in 15 minutes 12 km in 60 minutes Average speed = 12 km/h EXPLANATION The unknown value is speed. Write the formula with s as the subject. Distance travelled = 140 km Time taken = 5h Speed unit is km/h. Write down the rate provided in the question. Divide both quantities by 5. Distance travelled = 3 km Time taken = 15 min or 14 h Dividing by 14, is the same as multiplying by 41. Write down the rate provided in the question. Multiply both quantities by 4. ÷ 5 × 4 Now you try Find the average speed in km/h of: a a driver who travels 260 km in 4 hours b a walker who travels 2 km in 20 minutes. ISBN 978-1-108-77271-6 © Greenwood et al. 2019 Cambridge University Press Photocopying is restricted under law and this material must not be transferred to another party. Updated May 2023 6G Speed 399 Example 17 Finding the distance travelled Find the distance travelled by a truck travelling for 15 hours at an average speed of 95 km/h. SOLUTION d = s × t = 95km/h × 15h = 1425 km Alternative method: × 15 95 km in 1 hour EXPLANATION The unknown value is distance. Write the formula with d as the subject. Distance unit is km. Write the rate provided in the question. Multiply both quantities by 15. × 15 Truck travels 1425 km in 15 hours. 1425 km in 15 hours Now you try Find the distance travelled by a car driving for 3 hours at an average speed of 85 km/h. Example 18 Finding the time taken Find the time taken for a hiker walking at 4 km/h to travel 15 km. SOLUTION t = ds = 15 km 4 km/h = 3.75 h = 3 h 45 min Alternative method: 4 km in 1 hour ÷ 4 1 1 km in 4 hour × 15 15 km in 15 hours 4 ÷ 4 × 15 EXPLANATION The unknown value is time. Write the formula with t as the subject. The time unit is h. Leave answer as a decimal or convert to hours and minutes. 0.75h = 0.75 × 60 = 45min Express the rate as provided in the question. Divide both quantities by 4. Multiply both quantities by 15. 15 3 (Note: 4 = 34 as a mixed number.) It takes 3 h 45 min to travel 15 km. Now you try Find the time taken for a jogger to jog 4 km at 12 km/h. 400 Chapter 6 Ratios and rates FLUENCY 1, 2--4(1/2) 2--4(1/2), 5 2--4(1/2), 5 Example 16 Example 16 Example 17 Example 18 Speed, s Distance, d Time, t PROBLEM-SOLVING 6, 7 7--9 8--10 Exercise 6G 1 Find the average speed in km/h of: a a car travelling 180 km in 3 hours b a train travelling 60 km in 30 minutes. 2 Calculate the average speed of: a a sprinter running 200 m in 20 seconds b a skateboarder travelling 840 m in 120 seconds c a truck travelling 400 km in 8 hours d a tram travelling 15 km in 20 minutes. 3 Calculate the distance travelled by: a a cyclist travelling at 12 m/s for 90 seconds b an ant travelling at 2.5 m/s for 3 minutes c a bushwalker who has walked for 8 hours at an average speed of 4.5 km/h d a tractor ploughing fields for 2.5 hours at an average speed of 20 km/h. 4 Calculate the time taken by: a a sports car to travel 1200 km at an average speed of 150 km/h b a bus to travel 14 km at an average speed of 28 km/h c a plane to fly 6900 km at a constant speed of 600 km/h d a ball moving through the air at a speed of 12 m/s to travel 84 m. 5 Complete the following table. a b c d e 50km 2h 30m/s 1200m 5km/h 210 km/h 600 m 12h 5 min 20 min f 100m/s 10km 6 A plane is flying at a cruising speed of 900 km/h. How far will the plane travel from 11:15 a.m. to 1:30 p.m. on the same day? 7 The wheels on Charlie's bike have a circumference of 1.5 m. When Charlie is riding fastest, the wheels rotate at a speed of five turns per second. a What is the fastest speed Charlie can ride his bike, in km/h? b How far would Charlie travel in 5 minutes at his fastest speed? 6G Speed 401 8 The back end of a 160-metre-long train disappears into a 700-metre-long tunnel. Twenty seconds later the front of the train emerges from the tunnel. Determine the speed of the train in m/s. 9 Anna rode her bike to school one morning, a distance of 15 km, at an average speed of 20 km/h. It was raining in the afternoon, so Anna decided to take the bus home. The bus trip home took 30 minutes. What was Anna's average speed for the return journey to and from school? 10 The Ghan train is an Australian icon. You can board The Ghan in Adelaide and 2979 km later, after travelling via Alice Springs, you arrive in Darwin. (Round the answers correct to one decimal place.) a If you board The Ghan in Adelaide on Sunday at 2:20 p.m. and arrive in Darwin on Tuesday at 5:30 p.m., what is the average speed of the train journey? b There are two major rest breaks. The train stops for 414 hours at Alice Springs and 4 hours at Katherine. Taking these breaks into account, what is the average speed of the train when it is moving? 11 Nina, Shanti and Belle run a 1000 m race at a constant speed. When Nina crossed the finish line first, she was 200 m ahead of Shanti and 400 m ahead of Belle. When Shanti crossed the finish line, how far ahead of Belle was she? 12 Julie and Jeanette enjoy finishing their 6 km morning run together. Julie runs at an average speed of 10 km/h and Jeanette runs at an average speed of 3 m/s. If Julie leaves at 8 a.m., at what time should Jeanette leave if they are to finish their run at the same time? 13 Carry out research to find answers to the following questions. Light and sound a What is the speed of sound in m/s? b What is the speed of light in m/s? c How long would it take sound to travel 100 m? d How long would it take light to travel 100 km? e How many times quicker is the speed of light than the speed of sound? f What is a Mach number? Spacecraft g What is the escape velocity needed by a spacecraft to 'break free' of Earth's gravitational pull? Give this answer in km/h and also km/s. h What is the orbital speed of planet Earth? Give your answer in km/h and km/s. i What is the average speed of a space shuttle on a journey from Earth to the International Space REASONING 11 11, 12 11, 12 ENRICHMENT: Speed research -- -- 13 Station? Knots Wind speed and boat speed are often given in terms of knots (kt). j What does a knot stand for? k What is the link between nautical miles and a system of locating positions on Earth? l How do you convert a speed in knots to a speed in km/h? 402 Chapter 6 Ratios and rates Applications and problem-solving The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions. Bad news, good news 1 For a media project, Kima looks through the newspaper and finds 16 bad news stories and 12 good news stories. Kima is interested in the ratio of negative to positive news stories in the media and the best method for calculating these ratios. a Write down the ratio of good news stories to bad news stories from Kima's newspaper in simplest form. b If this ratio continued and Kima categorised a total of 140 newspaper stories, how many would be bad news? Kima thinks it would be fairer to include a third category, which she classifies as 'neutral' -- these include predominantly information stories and some sport reports. Kima analyses five newspapers and classifies 180 stories. Kima found the ratio of bad : neutral : good news stories to be 7 : 3 : 5. c How many neutral stories did Kima find in the five newspapers? d If Kima analysed another five newspapers and found 46 bad stories, 44 neutral stories and 30 good news stories, what would the ratio of bad : neutral : good news stories be across the ten different newspapers? e The newspaper decided to produce a good news only newspaper for one day. If Kima added this paper to her current set of ten newspapers, how many articles would need to be published for her overall ratio of good : bad news stories to become 1 : 1? Kima feels that just counting the number of stories is not the best way to determine the ratio of good to bad news. She finds that most major stories are bad news stories and that these articles can sometimes take up multiple pages of the newspaper, compared to sometimes a small good news story that only takes up a little corner of a page. Kima decides to measure the area of each news story for one newspaper. Her results are provided in the table below: Category Area (cm2) Category Area (cm2) Category Area (cm2) Category Area (cm2) Bad 2500 Bad 1600 Neutral 400 Good 200 Bad 800 Bad 400 Good 150 Neutral 1000 Neutral 800 Good 150 Bad 150 Bad 450 Bad 600 Neutral 200 Bad 1000 Good 300 Bad 400 Neutral 600 Bad 100 Good 200 f Calculate the ratio of area given to the coverage of bad : neutral : good news stories in Kima's newspaper. Give your answer in simplest form. g Calculate the ratio of bad : neutral : good news stories in this newspaper. Give your answer in simplest form. h Compare the two ratios in parts f and g. Which do you think is a more accurate ratio to report? Applications and problem-solving 403 i Choose a different media channel (social media, TV news, school newsletter) and analyse the ratio of good : bad news stories. You may like to consider other categories and you may wish to consider time, length of text, images, order or other variables to increase the sophistication of your analysis. Designing maps 2 Faibian has been tasked with designing several maps of Australia for his school. When designing a map, a designer must determine the actual size of the area to be represented, the desired size of the map and the scale to fit this area onto the map. Faibian is interested in calculating scales for maps of Australia depending on the size of the map required for different scenarios. Determine an appropriate scale for the following situations. a Faibian wishes to have a map of Australia the size of an A4 piece of paper. Note that Australia is approximately 4000 km from the east coast to the west coast and that an A4 piece of paper in landscape format is about 20 cm wide. b Faibian's school wishes to have a map of Australia which is approximately 10 m wide to go on the outside of their new mathematics building. c Faibian wishes to have a map of greater Adelaide the size of a large 1 m poster. Note that the diameter of greater Adelaide is about 100 km. Faibian turns his interest in scaled drawings to Google Maps where the scale of their maps instantly changes as touch screen users move their fingers to zoom in or out of particular maps. d Faibian gets a map of Australia to show on his phone on Google Maps. He observes the scale located at the bottom of the map. Using a ruler, determine the scale of this map. e Using Google Maps, determine the scale used to show a map that includes both your home and your school. f Using Google Maps, how many times larger is the size of your school compared to the map on your phone or computer? 404 Chapter 6 Ratios and rates Four seasons in one day 3 Melbourne is known for its changing weather patterns and some visitors refer to the weather as four seasons in one day. For a period of ten consecutive days over summer, Stuart records the following Melbourne temperature information. All temperatures are measured in °C and given to the nearest whole degree. Day Temp. at 9 a.m. Temp. at 3 p.m. Daily min. Daily max. 1 12 2 16 3 14 4 21 5 25 6 17 7 14 8 18 9 32 10 22 24 10 28 28 14 28 21 14 25 34 16 36 38 24 39 29 15 30 25 12 30 27 16 29 40 22 45 41 16 41 Stuart is interested in investigating the rate of change in Melbourne's temperature across the day in summer. a On which day did Stuart record the highest: i temperature at 9 a.m.? ii temperature at 3 p.m.? iii daily minimum? iv daily maximum? b Using the daily temperature recordings at 9 a.m. and 3 p.m., calculate the average rate of change in temperature in degrees per hour for each of the ten days. c i Which day had the highest average rate of change in temperature? ii Which day had the lowest average rate of change in temperature? d i What would the temperature at 3 p.m. need to be on day 5 if the average rate of change in temperature on this day was 2.5°C/hour? ii What would the temperature at 9 a.m. need to be on day 2 if the average rate of change in temperature on this day was 1.5°C/hour? Over these ten days, Stuart determines that the daily minimum occurs on average at 5:30 a.m. and the time of the daily maximum occurs on average at 4:15 p.m. e Using this additional time information and by calculating the average daily minimum and average daily maximum for the ten days, determine the average rate of change in °C/hour for Melbourne temperature across these ten days. f Investigate the rate of change in temperature for your local area. i Choose one day and record the temperature to the nearest degree each hour from 9 a.m. to 9 p.m. ii Calculate the rate of change in temperature for each hour. iii Calculate the average rate of change in temperature across the twelve hours. iv Calculate the average rate of change in temperature between the daily minimum and the daily maximum. v Discuss your findings in comparison to Melbourne's temperature. 6H Ratios and rates and the unitary method 405 6H Ratios and rates and the unitary method EXTENDING Learning intentions To understand that the unitary method involves finding the value of 'one unit' first To be able to solve ratio and rates problems using the unitary method To be able to convert rates between different units using the unitary method The concept of solving problems using the unitary method was introduced in Chapter 3. The unitary method involves finding the value of 'one unit'. This is done by dividing the amount by the given quantity. Once the value of 'one unit' is known, multiplying will find the value of the number of units required. LESSON STARTER Finding the value of '1 unit'? For each of the following, find the value of 1 unit or 1 item. 8 basketballs cost \$200. 4 cricket bats cost \$316. 5 kg of watermelon cost \$7.50. For each of the following, find the rate per 1 unit. Car travelled 140 km in 2 hours. 1000 L of water leaked out of the tank in 8 hours. \$51 was the price paid for 3 kg of John Dory fish. For each of the following, find the value of 1 'part'. Ratio of books to magazines read was 2 : 5. Milli had read 14 books. Ratio of pink to red flowers is 7 : 11. A total of 330 red flowers are in bloom. Ratio of girls to boys is 8 : 5. There are 40 girls in a group. Southern Bluefin Tuna are scientifically farmed in South Australia and fed on sardines. The FIFO ratio = Fish In (sardines) : Fish Out (tuna) is a measure ofefficiency.IfFIFO=5:1=1:0.2, this shows 1 kg of fish food produces 0.2 kg of tuna. KEY IDEAS The unitary method involves finding the value of 'one unit' and then using this information to solve the problem. When dealing with ratios, find the value of 1 'part' of the ratio. For example: If the ratio of phones to televisions is 5 : 2, find the number of televisions for 15 phones. 5 parts = 15 phones ÷5 ÷5 1 part = 3 devices ×2 ×2 2 parts = 6 televisions When dealing with rates, find the value of the rate per 1 'unit'. For example: Pedro earned \$64 for a 4 hour shift at work. Therefore, wage rate = \$64 per 4 hours = \$16 per hour = \$16/h. 406 Chapter 6 Ratios and rates Once the value of one 'part' or the rate per one 'unit' is known, the value of any number of parts or units can be found by multiplying. The technique of dividing and/or multiplying values in successive one-step calculations can be applied to the concept of converting rates from a set of given units to a different set of units. BUILDING UNDERSTANDING 1 12 packets of biscuits cost \$18.60. a What is the cost of one packet? b What is the cost of 7 packets? 2 The ratio of books to magazines is 2 : 3 and there are a total of 25 books and magazines altogether. a If 5 parts = 25 books and magazines, find the value of one part. b How many magazines are there? Example 19 Reviewing the unitary method Andy travels 105 km in 7 identical car trips from home to school. How far would she travel in 11 such car trips? SOLUTION ÷7 7 car trips = 105 km ÷7 ×11 1 car trip = 15 km ×11 11 car trips = 165 km Andy travels 165 km. EXPLANATION Find the value of 1 unit by dividing both quantities by 7. Solve the problem by multiplying both quantities by 11. Now you try Fred paid \$100 for 4 identical shirts. How much would 7 of these shirts cost? Example 20 Solving ratio problems using the unitary method The ratio of apples to oranges is 3 : 5. If there are 18 apples, how many oranges are there? SOLUTION ÷ 3 3 parts = 18 apples ×5 1 part = 6 pieces 5 parts = 30 oranges There are 30 oranges. ÷ 3 ×5 EXPLANATION Apples = 3 'parts', Oranges = 5 'parts' Need to find the value of 1 'part'. To find 5 'parts' multiply the value of 1 'part' by 5. 6H Ratios and rates and the unitary method 407 Now you try The ratio of bananas to mandarins is 4 : 7. If there are 24 bananas, how many mandarins are there? Example 21 Solving rate problems using the unitary method A truck uses 4 L of petrol to travel 36 km. How far will it travel if it uses 70 L of petrol? SOLUTION Rate of petrol consumption EXPLANATION Find the petrol consumption rate of 1 unit by dividing both quantities by 4. Solve the problem by multiplying both quantities by 70. ÷ 4 36 km for 4 L × 70 9 km for 1 L ÷ 4 × 70 630 km for 70 L Truck will travel 630 km on 70 L. Now you try A car uses 6 L of petrol to travel 72 km. How far will it travel if it uses 20 L of petrol? Example 22 Converting units using the unitary method Melissa works at the local supermarket and earns \$57.60 for a 4 hour shift. How much does she earn in c/min? SOLUTION EXPLANATION Write down Melissa's wage rate. Find the rate of \$ per 1 hour. Convert \$ to cents and hours to minutes. Divide rate by 60 to find rate of cents per minute. ÷ 4 ÷ 60 \$57.60 for 4 hours \$14.40 for 1 hour 1440c for 60 minutes 24c for 1 minute Melissa earns 24 c/min. ÷4 ÷ 60 Now you try Simone earns \$90 for a 5 hour shift working at a cafe. How much does she earn in c/min? 408 Chapter 6 Ratios and rates a Convert 72 km/h to m/s. b Convert 8 m/s to km/h. SOLUTION a 72 km in 1 hour EXPLANATION Express rate in kilometres per hour. Convert km to m and hour to minutes. Divide both quantities by 60. Convert 1 minute to 60 seconds. Divide both quantities by 60. Shortcut for converting km/h → m/s ÷ 3.6. Express rate in metres per second. Multiply by 60 to find distance in 1 minute. Multiply by 60 to find distance in 1 hour. Convert metres to kilometres. Shortcut: m/s × 3.6 → km/h. 8 m/s × 3.6 = 28.8 km/h ÷ 60 ÷60 72 000 m in 60 minutes 1200 m in 1 minute 1200 m in 60 seconds 20 m in 1 second ÷ 60 ÷60 28 800 m in 1 hour ∴ 28.8 km in 1 hour ∴ 72km/h = 20m/s b 8 m in 1 second × 60 8 m in 1 second × 60 ×60 ×60 480 m in 1 minute Now you try a Convert 18 km/h to m/s. b Convert 10 m/s to km/h. FLUENCY 1, 2--4(1/2), 6--8(1/2) 2--4(1/2), 5, 6--8(1/2) 3--4(1/2), 5, 6--8(1/2) Example 19 Example 19 Example 20 Example 23 Converting units of speed Exercise 6H 1 Marissa travels 117 km in 9 identical car trips. How far would she travel in 7 such car trips? 2 Solve the following problems. a If 8 kg of chicken fillets cost \$72, how much would 3 kg of chicken fillets cost? b If one dozen tennis balls cost \$9.60, how much would 22 tennis balls cost? c If three pairs of socks cost \$12.99, how much would 10 pairs of socks cost? d If 500 g of mince meat costs \$4.50, how much would 4 kg of mince meat cost? 3 Solve the following ratio problems. a The required staff to student ratio for an excursion is 2 : 15. If 10 teachers attend the excursion, what is the maximum number of students who can attend? b The ratio of commercials to actual show time for a particular TV channel is 2 : 3. How many minutes of actual show were there in 1 hour? c A rectangle has length and width dimensions in a ratio of 3 : 1. If a particular rectangle has a length of 21 m, what is its width? d Walter and William have a height ratio of 7 : 8. If William has a height of 152 cm, how tall is Walter? 6H Ratios and rates and the unitary method 409 Example 21 Example 22 Example 23a Example 23b PROBLEM-SOLVING 9, 10 9--11 10--12 4 Solve the following rate problems. a A tap is dripping at a rate of 200 mL every 5 minutes. How much water drips in 13 minutes? b A professional footballer scores an average of 3 goals every 6 games. How many goals is he likely to score in a full season of 22 games? c A snail travelling at a constant speed travels 400 mm in 8 minutes. How far does it travel in 7 minutes? d A computer processor can process 500 000 kilobytes of information in 4 seconds. How much information can it process in 15 seconds? 5 Leonie, Spencer and Mackenzie have just won a prize. They decide to share it in the ratio of 4 : 3 : 2. If Spencer receives \$450, how much do Leonie and Mackenzie receive, and what was the total value of the prize? 6 Convert the following rates into the units given in the brackets. a \$15/h (c/min) c 3.5 L/min (L/h) e 0.5 kg/month (kg/year) g 60 g/c (kg/\$) i 108 km/h (m/s) 7 Convert the following speeds to m/s. a 36 km/h b 180 km/h 8 Convert the following speeds to km/h. a 15m/s b 2m/s b \$144/h (c/s) d 20 mL/min (L/h) f 120 g/day (kg/week) h \$38/m (c/mm) j 14 m/s (km/h) c 660 m/min c 12 m/min d 4 km/s d 1 km/s 9 The Mighty Oats breakfast cereal is sold in boxes of three different sizes: small (400 g) for \$5.00, medium (600 g) for \$7.20, large (750 g) for \$8.25 a Find the value of each box in \$/100 g. b What is the cheapest way to buy a 4 kg of the cereal? 10 In Berlin 2009, Jamaican sprinter Usain Bolt set a new 100 m world record time of 9.58 seconds. Calculate Usain Bolt's average speed in m/s and km/h for this world record. (Round the answers correct to one decimal place.) 11 Zana's hair grew 6 cm in 5 months. a Find Zana's average rate of hair growth in cm/month and in m/year. b How long would it take for Zana's hair to grow 30 cm? 12 Maria can paint 15 m2 in 20 minutes. a What is the rate at which Maria paints in m2/h? b What area can Maria paint in 20 hours? c Maria must paint 1000 m2 in 20 hours. Find the rate at which she will need to paint in m2/min. 410 Chapter 6 Ratios and rates REASONING 13 13, 14 13--15 13 If x doughnuts cost \$y: a how much would 1 doughnut cost? b how much would one dozen doughnuts cost? c how much would z doughnuts cost? 14 a b A triangle has side lengths in a ratio of 3 : 5 : 4. If the longest side is 35 cm, find the lengths of the A triangle has side lengths in a ratio of 19 : 22 : 17. If the shortest side is 17 cm, find the lengths of the other two sides and the perimeter of the triangle. other two sides and the perimeter of the triangle. 15 In a faraway galaxy, a thriving alien colony uses the following units: For money they have puks and paks: 1 puk (pu) = 1000 pak (pa) For length they have doits and minidoits: 1 doit (D) = 80 minidoits (mD) Polynaute rope is priced at 4 pu/D. Find the cost of the rope in terms of pa/mD. 16 Phil lives in Perth and his friend Werner lives in Sydney. The distance, by road, between their two houses is 4200 km (rounded to the nearest 100 km). Phil decides to drive to Sydney and Werner decides to drive to Perth. They leave home at the same time and travel the same route, but in opposite directions. Phil drives at a constant speed of 75 km/h and Werner drives at a constant speed of 105 km/h. a Will they meet on the road at a spot closer to Sydney or closer to Perth? b How long will it take Phil to travel to Sydney? c How long will it take Werner to travel to Perth? d State the location of each friend after they have been driving for 15 hours. e At what location (distance from Sydney and/or Perth) will they meet? When they meet, Phil decides to change into Werner's car and they drive back to Werner's home at an average speed of 105 km/h. f How long did it take Phil to travel to Sydney? g Design a similar problem for two friends travelling at different constant speeds between two different capital cities in Australia. ENRICHMENT: Where will we meet? -- -- 16 Problems and challenges 411 1 This diagram is made up of 8 equal-sized squares. How many squares need to be shaded if the ratio of shaded squares to unshaded squares is: a 1:3? b 2:3? c 1:2? 2 Bottle A has 1 L of cordial drink with a cordial to water ratio of 3 : 7. Bottle B has 1 L of cordial drink with a cordial to water ratio of 1 : 4. The drink from both bottles is combined to form a 2 L drink. What is the new cordial to water ratio? 3 Brothers Marco and Matthew start riding from home into town, which is 30 km away. Marco rode at 10 km/h and Matthew took 20 minutes longer to complete the trip. Assuming that they both rode at a constant speed, how fast was Matthew riding? 4 a If 1 person takes 1 hour to dig 1 post hole, how long will it take 2 people to dig 2 post holes? b If 3 people take 3 hours to make 3 wooden train sets, how many train sets can 6 people make in 6 hours? 5 At a market you can trade 2 cows for 3 goats or 3 goats for 8 sheep. How many sheep are 3 cows worth? 6 Two cars travel toward each other on a 100 km straight stretch of road. They leave at opposite ends of the road at the same time. The cars' speeds are 100 km/h and 80 km/h. How long does it take for the cars to pass each other? Car 1 100 km/h Car 2 80 km/h 7 A river is flowing downstream at a rate of 1 km/h. In still water Michael can swim at a rate of 2 km/h. Michael dives into the river and swims downstream then turns around and swims back to the starting point, taking 0.5 hours in total. How far did he swim? 8 A fitness fanatic walks at 4 km/h for time t1 , and then runs at 7 km/h for time t2. He travels a total of 26 km. If he had run for the same time that he had walked (t1 ) and walked for the same time that he had run (t2 ), then he would have travelled 29 km. What was the total time spent walking and running? 9 A top secret goo has been developed so that it doubles in volume every 60 seconds. At midnight a small amount is placed in a beaker and its growth observed. At 1 a.m. the beaker is full. At exactly what time did the goo fill only 18 of the beaker? 10 A car averages 60 km/h for 20 km and then increases its speed averaging 80 km/h for the next 20 km. Calculate the car's average speed (to two decimal places) for the full 40 km trip. 11 Max travels at 50 km/h for 10 km. For the next 10 km, Max wants to drive fast enough to make his average speed 100 km/h over the 20 km trip. Advise Max about how to reach this average speed. Up for a challenge? If you get stuck on a question, check out the 'Working with unfamiliar problems' poster at the end of the book to help you. 100 km Problems and challenges Press 2023 412 Chapter 6 Ratios and rates Ratios are written in simplest form with whole numbers Simplest form Equivalent ratios 2 : 3 a:b a is to b a to b Ratios Comparison of two quantities of the same type and same unit Finding a quantity from a ratio Cows to horses in ratio 5 : 2 There are 8 horses, how many cows? Equivalent ratio method Dividing a quantity in a given ratio × 4 ÷2 ÷2 × 4 8 : 12 4 : 6 Divide \$180 in the ratio 4 : 5 Fraction method Unitary method 59 × 180 Eliminate decimals HCF 4 Recurring decimals to fractions Same units 0.8 : 1.2 Total number of parts Each fraction Each amount 4+5=9 4 and 5 × 10 ÷4 ÷4 × 10 8 : 12 2: 3 5.3 : 2.6 5 1 : 2 2 99 49 × 180 and = \$80 \$180 divided in the ratio 4 : 5 is \$80 and \$100 = \$100 Sentence answer ×3 ÷8 ×3 ÷8 Total number of parts One part Sentence answer Scale ratios Drawing : Actual length length Divide \$180 in the ratio 4 : 5 4 + 5 = 9 9 parts = \$180 3 16 : 8 3 3 16 : 8 2: 1 3 ÷9 × 4 ÷9 × 4 1 25 minutes: 1 4 hours ÷25 25 : 75 ÷25 1 : 3 Unitary method C: H 5 : 2 2 parts = 8 horses 1 part = 4 horses 5 parts = 20 cows C : H= 5 : 2 C : 8 = ×4 5 : 2 ×4 ×4 Model car 17 cm long Scale 1 : 25 Scale factor = 25 Actual car = 25 × 17 cm = 425 cm = 4.25 m × 5 same units = \$20 \$180 divided in the ratio 4 : 5 is \$80 and \$100 Actual size from model 1 part = \$20 4 parts = \$80 × 5 1 part 5 parts = \$100 = ×4 There are 20 cows. 20 : 8 Length units ÷ 1000 ÷ 100 ÷ 10 Average rates 720 km driven in 10 hours Average speed = 720 10 = 72 km/h = Drawing (or map) length 1 : scale factor × 1000 km × 100 m Map length from actual length Actual distance = 1 km × 10 cm mm Scale ratio 4 : 500 1 : 125 50 : 1 1 1 : 50 Scale factor 1 50 Simplified rates Actual object 125 times larger than drawing 1 th of the 50 drawing size times scale factor divide scale factor Rates Map scale = Scale factor = 1 : 25000 Actual length Map length 25000 = 1 km ÷ 25000 = 1 km same 25000 units = 1000 × 100 cm 25000 = 4 cm 25 km/h many km per one hour? \$12/kg many \$ per one kg? s s s 1 ÷1 2 ÷60 18 km in 112 hours 18 km in 1 hour 112 1 ÷1 2 ÷ 80 ×165 4 L for 80 m2 4 2 80 L for 1 m 4 × 165 for 165 m2 80 8.25 L for 165 m2 ÷ 80 1 part ×165 s : speed d : distance t : time Unitary method to change units (Ext) For a 18 km jog in 112 hours determine average rate in m/min Comparing two quantities with different units Unitary method with rates (Ext) d d d t t t s = dt d = s × t d t = s 4 L paint to cover 80 m2 ? L to cover 165 m2 12 000 m in 60 min 200 m in 1 min 1 part 1 part ÷60 Average speed = 200 m/min Chapter summary Chapter checklist 413 Chapter checklist: Success criteria 6A 6A 6B 6B 6B 6C 6C 6D 6D 6D 6E 6E 1\. I can write a ratio from a description. e.g. A sample of mixed nuts contains 5 cashews, 12 peanuts and 2 macadamia nuts. Write down the ratio of peanuts to other nuts. 2\. I can produce a ratio that is equivalent to a given ratio. e.g. State the missing number in the equivalence 30 : 15 = ? : 5. 3\. I can simplify ratios involving whole numbers. e.g. Simplify 450 : 200. 4\. I can simplify ratios involving fractions. e.g. Simplify 2 1 : 1 1. 34 5\. I can write simplified ratios involving quantities by first converting units. e.g. Write the relationship '25 minutes to 2 hours' as a ratio by first changing the quantities to the same unit. 6\. I can divide a quantity in a ratio with two or three terms. e.g. Divide 54 m in a ratio of 4 : 5 and divide \$300 in the ratio of 2 : 1 : 3. 7\. I can find a total quantity from a given ratio and the actual size of one component. e.g. The ratio of boys to girls at Birdsville College is 2 : 3. If there are 246 boys at the school, how many students attend Birdsville College? 8\. I can convert from scale distance to actual distance using a scale. e.g. A map has a scale of 1 : 20 000. Find the actual distance for a scale distance of 5 mm. Answer in metres. 9\. I can convert from actual distance to scale distance using a scale. e.g. A model boat has a scale of 1 : 500. Find the scaled length for an actual length of 75 cm. Answer in millimetres. 10\. I can determine the scale factor. e.g. Determine the scale factor if an actual length of 0.1 mm is represented by 3 cm on a scale drawing. 11\. I can write simplified rates. e.g. Express \$28 for 4 kilograms as a simplified rate. 12\. I can find average rates. e.g. Tom was 120 cm tall when he turned 10 years old, and 185 cm when he turned 20 years old. Find Tom's average rate of growth per year over this period. Chapter checklist Press 2023 414 Chapter 6 Ratios and rates 6F 6G 6G 6G 6G 6H 6H 6H 14\. I can find an average speed. e.g. Find the average speed in km/h of a runner who travels 3 km in 15 minutes. 15\. I can find the distance travelled. e.g. Find the distance travelled by a truck travelling for 15 hours at an average speed of 95 km/h. 13\. I can solve rate problems. e.g. Rachael can type at 74 words/minute. How many words can she type in 15 minutes? 16\. I can find the time taken to travel a given distance at a given speed. e.g. Find the time taken for a hiker walking at 4 km/h to travel 15 km. 17\. I can convert units of speed. e.g. Convert 72 km/h to m/s. 18\. I can solve ratio problems using the unitary method. e.g. The ratio of apples to oranges is 3 : 5. If there are 18 apples, how many oranges are there? Ext 19\. I can solve rate problems using the unitary method. e.g. A truck uses 4 L of petrol to travel 36 km. How far will it travel using 70 L? Ext 20\. I can convert units for rates using the unitary method. e.g. Melissa works at the local supermarket and earns \$57.60 for a 4 hour shift. How much does she earn in c/min? Chapter checklist Chapter review 415 6A 6B 6B 6B 6B 6C 6C 6E 6E Short-answer questions 1 In Lao's pencil case there are 6 coloured pencils, 2 black pens, 1 red pen, 5 textas, 3 lead pencils and a ruler. Find the ratio of: a lead pencils to coloured pencils b black pens to red pens c textas to all pencils. 2 State whether the following are true or false. a 1:4=3:6 b The ratio 2 : 3 is the same as 3 : 2. c The ratio 3 : 5 is written in simplest form. d 40 cm : 1 m is written as 40 : 1 in simplest form. e 11:2=5:8 4 3 Copy and complete. a 4:50=2: c 23:4=1: 4 Simplify the following ratios. b 1.2:2= :20 a 10:40 f 5:25 a 2cm:8mm e 400kg:2t d 1: c 75:100 h52:26 m21:3 :5=5:15:25 d 8:64 k 2:5 7 7 b 36:24 g6:4 b 5mm:1.5cm f 6h:1day i 6b:9b n 12:36:72 e 27:9 j 8a:4 d 30min:1h h 45min:21h l 11:2 10 10 2 4 5 Simplify the following ratios by first changing to the same units. 6 Divide: a \$80 in the ratio 7 : 9 c 40 m in the ratio 6 : 2 e \$1 in the ratio 3 : 1 : 1 b 200 kg in the ratio 4 : 1 d \$1445 in the ratio 4 : 7 : 6 calculate its retail price. b The ratio of Sally's height to Ben's height is 12 : 17. If the difference in their heights is 60 cm, how tall is Sally? c Orange juice, pineapple juice and guava juice are mixed in the ratio 4 : 3 : 2. If 250 mL of guava juice is used, how many litres of drink does this make? 8 Express each rate in simplest form. a 10kmin2hours b \$650for13hours c 2800kmin20days 9 Copy and complete. c 3L:7500mL g 120m:1km 2 7 a The ratio of the cost price of a TV to its retail price is 5 : 12. If its cost price is \$480, a c 400 km on 32 litres = \_\_\_\_\_ km/L = \_\_\_\_\_ L/100km \$1200 in 12 day = \_\_\_\_\_ \$/day = \_\_\_\_\_ \$/h b 5 grams in 2 min = \_\_\_\_\_ g/min = \_\_\_\_\_ g/h Chapter review 416 Chapter 6 Ratios and rates 10 For a scale of 1 : 1000, find the real length (in metres) if the scale length is given as: a 0.0002m b 2.7cm c 140mm 11 Two cities are 50 km apart. How many millimetres apart are they on a map that has a scale of 1 : 100 000? 12 Find the scale factor for the following pairs of similar figures, and find the value of x. a 4 cm 6D 6D 6D b 3 cm 13 a b Samira earns \$67.20 for a 12-hour shift. How much will she earn for a 7-hour shift? c Tap 1 fills the pool in 12 hours, while tap 2 fills the same pool in 15 hours. How long does it 6 cm 5 cm 4 cm 9 cm x cm 6 cm 15 cm x cm 14 a take to fill this pool if both taps are used? Sandra drives to her mother's house. It takes 45 minutes. Calculate Sandra's average speed if A truck uses 12 litres of petrol to travel 86 km. How far will it travel on 42 litres of petrol? 6F 6G 6H Ext 6A 6A her mother lives 48 km away. b How long does it take Ari to drive 180 km along the freeway to work if he manages to average 100 km/h for the trip? c How far does Siri ride his bike if he rides at 4.5 km/h for 90 minutes? 15 Copy and complete. a \$120/h = \_\_\_\_c/min b 6m/s = \_\_\_\_km/h c 720km/h = \_\_\_\_m/s Multiple-choice questions 1 A school has 315 boys, 378 girls and 63 teachers. The ratio of students to teachers is: A 11:1 B 1:11 C 5:6 D 6:5 2 The ratio of the shaded area to the unshaded area in this triangle is: A 3:5 C 5:3 B 8:5 D 5:8 Chapter review Chapter review 417 6B 6B 6C 6C 6D 6E 6F 6H Ext 3 The ratio 500 mm to 15 m is the same as: A 50:2 B 2500:1 C 2:5 D 5:2 4 The ratio 1 12 : 34 simplifies to: A2:1 B1:2 C3:4 D4:3 5 \$750 is divided in the ratio 1 : 3 : 2. The smallest share is: A \$250 B \$125 C \$375 D \$750 6 The ratio of the areas of two triangles is 5 : 2. The area of the larger triangle is 60 cm2. What is the area of the smaller triangle? A 12cm2 B 24cm2 C 30cm2 D 17cm2 7 On a map, Sydney and Melbourne are 143.2 mm apart. If the cities are 716 km apart, what scale has been used? A 1:5 B 1:5000 C 1:50000 D 1:5000000 8 Callum fills his car with 28 litres of petrol at 142.7 cents per litre. His change from \$50 cash is: A \$10 B \$39.95 C \$10.05 D \$40 120 cm D 75m/s 9 A flag is created by enlarging the shaded rectangle as shown. What is the length of the original rectangle? A 20cm B 30cm C 40cm D 13 cm 20 cm 10 45 km/h is the same as: A 0.25m/s 80 cm B 25m/s Extended-response question C 12.5m/s The Harrison family and the Nguyen family leave Wollongong at 8 a.m. on Saturday morning for a holiday in Melbourne. The Harrisons' 17-year-old son drives for the first 2 hours at 80 km/h. They then stop for a rest of 1 12 hours. Mr Harrison drives the rest of the way. The Nguyen family drives straight to Melbourne with no stops. It takes them 6 hours and 15 minutes to drive the 627 km to Melbourne. a At what time did the Nguyen family arrive in Melbourne? b Calculate the average speed of the Nguyen's car. Round your answer to the nearest whole number. c At what time did the Harrisons resume their journey after their rest stop? d How many kilometres did the Harrisons still have to cover after their rest break before arriving in Melbourne? e If the Harrisons arrive in Melbourne at 4:30 p.m., for how long did Mr Harrison drive and at what speed? f Calculate the cost of the petrol for each family's trip if petrol cost 125 c/L and the Harrison car's consumption is 36 L/100 km, while the Nguyen's car uses 40 L/100 km.