250 11 Graphs In this unit, you have looked at the graphs of linear functions. These can be written as y=mx+c, where m and c are integers. What general conclusions have you found? 12 Water is flowing out of a tank. The amount of water is given by the function y=−3x+40 where there are y litres of water in the tank after x minutes. a Copy and complete this table of values. x 0 1 2 3 4 5 6 7 8 y 28 b Plot a graph to show how the amount of water in the tank changes over time. c What does the y-intercept tell you about the water in the tank? d What does the gradient tell you about the water in the tank? 13 Here is a function: y=10x+30 a Construct a table of values with x going from 0 to 6. b Use your table of values to draw a graph of y=10x+30 c Describe a situation that the function y=10x+30 could represent. i Explain what the variables x and y and the numbers 10 and 30 represent. ii Explain how the numbers 10 and 30 are linked to the graph. d Look at a partner’s answer to this question. Is their explanation clear? Think like a mathematician 14 Here are three functions. y=x+6 y=2x+6 y=3x+6 a Plot a graph of each function. Plot them all on the same axes. b Find the y-intercept for each line. c The x-intercept is the x-coordinate where the line crosses the x-axis. Find the x-intercept for each line. d Use your answers to part c to predict the x-intercept of the graph of y=4x+6 Draw the graph to see if you are correct. e Can you generalise your results and predict the x-intercept for the graph of y=mx+6 where m is a positive integer? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
251 11.4 Interpreting graphs Summary checklist I know that equations of the form y=mx+c correspond to straight-line graphs. I know that m is the gradient and c is the y-intercept. In this section you will … • read and interpret graphs with several components • understand why graphs have specific shapes. 11.4 Interpreting graphs Graphs give information in a visual form. In real-life contexts, this can help you to understand a situation. For example, a graph that shows how the distance travelled by a car changes with time can help you to see how fast the car is travelling. Worked example 11.4 This graph shows the fares charged by two different taxis. Each taxi has a fixed charge and a charge per kilometre. a How much does each taxi charge for a journey of 7km? b Find the fixed charge for each taxi. c Find the charge per kilometre for each taxi. d What distance will cost the same amount in either taxi? Answer a $26 for taxi A and $32 for taxi B Find the y-coordinate on each line when the x-coordinate is 7. b $12 for taxi A and $4 for taxi B This is the y-intercept of each line. c $2 for taxi A and $4 for taxi B This is the gradient of each line. d 4km This is the x-coordinate where the two lines cross. x y 5 0 1 10 15 25 35 2 3 5 7 Cost ($) Distance (km) 20 30 40 4 6 8 Taxi A Taxi B We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
252 11 Graphs Exercise 11.4 1 Zalika and Tanesha are cycling on the same route. The graph shows their journeys. a Zalika started at 09 :00. What time did Tanesha start? b How far did Zalika travel in the first hour? c How long was Tanesha cycling before he caught up with Zalika? 2 Lucas is driving from Ackult to Bibas. Simone is driving from Bibas to Ackult. a How long did Lucas take to get to Middja? b For how long did Lucas stop in Middja? c How long did Simone take to get from Bibas to Ackult? d How far were the cars from Bibas when they passed one another? 3 Razi and Jake are running laps of a running track. a How do you know from the graph that Jake is running faster than Razi? b For how long had Razi been running before Jake started? c Where were the runners 9 minutes after Razi started running? 4 This graph shows the journeys of a van and a car. The van is travelling at a constant speed. a i What is the speed of the van? ii For how long does the van travel at that speed? The speed of the car increases steadily from 0 m/s for 20 seconds. b What is the speed of the car after 20 seconds? After the first 20 seconds, the car travels at a constant speed for 20 seconds. Then the speed steadily decreases to 0m/s in 10 seconds. c Copy the graph and plot the rest of the journey of the car. d Give your graph to a partner so they can check your answer to part c. 09:00 0 5 10 15 20 25 30 35 09:30 10:00 10:30 Time (24-hour clock) Tanesha Zalika Distance (km) Ackult 0 1 2 3 4 5 100 0 Middja 200 300 Lucas Simone Bibas Time (hours) Distance from Ackult (km) 10 0 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 10 Time (minutes) Jake Laps run Razi 0 0 10 15 5 20 25 30 10 20 30 40 50 Time (seconds) Speed (m/s) 60 x y Car Van We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
253 11.4 Interpreting graphs 0 0 10 20 30 40 1 2 3 4 5 Weeks Height (cm) 6 Plant Y Plant X x y e When is the car travelling at 15m/s? f For how many seconds is the car travelling faster than the van? 5 Two plumbers charge different rates. Each plumber has a fixed charge and a charge per hour. a Find the cheaper plumber for a job that takes 2 hours. b Plumber B charges $250 for a job. Find the time for the job. c Copy and complete this table to show the total cost for plumber A. hours 0 1 2 3 4 5 6 cost ($) 230 260 d Find the fixed charge for plumber A. e Find the charge per hour for plumber A. f Find the fixed charge for plumber B. g Find the charge per hour for plumber B. 6 There are two different tariffs for a long-distance phone call. Each tariff has a connection charge plus a charge for each minute. The tariffs are shown on this graph. a Which tariff is cheaper for a 5-minute call? b A call costs 200 rupees on tariff B. How long does it last? c What length of call costs the same on both tariffs? d Find the fixed charge for each tariff. e Work out the charge per minute for each tariff. 7 This graph shows the growth of two plants over a period of 6 weeks. a Which plant grew more quickly? Explain how the graph shows this. b Work out the initial height of plant X. c When were the plants the same height? d Work out how many centimetres plant Y grew each week. 0 0 100 200 300 400 1 2 3 4 5 Time (minutes) Cost (rupees) 6 Tariff B Tariff A x y 0 0 100 200 300 400 1 2 3 4 5 Time (hours) Cost (dollars) 6 Plumber A Plumber B x y We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
254 11 Graphs 8 A car and a van travel 70km. This graph shows the fuel in the tank of each vehicle. a Work out how much fuel each vehicle had at the start of the journey. b Work out how much fuel each vehicle used to travel 70km. c The two lines cross at one point. What does this indicate? 9 Arun and Marcus are walking along the same path but in different directions. The graph shows how far they are from home. a How far from home is each person when they start walking? b How far does Arun walk in 5 hours? c Arun is ykm from home after xhours. Write an equation for the line that shows Arun’s journey. d Describe Marcus’s journey. Give as much detail as you can. e The lines cross. What does the point where they cross indicate? 0 0 5 10 15 20 10 20 30 40 50 Kilometres Litres 60 70 Car Van x y 0 0 10 20 30 40 1 2 3 4 5 Time (hours) Distance from home (km) 50 Arun Marcus x y Summary checklist I can interpret a real-life graph that shows a situation with several distinct sections or shows more than one component. Think like a mathematician 10 This graph shows the changing temperatures of two liquids. a Describe how the temperature of each liquid changes. Give as much detail as you can. b When are the two liquids at the same temperature? 0 0 10 20 30 40 35 2 3 6 8 Time (minutes) Temperature (°C) 15 5 25 1 4 5 7 Liquid A Liquid B x y We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
255 11 Graphs Check your progress 1 The cost of a holiday is $200 for travel plus $150 per night for a hotel. a Work out the cost of a 7-night holiday. b A holiday that lasts n nights costs $c. Write a function for c in terms of n. 2 The perimeter of this shape is p cm. a Write a function for p in terms of w. b Copy and complete this table of values. w 0 1 2 3 4 5 6 p c Use the table to draw a graph to show the perimeter for different values of w. 3 Here is a function: y=3x+6 a Copy and complete this table of values. x −3 −2 −1 0 1 2 3 y=3x+6 b Use the table to draw a graph of y=3x+6 c Find the gradient and the y-intercept of the line y=3x+6 d Write the equation of a line parallel to y=3x+6 that passes through the origin. 4 Here is the equation of a line: y=12−2x a Where does the line cross the y-axis? b What is the gradient of the line? 5 The depth of water in two flasks is changing. This graph shows the changes. a Describe how the depth in flask 1 is changing. b When do the two flasks have the same depth of water? c The depth of water in flask 2 is dcm after t minutes. Choose the correct equation of the line for flask 2. d=5t+30 d=6t+30 d=30−5t d=30−6t wcm wcm 3cm 4 cm 3 cm 0 0 10 20 30 40 1 2 3 4 5 Time (minutes) Depth (cm) 6 Flask 1 Flask 2 x y We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
256 Straight line mix-up Here are nine function cards: x y 2 1 0 –1 –4 –1–2–3 21 –2 –3 –4 3 y = −2x − 4 x y y = x − 1 2 1 0 –1 –1–2–3 2 31 –2 –3 –4 3 x y y = 2x 2 1 0 –1 –1–2–3 2 31 –2 3 4 5 y=4−2x y=3x−4 y=x You may wish to sketch the graphs, work out the equations, and work out a table of values for each card. Here are six property cards: The gradient is positive The gradient is negative The y-intercept is negative The line passes through (0, 0) The line is parallel to y=x+2 The y-intercept is positive Can you find a way to arrange the property cards and the function cards in a grid, so that each function satisfies the property at the top of its column and at the left of its row? For example, in the grid below, y=3x−4 has a positive gradient and a negative y-intercept. The gradient is positive The y-intercept is negative y=3x−4 Is there more than one way to arrange the cards? x −1 0 1 2 y 3 4 5 6 x −1 0 1 2 y 2 0 −2 −4 x −1 0 1 2 y 0 3 6 9 Project 4 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Getting started 1 For each of these shapes, write the ratio of green squares to blue squares. Write each ratio in its simplest form. a b 2 Write each of these ratios in its simplest form. a 2 : 4 b 18 :6 c 6: 9 d 32:24 3 Share these amounts between Tim and Chan in the ratios given. a $18 in the ratio 1:2 b $25 in the ratio 2 :3 4 Write the missing numbers in these conversions. a 4m= cm b 6.5cm= mm c 5 t= kg d 0.8kg= g e 2.3 l= ml f 0.75km= m 12 Ratio and proportion Ratios are used to compare two or more numbers or quantities. Every day, ratios are used in all sorts of ways to work out all sorts of things. For example, builders use ratios to work out the amounts of ingredients needed to mix together, to make concrete or mortar. The ratio and ingredients vary, depending on what the builder will do with the concrete or mortar. 257 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
258 12 Ratio and proportion To make the mortar for laying brickwork or block pavements, a builder would use cement and sand in the ratio 1:4. This means that every 1kg of cement must be mixed with 4kg of sand. Builders often use a shovel or bucket to measure their ingredients. For this mortar, they would need one shovel (or bucket) of cement for every four shovels (or buckets) of sand. To make a medium-strength concrete for a floor, a builder would use three ingredients: cement, sand and gravel, mixed in the ratio 1 :2 : 4. This means that every 1kg of cement must be mixed with 2kg of sand and 4kg of gravel. It is important that a builder uses the correct ratio of ingredients for each job, otherwise walls may fall down or floors may crack. Tip When the units are the same, you do not need to write the units with the numbers. In this section you will … • simplify and compare ratios. 12.1 Simplifying ratios A ratio is a way of comparing two or more quantities. In this pastry recipe, the ratio of flour to butter is 0.5kg:250 g. Before you simplify a ratio, you must write all quantities in the same units. 0.5kg:250 g is the same as 500 g : 250g, which you write as 500:250. You can now simplify this ratio by dividing both numbers by the highest common factor. In this case the highest common factor is 250. Divide both numbers by 250 to simplify the ratio to 2 :1. If you cannot work out the highest common factor of the numbers in a ratio, you can simplify the ratio in stages. Divide the numbers in the ratio by common factors until you cannot divide any more. In the example above you could start by: • dividing by 10 • then dividing by 5 • then dividing by 5 again • giving you the same answer of 2 :1. Key words adapt common factor highest common factor ratio simplify Pastry recipe 0.5kg flour 250g butter water to mix 500 : 250 2 : 1 ÷250 ÷250 500 : 250 50 : 25 10 : 5 2 : 1 ÷10 ÷5 ÷5 ÷10 ÷5 ÷5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
259 12.1 Simplifying ratios Worked example 12.1 Simplify these ratios. a 12 :20 b 12:30:24 c 2m: 50cm Answer a 12 : 20 3 : 5 ÷4 ÷4 The highest common factor of 12 and 20 is 4, so divide both numbers by 4. b 12 : 30 : 24 2 : 5 : 4 ÷6 ÷6 ÷6 The highest common factor of 12, 30 and 24 is 6, so divide all three numbers by 6. c 2 m : 50 cm 200 : 50 4 : 1 ÷50 ÷50 First, change 2 metres into 200 centimetres. The highest common factor of 200 and 50 is 50, so divide both numbers by 50. Exercise 12.1 1 Simplify these ratios. a 2: 10 b 3 : 18 c 5:25 d 30 : 5 e 36 :12 f 180: 20 g 4: 6 h 9: 15 i 10 :35 j 75 : 10 k 72:20 l 140 : 112 2 Simplify these ratios. a 5: 10 : 15 b 8 : 10 : 12 c 20:15:25 d 18 :15 : 3 e 27: 9: 45 f 72: 16:32 3 This is part of Ben’s classwork. a Explain the mistake that Ben has made. b Work out the correct answer. Question Simplify the ratio 6 :12 : 3 Answer 6÷6=1 and 12÷6=2 So the ratio is 1: 2 : 3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
260 12 Ratio and proportion 5 Simplify these ratios. a 500m: 1km b 36 seconds: 1 minute c 800ml:2.4 l d 1.6kg: 800 g e 3 cm:6mm f 2 days: 18 hours g 2 hours: 48 minutes h 8 months:1 year 6 Zara uses this recipe for orange preserve. The ratio of oranges to sugar is 2:1. Is Zara correct? Explain your answer. Think like a mathematician 4 Arun and Sofia compare methods to simplify the ratio 4mm:6cm. My first step is to change 4 mm into 0.4 cm. My first step is to change 6 cm into 60 mm. a Who do you think has the better first step? Explain why. b Discuss your answer with other learners in the class. Tip Remember that both quantities must be in the same units before you simplify. Orange preserve 750g oranges 1.5kg sugar juice of one lemon Activity 12.1 On a piece of paper, write two ratios similar to those in Question 5 and write two ratios similar to those in Question 7. Make sure each ratio can be simplified. Write the answers on a separate piece of paper. Exchange questions with a partner. Work out the answers to your partner’s questions. Exchange back and mark each other’s work. Discuss any mistakes that have been made. 7 Simplify these ratios. a 600 m: 1km: 20m b 75cm:1m:1.5m c 300ml: 2.1l: 900ml d 3.2kg: 1600g : 0.8kg e $1.08: 90 cents: $9 f 4 cm:8mm:0.2m We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
261 12.1 Simplifying ratios 8 Marcus and Sofia are mixing paint. They mix 250ml of white paint with 750ml of red paint and 1.2 litres of yellow paint. The ratio of white to red to yellow paint is 1:3:5. The ratio of white to red to yellow paint is 25:75:12. Is either of them correct? Explain your answer. 9 Preety answers this question. Five cups hold 1.2 litres and three mugs hold 900ml. Which holds more liquid, one cup or one mug? This is what she writes. 1.2 litres=1200ml Ratios: 5 cups : 1200ml 1 cup : 240ml ÷5 ÷5 3 mugs : 900ml 1 mug : 300ml ÷3 ÷3 A mug holds 60ml more than a cup. Use Preety’s method to answer these questions. a Four bags of sugar have a mass of 1.3kg and three bags of flour have a mass of 960 g. Which has a greater mass, one bag of sugar or one bag of flour? b Eight pens have a total length of 1.2m. Five pencils have a total length of 90 cm. Which is longer, a pen or a pencil? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
262 12 Ratio and proportion 11 Use Jed’s method to simplify these ratios. a 0.5:2 b 1.5:3 c 1.2: 2.4 d 3.6:0.6 e 7.5:1.5 f 2.4: 4 g 1.8 : 6.3 h 2.1 :0.7 :1.4 12 Oditi goes for a run three times a week. Her notebook shows the time she took for each run one week. a Oditi thinks the ratio of her times for Monday to Wednesday to Friday is 1:2 :3. Without doing any calculations, explain how you know Oditi is wrong. b Oditi’s mum uses this method to work out the ratio of Oditi’s times. Monday : Wednesday : Friday 1 hour 40 mins : 50 mins : 2.5 hours 1.4 : 0.5 : 2.5 ×10 14 : 5 : 25 ÷5 14 : 1 : 5 Explain the mistakes Oditi’s mum has made. c Work out the correct ratio of Oditi’s times. Show all your working. Think like a mathematician 10 Work with a partner or in a small group to answer this question. This is part of Jed’s homework. a Explain why Jed’s first step is to multiply both of the numbers in the ratio by 10. b What are the advantages of Jed’s method? Can you think of any disadvantages? c How could you adapt Jed’s method to simplify the ratio 0.03:0.15? d Discuss your answers with other groups in the class. Question Simplify these ratios. a 1.5 : 2 b 0.8 : 3.6 Answer a 1.5 : 2 15 : 20 3 : 4 ×10 ×10 ÷5 ÷5 b 0.8 : 3.6 8 : 36 2 : 9 ×10 ×10 ÷4 ÷4 Monday 1 hour 40 mins Wednesday 50 mins Friday 2½ hours We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
263 12.2 Sharing in a ratio In this exercise you have answered questions on: • simplifying ratios with two or three numbers • simplifying ratios with quantities in different units • simplifying ratios with decimal numbers. a Which questions have you found i the easiest ii the hardest? Explain why. b How can you improve your skills in simplifying ratios? Summary checklist I can simplify ratios when the quantities have different units. I can compare ratios when the quantities have different units. I can simplify a ratio with more than two parts. Sometimes you need to share an amount in a given ratio. For example, Zara, Sofia and Marcus buy a painting for $600. Zara pays $200, Sofia pays $300 and Marcus pays $100. You can write the amounts they pay as a ratio like this: Zara : Sofia : Marcus 200 : 300 : 100 Simplify the ratio by dividing by 100 to give: 2 : 3 : 1 You can see that Zara paid twice as much as Marcus, and Sofia paid three times as much as Marcus. There are now 6 equal parts in total (2+3+1=6). When they sell the painting, they need to share the six parts of the money fairly between them. They can do this by using the same ratio of 2 : 3:1. 12.2 Sharing in a ratio In this section you will … • divide an amount into two or more parts in a given ratio. Key words profit share We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
12 Ratio and proportion Exercise 12.2 1 Copy and complete the workings to share $80 between So, Luana and Kyra in the ratio 3: 2 : 5. Total number of parts: 3+2+5= Value of one part: $80÷ = So gets: 3× = Luana gets: 2× = Kyra gets: 5× = 2 Share these amounts between Mia, Beth and Fen in the given ratios. a $90 in the ratio 1:2:3 b $225 in the ratio 2: 3:4 c $432 in the ratio 3:5:1 d $396 in the ratio 4:2:5 Worked example 12.2 Share $840 between Alan, Bob and Chris in the ratio 2 :3: 1 Answer 2+3+1=6 First, add the numbers in the ratio to find the total number of parts. 840÷6=140 1 part=$140 Then divide the amount to be shared by the total number of parts to find the value of one part. Alan gets 2×140=$280 Bob gets 3×140=$420 Chris gets 1×140=$140 Finally, work out the value of each share using multiplication. Make sure you write the name of the person with each amount. Follow these steps to share an amount in a given ratio. 1 Add all the numbers in the ratio to find the total number of parts. 2 Divide the amount to be shared by the total number of parts to find the value of one part. 3 Use multiplication to work out the value of each share. 264 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
265 12.2 Sharing in a ratio Think like a mathematician 3 Look again at your answers to questions 1 and 2. a Think about a method you can use to check you have shared each amount correctly. b Discuss your method with a partner. Do they have the same method? If they have a different method, which method do you think is better? Explain your answer. 4 Dave, Ella and Jia share their electricity bills in the ratio 3 : 4 : 5. a How much does each of them pay when their electricity bill is i $168 ii $192 iii $234? b Show how to check your answers to part a. 5 A choir is made up of men, women and children in the ratio 5 : 7 :3 Altogether, there are 285 members of the choir. a How many members of the choir are i men ii women iii children? b How many more women than men are there in the choir? c How many more men than children are there in the choir? 6 A box of fruits contains oranges, apples and peaches in the ratio 4 : 2:3. The box contains 72 fruits altogether. a How many fruits in the box are i oranges ii apples iii peaches? b The ratio of the number of oranges, apples and peaches is changed to 3 : 1 : 4. There are still 72 fruits in the box. How many fruits in this box are i oranges ii apples iii peaches? 7 Aden, Eli, Lily and Ziva run their own business. They share the money they earn from a project in the ratio of the number of hours they put into the project. On the right is the time-sheet for one of their projects. How much does each of them earn from this project? Project earnings: $450 Time spent working on project: Aden: 6 hours Eli: 4 hours Lily: 3 hours Ziva: 5 hours We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
266 12 Ratio and proportion 8 Here is a set of ratio cards. $50 $14 $60 $55 $15 $11 $28 $66 Share $150 … … in the ratio 2:3:1 … in the ratio 2:6:1 … in the ratio 3:1:4 … in the ratio 1:5:6 Share $120 … Share $132 … Share $126 … $84 $75 $25 $45 Sort the cards into their correct groups. Each group must have one pink, one yellow and three blue cards. 9 The angles in a triangle are in the ratio 2 : 3:5. Work out the size of the angles. Tip The angles in a triangle add up to degrees. Think like a mathematician 10 In pairs or groups, look at the following question and answer. This is part of Zara’s homework. Question A grandmother leaves $2520 in her will, to be shared among her grandchildren in the ratio of their ages. The grandchildren are 6, 9 and 15 years old. How much does each child receive? Answer Ratio for grandchildren is 6 : 9 :15 Total number of parts=6+9+15=30 Value of one part=$2520÷30=84 6-year-old child gets: $84 × 6=$504; 9-year-old child gets $84 × 9=$756, 15-year-old gets: $84 × 15=$1260 Check: $504+$756+$1260=$2520 ✓ We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
267 12.2 Sharing in a ratio Continued Zara has got the answer correct. However, some of her calculations were difficult and she had to use a calculator. a How can she make the calculations easier? b Rewrite the solution for her. Do not use a calculator. c Compare you answers to parts a and b with other groups in your class. Did you come up with the same idea or different ideas? d What extra step could she add to simplify her solution? 11 Every year, on his birthday, David shares $300 among his children in the ratio of their ages. This year the children are aged 4, 9 and 11. Show that, in two years time, the oldest child will receive $7.50 less than he receives this year. 12 Zhi, Zhen and Lin buy a house for $180 000. Zhi pays $60 000, Zhen pays $90 000 and Lin pays the rest. Five years later, they sell the house for $228 000. They share the money in the same ratio that they bought the house. Lin thinks he will make $9000 profit on the sale of the house. Is Lin correct? Show all your working. 13 Akello, Bishara and Cora are going to share $960, either in the ratio of their ages or in the ratio of their heights. Akello is 22 years old and has a height of 168 cm. Bishara is 25 years old and has a height of 152 cm. Cora is 33 years old and has a height of 160 cm. a Without working out the answer, which ratio do you think will be better for Bishara, age or height? Explain your decision. b Work out whether your decision was correct. If it was not, explain why you think you made the wrong decision. Summary checklist I can share an amount in a given ratio with two or more parts. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
268 12 Ratio and proportion You can see ratios in a variety of situations, such as mixing ingredients in a recipe or sharing an amount among several people. Ratios can also be used to make comparisons. For example, suppose you wanted to compare two mixes of paint. Pink paint is made from red and white paint in a certain ratio (red:white). If two shades of pink paint have been mixed from red and white paint, how do you decide which shade is darker? The shade which is darker is the shade with the greater proportion of red paint. You can change the ratios into fractions, decimals or percentages to compare the proportions of red paint in each shade. 12.3 Ratio and direct proportion In this section you will … • use the relationship between ratio and direct proportion. Key words comparison justify proportion shades Pink Pink Worked example 12.3 Pablo mixes two shades of pink paint in the ratios of red:white paint shown below. Perfect pink 3:4 Rose pink 2:3 a What fraction of each shade of pink paint is red? b Which shade is darker? Justify your choice. Answer a Perfect pink: 3+4=7 Fraction red=3 7 Add together the numbers in the ratio to find the total number of parts. Three parts out of seven are red. Four out of seven are white. Rose pink 2+3=5 Fraction red=2 5 Add together the numbers in the ratio to find the total number of parts. Two parts out of five are red. Three out of five are white. b Perfect pink: 3 5 7 5 15 35 × × = Rose pink: 2 7 5 7 14 35 × × = Perfect pink is darker, because it contains more parts of red. For each shade, write the fraction that is red as an equivalent fraction with common denominator 35. For perfect pink, 15 parts out of 35 parts are red. For rose pink, 14 parts out of 35 parts are red. There are more parts of red in perfect pink, so this shade is darker. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
269 12.3 Ratio and direct proportion Exercise 12.3 1 Copy and complete the workings to change each ratio into a fraction. a A bag of nuts contains cashew nuts and peanuts in the ratio 2 :7. What fraction of the nuts are i cashew nuts ii peanuts? Total number of parts=2+7= i fraction that are cashew nuts= 2 ii fraction that are peanuts= 7 b A box of toys has plastic and paper toys in the ratio 3 : 5. What fraction of the toys are i plastic ii paper? Total number of parts=3+5= i fraction that are plastic= 3 ii fraction that are paper= 5 c A basket of fruit has apples and bananas in the ratio 3 :1. What fraction of the fruit are i apples ii bananas? Total number of parts=3+1= i fraction that are apples= ii fraction that are bananas= We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
12 Ratio and proportion 270 3 A tin of biscuits contains coconut and ginger biscuits in the ratio 3 :7. The tin contains 50 biscuits. a What fraction of the biscuits in the tin are coconut biscuits? b How many coconut biscuits are in the tin? 4 A school tennis club has 35 members. The ratio of boys to girls is 4 :3. a What fraction of the club members are girls? b How many girls are in the club? Think like a mathematician 2 Tio and Kai work out the answer to this question. A school choir is made up of girls and boys in the ratio 2:1. There are 36 students in the choir altogether. How many of the students are girls? Tio uses the ‘sharing in a ratio’ method. Kai uses the ‘fraction of an amount’ method. Tio Total number of parts=2+1=3 Value of one part=36÷3=12 Number of girls=2 × 12=24 Kai Fraction of the choir that are girls= 2 2+1 = 2 3 Number of girls= 2 3 × 36 = 36÷3 × 2=24 Work with a partner or in a small group to discuss these questions. a Compare Tio and Kai’s methods. How are they similar? How are they different? b Whose method do you prefer? Explain why. Activity 12.3 On a piece of paper, write two questions similar to questions 3 and 4 in this exercise. Write the answers on a separate piece of paper. Make sure the questions can be answered without using a calculator. Exchange questions with a partner. Work out the answers to your partner’s questions. Exchange back and mark each other’s work. Discuss any mistakes that have been made. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
271 12.3 Ratio and direct proportion 6 The ratio of boys to girls in class 8C is 5 :7. Which of these cards shows the number of learners that could be in class 8C? A B C D E Justify your choice. 7 The ratio of men to women in a book club is 3 :5. The number of adults in the book club is greater than 20 but fewer than 30. How many adults are in the book club? 8 This is part of Jan’s classwork. Question A bag contains blue and yellow cubes. 3 11 of the cubes are blue. What is the ratio of blue to yellow cubes? Answer 3 11 are blue, so 1 – 3 11=11 11 – 3 11= 8 11 are yellow. So the ratio of blue to yellow is 3 11 : 8 11=3 : 8. 25 28 32 36 38 Think like a mathematician 5 Work with a partner or in a small group to discuss this question. The ratio of red to blue counters in a bag is 5:4. Read what Sofia and Zara say. It is possible that there are 62 counters in the bag. It is possible that there are 72 counters in the bag. Is either of them correct? Explain how you know. Discuss your answers with other groups in the class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
12 Ratio and proportion 272 Use Jan’s method to work out the following. a A bag contains green and red counters. 2 3 of the counters are green. What is the ratio of green to red counters? b A box of books contains history and science books. 3 7 of the books are science books. What is the ratio of history to science books? c A café sells sandwiches and cakes. 4 9 of the items they sell are cakes. What is the ratio of sandwiches to cakes that the café sells? 9 Shani mixes two shades of blue paint in the following ratios of blue :white. Sky blue 3 : 2 Sea blue 7:3 a What fraction of each shade of blue paint is white? b Which shade of blue paint is lighter? Show all your working. Justify your choice. 10 Angelica mixes a fruit drink using mango juice and orange juice in the ratio 3:5. Sanjay mixes a fruit drink using mango juice and orange juice in the ratio 5:11. a What fraction of each fruit drink is orange juice? b Whose fruit drink, Angelica’s or Sanjay’s, has the higher proportion of orange juice? Show all your working. Justify your choice. 11 In the Seals swimming club there are 13 girls and 17 boys. a What fraction of the children are boys? In the Sharks swimming club there are 17 girls and 23 boys. b What fraction of the children are boys? c Which swimming club has the higher proportion of boys? Show all your working. Justify your choice. Blue Blue Tip The paint which is lighter has a greater proportion of white. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
12.3 Ratio and direct proportion 13 Liam and Hannah collect coins and stamps. Liam has 20 coins and 320 stamps. Hannah has 15 coins and 270 stamps. Use your favourite method from Question 12 to decide who has the greater proportion of stamps. Justify your choice. 14 Two jewellery shops sell watches and rings. Bright Jewellery has 12 watches and 180 rings for sale. Mega-Jewellery has 30 watches and 438 rings for sale. Which shop has the greater proportion of watches? Justify your choice. Think like a mathematician 12 a Work with a partner or in a small group to discuss the different methods you could use to answer this question. Lin has black and white counters in the ratio 40:840 Ian has black and white counters in the ratio 25:535 Who has the greater proportion of black counters, Lin or Ian? b Compare and discuss the different methods with other groups in the class. What do you think is the best method? Explain why. Summary checklist I can use the relationship between ratio and direct proportion. 273 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Check your Progress Nimrah thinks of a number, n . Write an expression for the number Nimrah gets each time. a She multiplies the number by 4. b She subtracts 6 from the number. c She multiplies the number by 3 d She divides the number by 6 then adds 5. then subtracts 1. Copy the number line and show the inequality on the number line. Write down the inequality that this number line shows. Work out the value of each expression. Loli lives with 3 friends. They share the electricity bill equally between the four of them. Write a formula to work out the amount they each pay, in: i words ii letters Use your formula in part a ii to work out the amount they each pay when the electricity bill is $96. 6 Simplify these expressions. a n + n + n b 3c + 5c c 9x − x 7 Simplify these expressions by collecting like terms. a 5c + 6c + 2d b 6c + 5k + 5c + k c 3xy + 5yz − 2xy + 3yz 8 Work these out. a 3 + (x × 2) b 6(3 − w) c 4(3x + 2) d 3(7 − 4v) 9 Solve each of these equations and check your answers. a n + 3 = 8 b m − 4 = 12 c 3p = 24 d x = 3 5 10 Shen has set a puzzle. Write an equation for the puzzle. Solve the equation to find the value of the unknown number. 274 12 Ratio and proportion Check your progress 1 Simplify these ratios. a 6cm:5mm b 12 seconds:1 minute c 400ml:1.6l 2 Five bags of peanuts have a mass of 1.375kg and two bags of walnuts have a mass of 540 g. Which has a greater mass, one bag of peanuts or one bag of walnuts? 3 A running club is made up of men, women and children in the ratio 8 :5:7. Altogether, there are 260 members of the running club. How many members of the running club are a men b women c children? 4 A school quiz club has 45 members. The ratio of boys to girls is 4 :5. a What fraction of the club members are boys? b How many boys are in the club? 5 Ellen mixes two shades of grey paint in the following ratios of black:white. Silver grey 2:5 Stone grey 3:8 a What fraction of each shade of grey paint is white? b Which shade of grey paint is lighter? Show all your working. Justify your choice. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
275 Getting started 1 This is a spinner. Each colour is equally likely. a Find the probability of green. b Find the probability of blue or yellow. 2 An unbiased 6-sided dice is thrown. Work out the probability of getting a 3 b 6 c an even number d less than 5. 3 Tomorrow at 11:00 it will be sunny, cloudy or wet. The probability it will be sunny is 25% and the probability it will be cloudy is 40%. Find the probability it will be wet. 4 A large number of drawing pins are dropped on the floor. 87 land point up and 135 land point down. Work out the experimental probability of landing point up. White Red Yellow Blue Green 13 Probability Do you know the game ‘rock, paper, scissors’? It is a very old game and is known by other names as well. Two people simultaneously show either a fist (rock), the first two fingers pointing forwards (scissors) or an open hand (paper). Scissors beats paper, paper beats rock and rock beats scissors. This is because scissors cut paper, paper wraps rock and rock blunts scissors. If both players choose the same thing it is a draw (neither wins) and they play again. scissors beats paper paper beats rock rock beats scissors We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
276 13 Probability This may seem a trivial game but in 2005 the Maspro Denkoh electronics corporation used it to decide whether to give the contract to auction its $20 million collection of paintings to Sotheby’s or to Christie’s auction houses. Christie’s won with paper, after taking the advice of Flora and Alice, the 11-year-old daughters of one of the directors of the company. Their argument was that for beginners, rock seems strongest, so they tend to start with that. Playing against a beginner, you should start with paper. This game illustrates two methods of finding probabilities. One method is to say that each different play – rock, scissors, paper – is equally likely. If the three outcomes are equally likely, each one has a probability of 1 3 . Flora and Alice realised that, for less experienced players, the outcomes are not equally likely. The probability of starting with rock is more than 1 3 . 13.1 Calculating probabilities In this section you will … • find the probability of complementary events • use lists and diagrams to show equally likely outcomes • use lists and diagrams of outcomes to calculate probabilities. This is a spinner. The probability that it points to red is 0.2. The probability that it points to blue is 0.15. We can write those probabilities as P(red)=0.2 and P(blue)=0.15 The sum of the probabilities for all six colours is 1. This means the probability the spinner does not point to red, P(not red)=1−0.2=0.8 The probability the spinner does not point to blue, P(not blue)=1−0.15=0.85 Getting blue and not getting blue are complementary events. One of them must happen and they cannot both happen. If A is an event and A′ is the complementary event, then P(A′ )=1−P(A) Key word complementary event Red Blue We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
277 13.1 Calculating probabilities Worked example 13.1b Two unbiased 6-sided dice are thrown. Find the probability of getting a the same number on both dice b a total of 6 c a total of 9 or more. Answer a The diagram shows all possible outcomes. There are 36 outcomes altogether. The loop shows the outcomes with the same number: (1, 1), (2, 2) and so on. There are 6 of them. The probability is 6 36 which is equivalent to 1 6 × × × × × × 1 2 3 4 5 6 × × × × × × × × × × × × × × × × × × × × × × × × × 1 2 Second dice First dice 3 4 5 6 × × × × × Worked example 13.1a The probability that it will be sunny tomorrow is 40%. The probability it will not rain tomorrow is 95%. Find the probability that tomorrow a will not be sunny b it will rain. Answer a P(not sunny)=1−P(sunny)=100%−40%=60% b P(rain)=1−P(not rain)=100%−95%=5% We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
278 13 Probability Continued b This table shows the total for each outcome. Five outcomes give a total of 6 (shown by a blue loop). The probability is 5 36 c Using the same table as for part b, ten outcomes give a total of 9, 10, 11 or 12 (shown by the red loop). The probability is 10 36 5 18 = Second dice 6 7 8 9 10 11 12 5 6 7 8 9 10 11 4 5 6 7 8 9 10 3 4 5 6 7 8 9 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 First dice For example, 5 on the first dice and 3 on the second gives a total of 8. Exercise 13.1 1 The probability that a football team will win a match is 0.3. The probability that the team will draw is 0.1. Work out the probability that the team will a not win b not draw c lose d not lose. 2 Tomorrow must be hotter, colder or the same temperature as today. The probability it will be hotter is 55%. The probability it will be colder is 25%. Work out the probability that it will a not be hotter b not be colder c not be the same temperature. 3 A spinner has five colours on it. The probability it shows green is 0.32. The probability it shows purple is 0.17. Find the probability that the colour is a not green b not purple. Tip All dice in this exercise are unbiased, 6-sided dice. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
279 13.1 Calculating probabilities 4 There are lots of coloured toys in a box. Here are the percentages of some of the colours. Colour yellow orange red green Percentage 15% 25% 30% 10% a Why do the percentages add up to less than 100%? A child takes a toy at random. b Find the probability that the toy is i not orange ii not green iii not red iv not yellow. 5 Two dice are thrown. Find the probability that a both dice show 5 b one dice shows a 5 and the other does not c neither dice shows a 5. 6 Two dice are thrown. The numbers are added together. a Draw a table to show all the possible outcomes. b Find the probability that the total is i 3 ii 7 iii 12 iv 9 c Copy and complete this table of probabilities. Total 2 3 4 5 6 7 8 9 10 11 12 Probability 7 Two dice are thrown. The numbers are added together. a Find the probability that the total is i 5 or less ii more than 5 iii 10 or more iv less than 10 v a prime number. b Find an event with a probability of 7 36 c Give your answer to part b to a partner to check it is correct. 8 A fair coin and a fair dice are thrown. This table shows the possible outcomes. Dice 1 2 3 4 5 6 Coin H H1 T T3 a Copy and complete the table. b How many outcomes are there? Are they all equally likely? Tip Use the diagram from part a of Worked Example 13.1b. Tip T3 stands for a tail on the coin and 3 on the dice. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
280 13 Probability c Find the probability of i 6 and a tail ii 4 and a head iii a head and an even number iv a tail and a number less than 3. d Find the probability of each of the events in part c not happening. e Describe an event with a probability of 5 12 f Give your answer to part e to a partner to check. 9 Here are two spinners. a The two spinners are spun. Draw a diagram to show all the outcomes. b Work out the probability that i both spinners show a 1 ii neither spinner shows a 1 iii both spinners show the same number iv the spinners do not show the same number. c The two scores are added together. Draw a table to show the possible totals. d Find the probability that the total is i 4 ii 5 iii not 7 iv a multiple of 3 v a factor of 12. e Now the scores on the spinners are multiplied. Draw a table to show the possible products. f Find the probability of each of the different possible products. g Find the probability that the product is i 6 or more ii less than 6 iii an odd number iv an even number. 10 a Two fair coins are flipped. Copy and complete this table to show the outcomes. Second coin H T First coin H HT T 4 1 23 3 1 2 Tip Use a table like the one in part b of Worked example 13.1b. Tip The product is the result of multiplying two numbers. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
281 13.1 Calculating probabilities b Read what Arun says: Explain why Arun is not correct. c Find the probability of i 2 heads ii 2 tails iii a head and a tail. d Another way to show the outcomes when two fair coins are thrown is a tree diagram. Copy the tree diagram and fill in the missing outcomes. e Explain how the table in part a and the tree diagram in part d show the same outcomes. f Three fair coins are thrown. One possible outcome is HHH, a head on all three coins. List all the possible outcomes in this way. g Draw a tree diagram to show the results of throwing three fair coins. Use it to check your answer to part f. h When three fair coins are thrown, find the probability of i 3 heads ii 3 tails iii not getting 3 heads iv 2 heads and 1 tail v 1 head and 2 tails. H H ... ... ... T HT H T First coin Second coin Outcome T Think like a mathematician 11 Investigate the possible outcomes when 4 fair coins are thrown. You should find all the possible outcomes and find probabilities of different events. Use your experience from Question 10 to help you. 12 Zara has three cards with numbers on them. 2 4 5 She puts the cards side by side in a random order to make a 3-digit number. a List all the possible numbers. Make sure you have found them all. b Find the probability that the number formed is i an odd number ii an even number iii more than 400. Zara adds an extra card. Now she has four cards. 2 4 5 8 Zara takes two cards at random and places them side by side to make a 2-digit number. c List all the possible numbers she can make. Make you sure you have found them all. d Find the probability that the 2-digit number i is 48 ii is not 48 iii is an odd number iv is an even number v includes the digit 2. When you throw two coins there are three outcomes. They are 2 heads, 2 tails or a head and a tail. So the probability of 2 heads is 1 3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
282 13 Probability In this exercise you have used different methods to find outcomes. What are they? Which do you prefer and why? Summary checklist I can find the probability of a complementary event. I can use a chart, a table or a list to find all possible outcomes. I can use lists and diagrams of outcomes to calculate probabilities. Now Zara takes three cards at random and places them side by side to make a 3-digit number. e List all the possible numbers she can make. f Find the probability that the 3-digit number is i an odd number ii an even number iii less than 500. 13.2 Experimental and theoretical probabilities You can use equally likely outcomes to calculate probabilities. When this is not possible you can do an experiment. A spreadsheet is used to simulate throwing a dice 200 times. Here are the results of the experiment. Score 1 2 3 4 5 6 Frequency 30 36 37 33 35 29 From the information in the table, we can work out the experimental probabilities: • The experimental probability of 1 is 30 200 =0.15 • The experimental probability of 2 is 36 200 =0.18 • The experimental probability of an even number is 36 33 29 200 98 200 =0.49 In this section you will … • calculate experimental probabilities and compare them to theoretical probabilities. Key words experimental probability theoretical probability We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
283 13.2 Experimental and theoretical probabilities We know that each number is equally likely with a fair dice so we can also calculate the theoretical probabilities: • The theoretical probability of 1 is 1 6 =0.167 to 3 d.p. • The theoretical probability of 2 is 1 6 =0.167 to 3 d.p. • The theoretical probability of an even number is 3 6 1 2 = =0.5 The experimental probabilities and the theoretical probabilities are very similar. This shows that the spreadsheet simulation is reliable. Worked example 13.2 Read what Marcus says. Event 2 heads 2 tails 1 head and 1 tail Frequency 17 14 19 a Calculate the experimental probability of each outcome. b Calculate the theoretical probability of each outcome. c Marcus’s teacher thinks Marcus has made up his results. What do you think? Give a reason for your answer. Answer a The experimental probability of 2 heads is 17 50 =0.34 The experimental probability of 2 tails is 14 50 =0.28 The experimental probability of 1 head and 1 tail is 19 50 =0.38 b There are four equally likely outcomes: HH, HT, TH, TT The theoretical probability of 2 heads is 1 4 =0.25 The theoretical probability of 2 tails is also 1 4 =0.25 There are two ways to get 1 head and 1 tail: HT or TH The theoretical probability is 2 4 1 2 = =0.5 c The experimental and theoretical probabilities are not similar. It looks as if Marcus may have made up his results. I have thrown 2 coins 50 times. The results are in this table. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
284 13 Probability Exercise 13.2 1 A learner throws a coin 50 times. This table shows the results. T H T T T H H T H T H T T H H T H H H T H H T H H H T H T H T T T T T H T T T T T H H T T T H T H T a Use the first row of the table to calculate the experimental probability of a head based on the first 10 throws. b Use the first two rows of the table to calculate the experimental probability of a head based on 20 throws. c In the same way, find the experimental probability of a head based on i 30 throws ii 40 throws iii 50 throws. d Compare the experimental probabilities you have found so far with the theoretical probability of a head. The learner throws the coin another 50 times. Here are the results. H H H H T T T H T H T T T T T H H T T T T T H T H H T T T T H T H H H H T H H T H T T T H H H H T T e Use the two sets of results to find the experimental probability of a head based on 100 throws. How close is it to the theoretical probability? 2 This spinner has 3 sectors. The probability of red, P(red)=0.6 The probability of white, P(white)=0.3 The probability of blue, P(blue)=0.1 Here are the results of 50 spins. R W R R B B B B R R W R W R W R R R R W R W R W R R R R R R R W R R R R R B B R R R R R W R R W R R Blue Red White We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
285 13.2 Experimental and theoretical probabilities a Use each row to find an experimental probability of red based on 10 spins. b Find two different sets of 25 spins and use them to find the experimental probability of red. c Use all 50 spins to find experimental probabilities of red, white and blue. d Here are the results of 800 spins. Colour red white blue Frequency 489 218 93 Use these results to find experimental probabilities for each colour. e Read what Marcus says: It is better to use a large number of spins to work out experimental probabilities. Do you agree? Give a reason for your answer. 3 This question is about throwing six dice together and seeing if there is at least one 6. Four learners each threw six dice together a number of times. Here are their results. Name Arun Sofia Marcus Zara Number of throws 10 20 40 50 Frequency of at least one 6 7 9 31 36 a Work out the experimental probability of at least one 6 for each learner. b Combine the four sets of results to get another experimental probability. c A computer simulated 500 throws. There was at least one six 333 times. Work out an experimental probability from this data. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
286 13 Probability d In fact, the theoretical probability of throwing at least one 6 is 0.6651. Compare the experimental probabilities with the theoretical probability. Activity 13.2 Work with another learner on this question. Each pair will need a dice. Design and carry out an experiment to answer this question: Is your dice unbiased? Before you start, you need to decide: • how many times to throw a dice • how to record your data • how to compare experimental probabilities and theoretical probabilities. Write your plan before you start. Give reasons for your conclusion based on your data. 4 Work with one or more other learners on this question. You learnt about the number π (pi) in Unit 8. It is the ratio of the circumference of a circle to its diameter. The value of π is a decimal that does not terminate and has no pattern to its digits. Here are the first 200 decimal places of π: 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 96 Look at this statement: All the digits from 0 to 9 are equally likely. a Devise and carry out an experiment to test this statement. Use experimental probabilities and compare them with theoretical probabilities. b Describe your experiment and your result. Give a reason for your conclusion. c Look at the results of another pair. How do they compare with yours? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
13.2 Experimental and theoretical probabilities 5 You need a spreadsheet to answer this question. You also need to know how to use it to generate random numbers. a Carry out a simulation to model throwing a coin 50 times. Find the experimental probability of throwing a head and compare it with the theoretical probability. b Repeat part a another 5 times. How much do the experimental probabilities vary? c You now have the results of 300 simulated throws. Use them all to find an experimental probability of throwing a head. d Experiment with larger numbers of throws, finding an experimental probability of throwing a head each time. Comment on your results. In some situations, you can find theoretical probabilities based on equal likelihood and you can also find experimental probabilities. What is the connection between the two? Summary checklist I can use the results of an experiment to find experimental probabilities and compare them to the theoretical probability. 287 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
288 13 Probability Check your progress 1 A spinner has a yellow section. The probability of landing on yellow is 0.27. Work out the probability of getting a different colour. 2 Zara writes the digits 3, 6 and 9 in a random order to make a 3-digit number. a List all the possible 3-digit numbers she could make. b Find the probability that Zara’s number is i less than 500 ii an odd number iii a multiple of 3. 3 An unbiased tetrahedral dice has four faces showing the numbers 1, 2, 3 and 4. a Two unbiased tetrahedral dice are thrown. Copy and complete this table to show the possible totals. First dice 4 3 2 5 1 1 2 3 4 Second dice b Work out the probability of i a total of 6 ii a total of less than 6 iii the same number on each dice. 4 Here are the results of a computer simulation of throwing a dice 40 times. 3 5 4 3 1 2 2 6 1 6 4 5 3 1 6 1 4 2 6 3 2 6 2 2 2 2 2 6 6 5 1 3 3 6 3 1 6 3 1 6 a Find the experimental probability of getting 3. b Compare the experimental and theoretical probabilities of getting a 3. 1 2 4 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
289 Project 5 High fives For this problem, you need to be able to generate random numbers between 1 and 5. You could do this by: • using the random number function on a calculator or spreadsheet • putting five counters in a bag and picking one out • making a five-sided spinner • rolling a ten-sided dice and subtracting 5 if you get an answer greater than 5. Imagine spinning two 1 to 5 spinners and writing down the higher of the two numbers. If you did this lots of times, how often would you expect to write each number? Let’s try an experiment to find out. Step 1: Generate some pairs of random numbers. Step 2: Write the larger number in each pair. For example, if you get the numbers 2 and 4, write 4; if you get 3 and 3, write 3. Step 3: Do this 100 times. (You may want to work in a group to do this, or use a spreadsheet to generate the numbers automatically.) Step 4: Display your results. You could use a bar chart to do this. Are your results what you expected? Sometimes, mathematicians predict the results of an experiment using theoretical distributions. Here is a sample space diagram that you could complete to identify the higher number for every possible combination: Spinner A 1 2 3 4 5 Spinner B 1 2 3 3 4 5 In 25 trials, how many times would you expect to write each number? How could you scale this up to work out how often each number would occur in 100 trials? Can you picture the sample space diagram if the spinners went from 1 to 6 instead of 1 to 5? How many times would you expect to write each number in this case? If your spinners went from 1 to 7, how many times would you expect to write each number? Can you work out how many times you would write the largest number if the spinners went from 1 to 10? Or from 1 to 100? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
290 Getting started 1 Use a protractor to a measure the size of this angle b draw an angle of 125°. 2 Work out the distance between these coordinates. a (9, 2) and (9, 11) b (3, 8) and (5, 8) 3 A point P has coordinates (6, 1). P is translated 3 squares right and 4 squares up to point P′. Work out the coordinates of P′. 4 Make a copy of this diagram. a Reflect shape A in the x-axis. Label the shape B. b Reflect shape A in the y-axis. Label the shape C. 5 Copy the diagram. Rotate the shape about the centre C a 90° anticlockwise b 180° 6 Copy this shape onto squared paper. Enlarge the shape using scale factor 2. C x y 2 1 0 –1 –4 –1–2–3 4321 –2 –3 –4 3 4 A 14 Position and transformation We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
291 14 Position and transformation In Stage 7 you learned how to transform 2D shapes by reflecting, translating or rotating them. Here is a summary of the key points: • The shape before any transformation is called the object. • The shape after the transformation is the image. • You need a mirror line to reflect a shape. • When you translate a shape, you move it a given distance, right or left and up or down. • When you rotate a shape, you turn it through a given number of degrees. You turn it about a fixed point, called the centre of rotation. You turn it either clockwise or anticlockwise. With any of these three transformations, only the position of the shape is changed. The shape and size of the shape are not changed. An object and its image are always identical. They are congruent. In stage 7 you also learned about enlargements. An enlargement of a shape is a copy of the object, but it is bigger. You can use a microscope to look at enlarged images of very small objects. In this picture, you can see a dust mite. These mites are about 0.04mm long so they cannot usually be seen without the use of a microscope. A typical mattress on a bed may have from 100 000 to 10 million mites inside it. This is not a very nice thought as you go to bed at night! We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
292 14 Position and transformation A bearing describes the direction of one object from another. It is an angle measured from north in a clockwise direction. A bearing can have any value from 0° to 360°. It is always written with three figures. A B 120° N A B 65° N In this diagram, the bearing In this diagram, the from A to B is 120°. bearing from A to B is 065°. In this section you will ... • use bearings as a measure of direction. 14.1 Bearings Key word bearing Worked example 14.1 The diagram shows three towns, A, B and C. a Write the bearing of B from A. b Write the bearing from A to C. c Write the bearing of B from C. Answer a Draw a north arrow from A, and a line joining A to B. Measure the angle from the north arrow clockwise to the line joining A to B. A B N 130° The bearing is 130°. b Draw a north arrow from A, and a line joining A to C. Measure the angle from the north arrow clockwise to the line joining A to C. A C N 210° The bearing is 210°. c Draw a north arrow from C, and a line joining C to B. Measure the angle from the north arrow clockwise to the line joining C to B. C B N 80° The bearing is 080°. C A B We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
293 14.1 Bearings Exercise 14.1 1 For each diagram, write the bearing of B from A. Use a protractor to measure the angle from north in a clockwise direction. a B A N b B A N c B A N d B A N 2 Draw diagrams similar to those in Question 1, to show these bearings of B from A. a 025° b 110° c 195° d 330° 3 This is part of Freya’s homework. Question Write the bearing of B from A in this diagram. Answer: The angle is 32°, so the bearing of B from A is 32°. B A N Is Freya correct? Explain your answer. Activity 14.1 a Draw four diagrams similar to those in Question 1, to show different bearings of B from A. On a different piece of paper, write the bearings you have drawn. b Exchange diagrams with a partner. Measure the bearings they have drawn. c Exchange back and check each other’s answers. Discuss any mistakes. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
294 14 Position and transformation 4 The diagram shows the positions of a shop and a school. School Shop N N a Write the bearing of the shop from the school. b Write the bearing of the school from the shop. Tip To find the bearing of the shop from the school (part a) you need to measure the angle at the school. To find the bearing of the school from the shop (part b) you need to measure the angle at the shop. Think like a mathematician 5 The diagram shows the position of a tree and a lake. Seren, Taylor and Ros are standing at the tree. Seren walks straight from the tree to the lake. a On what bearing must she walk? Taylor walks north from the tree. After a short distance she then walks to the lake. b Is the bearing she walks on to the lake, larger or smaller than the bearing from the tree to the lake? Explain your answer. Ros walks south from the tree. After a short distance she then walks to the lake. c Is the bearing she walks on to the lake, larger or smaller than the bearing from the tree to the lake? Explain your answer. d Discuss your answers to parts b and c with other learners in the class. What can you say about how bearings change as you move north or south from the original position before turning to walk towards another object? Tree Lake N N We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
295 14.1 Bearings 6 Arun goes for a walk. The diagram shows Arun’s initial position (A), a farm (F), a pond (P), a tree (T) and a bridge (B). Write the bearing Arun follows to walk from a A to F b F to P c P to T d T to B e B to A. N N N N N B A F P T Think like a mathematician 7 Work with a partner or in a small group to answer these questions. a For each diagram, write the bearing of Y from X and X from Y. i N N X Y ii N N X Y iii N N X Y b Draw two different diagrams of your own, plotting two points X and Y. In each diagram, the bearing of Y from X must be less than 180°. For each of your diagrams, write the bearing of Y from X and of X from Y. c What do you notice about each pair of answers in parts a and b? d Copy and complete this rule for two points X and Y, when the bearing of Y from X is less than 180°. When the bearing of Y from X is m°, the bearing of X from Y is ................°. e Discuss your answers to parts c and d with other groups in your class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
296 14 Position and transformation 8 This is part of Marcus’s homework. Question i Write the bearing of B from A. ii Work out the bearing of A from B. Answer i Bearing of B from A is 127° ii Bearing of A from B is 180°+127°=307° A B 127 ° N N A B 127 ° 127 ° 180 ° N N Marcus uses alternate angles to work out the bearing of A from B. For each diagram i write the bearing of B from A ii use Marcus’s method to work out the bearing of A from B. a A 77° B N N b A B 118° N N c A B 16° N N We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
297 14.1 Bearings 9 This is part of Sofia’s homework. Question i Write the bearing of P from Q. ii Work out the bearing of Q from P. Answer i Bearing of P from Q is 223° ii Bearing of Q from P is 223° – 180°=043° P 223 ° N N Q P 180 ° 43 ° 43 ° N N Q Sofia uses alternate angles to work out the bearing of Q from P. For each diagram i write the bearing of P from Q ii use Sofia’s method to work out the bearing of Q from P. a N P Q 244° N b N P Q 348° N c N P Q 204° N In this exercise, you have learned these three skills: • measuring bearings in diagrams • drawing bearing diagrams • working out bearings using alternate angles. a Which of these did you find the easiest? Explain why. b Which of these did you find the hardest? Explain why. c What could you do to improve these skills? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
298 14 Position and transformation The diagram shows two line segments, AB and CD. The midpoint of AB is halfway between A and B. You can see from the diagram that the midpoint of AB is (3, 3). You can see from the diagram that the midpoint of CD is (1, 0). 14.2 The midpoint of a line segment In this section you will ... • work out the coordinates of the midpoint of a line segment. Key words line segment midpoint x y 2 1 0 –1 –4 –1–2–3 4 5321 –2 –3 –4 3 4 A (1, 3) D (3, –3) (–1, 3) C B (5, 3) Summary checklist I can use bearings as a measure of direction. Worked example 14.2a The diagram shows two line segments, LM and PQ. x y L P M 2 Q 1 –1 –4 –1–2–3 0 4 5321 –2 –3 –4 3 4 a Write the coordinates of the midpoint of LM. b Work out the coordinates of the midpoint of PQ. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
299 14.2 The midpoint of a line segment Continued Answer a (3, 3) You can see that the y-coordinate of the midpoint is 3, because all the points on the line LM have a y-coordinate of 3. You can see that the x-coordinate of the midpoint is 3, because it is exactly halfway along the line LM. b (1, 0) x y P 2 Q 1 –1 –1 0 –2–3 4 5321 –2 –3 3 4 4 squares up 2 squares up 4 squares across 8 squares across To go from P to Q, you go 8 squares across and 4 squares up (shown by the red line). To go from P to the midpoint, you do half of this, so you go 4 squares across and 2 squares up (shown by the blue line). Exercise 14.2 1 Write the coordinates of the midpoint of each line segment. x y A B 2 1 0 4 53210 3 4 5 x y C D 2 1 0 4 53210 3 4 5 x y F E 2 1 0 4 5321 3 4 5 0 x y H 2 G 1 0 4 5321 3 4 5 0 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE