KS3 Maths Now Textbook

501 35.3 Solve problems The expression at the top of the pyramid is the product of the two expressions below it. Work out the missing expression. 16x – 20xy = 4x(4 – 5y) 16x – 20xy 4x ? 15 For each shape, write an expression for the perimeter. Then simplify it as much as possible. Factorise it if you can. a x + 2 x + 4 x + 6 b c 2n + 1 2n –3 2n + 5 2x – 3 x + 2 2x + 1 x + 4 16 For each shape, write an expression for the perimeter. Factorise the answer fully. t – 3 2r – 3 r + 3 r + 3 2r + 4 p + 4 p + 8 p + 5 p + 5 a b c p + 8 t + 6 2t + 1 17 The expression at the top of the pyramid is the product of the two expressions below it. Work out the missing expressions. 3a + 18ab 3a ? 25y – xy ? ? 18 a If the units digit of a number is a, give an expression for: i the tens column ii the hundreds column. b Find an expression for any two-digit number where the tens digit is twice the units digit, e.g. 63. c Find more two-digit numbers with this property. d 63 is divisible by 21. Simplify your expression to show that numbers where the tens digit is twice the units digit are always divisible by 21. 35.3 Equations with brackets ● I can solve equations with one or more sets of brackets You have already learnt how to solve simple equations with brackets. You can use either method in the worked examples. If there is more than one set of brackets, it is usually easier to multiply them both out first. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 501 2867_P001_680_Book_MNPB.indb 501 24/03/2020 15:33 4/03/2020 15:33

502 Chapter 35 Develop fluency Solve the equation 3(x − 4) = 16 a by first multiplying out the brackets b by first dividing by 3. a 3(x − 4) = 16 3x − 12 = 16 Multiply x and 4 by 3. 3x = 28 Add 12 to both sides. x = 9 1 3 Divide 28 by 3. b 3(x − 4) = 16 x − 4 = 5 1 3 Divide 16 by 3. x = 9 1 3 Add 4 to 5 1 3. 1 a Solve these equations. i 2x + 6 = 18 ii 2(x + 6) = 18 iii 4y − 2 = 20 iv 4(y − 2) = 20 b Solve ii and iv in a different way. (Hint: Look at the example above.) 2 Solve these equations. Show your method. a 2(y − 8) = 20 b 2(y − 8) = 10 c 40 = 5(f − 17) d 20 = 4(w − 9) 3 Solve these equations. Give your answers in fraction form. a 3(x − 2 ) = 5 b 4(y + 1) = 11 c 6(t − 4) = 19 d 8(c + 3) = 25 4 a Simplify 4(x + 2) + x – 5. b Solve the equation 4(x + 2) + x − 5 = 38. 5 a Simplify 2(x − 1) + 6(x − 2). b Solve the equation 2(x − 1) + 6(x − 2) = 30. 6 a Simplify 7(x − 1) + 6(x − 2). b Now solve the equation 7(x − 1) + 6(x − 2) = 30. 7 Solve these equations. a 2(x + 3) + 2(x + 4) = 28 b 3(x − 3) + 2(x + 4) = 34 c 4(t − 3) − (t − 5) = 20 8 Solve each of these equations. Start by multiplying out the brackets. a 5(x − 1) = 4(x + 1) b 4(x − 2) = 5(x − 3) c 6(x − 1) = 4(x + 5) 9 Solve these equations. Start by dividing both sides by 2. a 2(3 + 2v) = 2(v + 7) b 2(2d + 4) = 2(d − 7) c 2(5 − 2k) = 4(1 + 2k) 10 Solve these equations and show your working. a 52 − 4(m + 2) = 3(m − 1) b 7 (d + 2) = 21 + 2(2d + 1) c 3(3 + 2b) = 33 − 4(2 − b) Reason mathematically This solution contains an error. Describe the error and solve the equation correctly. 2x + 5 = 12 2x = 17 x = 8.5 Step 2 is incorrect. Instead of adding 5, subtract 5. 2x = 12 − 5 2x = 7 x = 3.5 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 502 2867_P001_680_Book_MNPB.indb 502 24/03/2020 15:33 4/03/2020 15:33

503 35.3 11 Each equation has been solved incorrectly. Spot the mistakes and solve the equations correctly. a 3(x + 5) = 12 x + 5 = 9 x = 4 b 5y – 11 = 2y + 25 7y = 25 – 11 7y = 14 y = 2 c 4(z + 3) = 8z + 10 z + 3 = 2z + 10 z = 7 12 Thomas solves this equation, as shown. 18 − 4(x + 1) = 2(x + 12) 18 − 4x − 4 = 2x + 24 Expand brackets. 14 − 4x = 2x + 24 Simplify. 4x = 2x + 10 Subtract 14 from both sides. 2x = 10 Subtract 2x from both sides. x = 5 Divide both sides by 2. Is Thomas correct? Explain your answer 13 Johnny solves this equation: 5(x − 3) − 15(x − 2) = 10 x – 3 − 3(x − 2) = 2 a Explain his first step. b Finish Johnny’s solution to find x. c Use Johnny’s method to solve 6(x +1) + 12(x − 2) = 0 14 a Show that the expression 5(x − 3) − 4(x − 2) simplifies to x − 7. b Use the result in part a to find a quick solution to the equation 5(x − 3) − 4(x − 2) = 10. c Solve the equation 5(x − 3) − 4(x − 2) = −7. Solve problems Form and solve an equation to work out the value of x. 2x 3(x – 10) 3(x – 10) + 2x + 90 = 180 The angles in a straight line add up to 180°. 3x – 30 + 2x = 180 – 90 Expand the brackets and subtract 90 from both sides. 5x – 30 = 90 Simplify the equation. 5x = 90 + 30 = 120 Add 30 to both sides. x = 120 ÷ 5 Divide both sides by 5. x = 24° © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 503 2867_P001_680_Book_MNPB.indb 503 24/03/2020 15:33 4/03/2020 15:33

504 Chapter 35 15 a Explain why a regular hexagon whose perimeter is 12h + 6 must have sides all equal to 2h + 1. b h = 3.5cm. What are the sides and perimeter of the hexagon? c If the hexagon’s perimeter is 30cm, what is the value of h? 16 The pentagon has the same perimeter as the square. Set up an equation and calculate the value of x. 17 I am thinking of a number n. I double it and subtract 11. Then I multiply the answer by 5. I get 35. Construct an equation and solve it to find the number I was thinking of. 18 The sum of three consecutive integers is 54. a Construct an equation and solve it to find the integers. b The sum of three consecutive odd integers is 63. Construct an equation and solve it to find the integers. 35.4 Equations with fractions ● I can solve equations involving fractions If there is a fraction in an equation you can remove it by multiplying the whole equation by the denominator of the fraction. When you cannot write the answer exactly as a decimal, it is better to leave it as a fraction. Equations with fractions can be written in different ways. Look at these equations. 3 5 x = 8 could be written as 3x 5 = 8. 2 3 (a − 4) = 5 could be written as 2(a – 4) 3 = 5. 2(x – 4) cm 3(x – 5) cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 504 2867_P001_680_Book_MNPB.indb 504 24/03/2020 15:33 4/03/2020 15:33

505 35.4 Develop fluency Solve the equations: a 2 3 (a − 4) = 5 b 12 x + 1 = 5. a 2 3 (a − 4) = 5 2(a − 4) = 15 Multiply both sides by 3. 15 is 3 × 5 and you now have an integer in front of the bracketed term. a − 4 = 7.5 Divide by 2. a = 11.5 Add 4. b 12 x + 1 = 5 x + 1 is in the denominator so multiply by it to remove the fraction. 12 = 5(x + 1) Solve this in the usual way. Expand the term with brackets first. 12 = 5x + 5 Now subtract 5. 7 = 5x Then divide by 5. x = 1 2 5 or 1.4 Solve these equations. 1 a x 5 = 6 b y 3 = 7 c t 4 = 4 d n 2 = 15 e m 15 = 2 2 a 1 5 x = 4 b 2 5 x = 4 c 3 5 x = 4 d 4 5 x = 4 3 a 20 x = 5 b 20 x = 10 c 15 y = 6 d 30 t = 5 4 a 3 4 a = 6 b 3 8 b = 6 c 5 8 c = 2 d 2 7 k = 4 e 3 5 t = 9 5 a 1 4 (x + 2) = 5 b 1 3 (x − 4) = 2 c 1 6 (x + 13) = 4 d 1 8 (x − 9) = 3 6 a 3d 5 = 6 b 4u 3 = −5 c − 3m 5 = 9 d 2t –3 = −5 7 a 16 x + 2 = 8 b 48 y – 1 = 8 c 30 k – 5 = 5 d 36 d + 3 = 4 8 a 2(x + 5) 3 = 16 b 3(t – 2) 4 = 4 c 3(2 + x) 10 = 4 d 5(c – 10) 9 = 2 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 505 2867_P001_680_Book_MNPB.indb 505 24/03/2020 15:33 4/03/2020 15:33

506 Chapter 35 Reason mathematically Jake and Isabella solved 6(r – 8) 7 = 12 in two different ways. Explain their methods. Jake’s method r – 8 7 = 2 r – 8 = 14 r = 14 + 8 r = 22 Isabella’s method 6(r – 8) = 12 × 7 6r – 48 = 84 6r = 132 r = 22 Jake divides both sides by 6, then solves the equation. Isabella multiplies both sides by 7, so that she removes the fraction first, then solves the equation. 9 Solve each equation in two different ways. Explain your methods. a 4(x + 1) 3 = 16 b 3(t – 3) 4 = 9 c 6(2 + x) 5 = 4 d 15(c – 1) 9 = 10 10 The width of this rectangle is 5cm and the area is 3n + 4cm2 . a Explain why the length of the rectangle is 3n + 4 5 cm. b The length of the rectangle is 13cm. Write down an equation and solve it to work out the value of n. 11 The width of this rectangle is 6cm and the area is 2n + 3cm2 . a Explain why the length of the rectangle is 2n + 3 6 cm. b The length of the rectangle is 9cm. Write down an equation and solve it to work out the value of n. 12 The width of a triangle is n + 5cm. The area is 17cm2 . a Explain why the height of the triangle is 34 n + 5 cm. b The width of the triangle is 3cm. Write down an equation and solve it to work out the value of n. 3n + 4 cm2 5 cm 6 cm 2n + 3 cm2 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 506 2867_P001_680_Book_MNPB.indb 506 24/03/2020 15:33 4/03/2020 15:33

507 35.4 Solve problems Three numbers are 2x, x + 4, x – 3. The mean of the numbers is x + 5. Work out the numbers. Mean = 2x + x + 4 + x – 3 3 = x + 5 Construct an equation. 4x + 1 3 = x + 5 Simplify the equation. 4x + 1 = 3 × x + 3 × 5 = 3x + 15 Multiply both sides by 3 4x + 1 = 3x + 15 Solve the equation: subtract 3x from both sides; subtract 1 from both sides. x = 14 2x = 28 Work out the three numbers. x + 4 = 18 x – 3 = 11 The numbers are 28, 18 and 11. 13 Four numbers are x, x + 4, x − 3, and 2x. The mean of the four numbers is 11. a Write down an equation and solve it to work out the value of x. b Work out the four numbers for the value of x you found in part b. 14 Asma is thinking of a number. She doubles her number and adds 1. She then calculates 2 3 of the answer and gets 6. Set up an equation and solve it to find Asma’s number. 15 Work out the value of x by constructing and solving an equation. 16 Owen has saved some money. He gives £2 to his brother and spends 3 5 of what is lef t at the school fair. Owen spends £9 at the school fair. Construct an equation and find how much money Owen had saved. 45° 3(x – 50) 2 Now I can… multiply out brackets solve equations with one or more sets of brackets solve equations involving fractions factorise expressions © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 507 2867_P001_680_Book_MNPB.indb 507 24/03/2020 15:33 4/03/2020 15:33

508 36 Prisms and cylinders 36.1 Metric units for area and volume ● I can convert from one metric unit to another You need to know the metric units for area, volume and capacity. They are listed in the table, which also gives the conversions between these units. Area Volume Capacity 10000m2 = 1 hectare (ha) 10000cm2 = 1m2 1m2 = 1000000mm2 100mm2 = 1cm2 1000000cm3 = 1m3 1000mm3 = 1cm3 1m3 = 1000 litres (l) 1000cm3 = 1 litre 1cm3 = 1 millilitre (ml) 10 millilitres = 1 centilitre (cl) 1000 millilitres = 100 centilitres = 1 litre The unit symbol for litres is the letter l, which is often written as l, as shown in the table. To avoid confusion with the digit 1 (one), it is also common to use the full unit name instead of the symbol. Remember: • to convert large units to smaller units, always multiply by the conversion factor • to convert small units to larger units, always divide by the conversion factor. Develop fluency Convert each of these units as indicated. a 72000cm2 to m2 b 0.3cm3 to mm3 c 4500cm3 to litres a Hint: You are converting smaller units to larger units, so divide by the conversion factor 10000. 72000cm2 = 72000 ÷ 10000 = 7.2m2 b You are converting larger units to smaller units, so multiply by the conversion factor 1000. 0.3cm3 = 0.3 × 1000 = 300mm3 c You are converting smaller units to larger units, so divide by the conversion factor 1000. 4 500cm3 = 4500 ÷ 1000 = 4.5 litres 1 Express each of these in square centimetres (cm2 ). a 4m2 b 7m2 c 20m2 d 3.5m2 e 0.8m2 f 540mm2 g 60mm2 2 Express each of these in square millimetres (mm2 ). a 2cm2 b 5cm2 c 8.5cm2 d 36cm2 e 0.4cm2 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 508 2867_P001_680_Book_MNPB.indb 508 24/03/2020 15:33 4/03/2020 15:33

509 36.1 3 Express each of these in square metres (m2 ). a 20000cm2 b 85000cm2 c 270000cm2 d 18600cm2 e 3480cm2 4 Express each of these in cubic millimetres (mm3 ). a 3cm3 b 10cm3 c 6.8cm3 d 0.3cm3 e 0.48cm3 5 Express each of these in cubic metres (m3 ). a 5000000cm3 b 7500000cm3 c 12000000cm3 6 Express each of these in litres. a 8000cm3 b 17000cm3 c 500cm3 d 3m3 e 7.2m3 7 Express each measure as indicated. a 85ml in cl b 1.2 litres in cl c 8.4cl in ml 8 Convert these measurements to cubic centimetres. a 2m3 b 0.5m3 c 78mm3 d 9300mm3 9 Convert these quantities (remember that cubic centimetres are the same as cm3 ). a 8400cm3 to litres b 65 cubic centimetres to litres c 4.8ml to cm3 d 200ml to litres e 9 litres to cm3 f 3.75 litres to ml Reason mathematically 25 litres of water flows into a pond every second. The pond is in the shape of a cuboid with dimensions 50m by 2m by 3m. How many hours and minutes does it take to fill the pond from empty? 1m3 = 1000000cm3 25 litres = 25000cm3 Volume of pond = 50m × 2m × 3m = 300m3 300m3 = 300000000cm3 Time to fill pond = 300000000 ÷ 25000 = 12000 seconds = 200 minutes = 3 hours 20 minutes. 10 a Explain, using areas of squares, why 1cm2 = 100mm2 . b Explain, using volumes of cubes, why 1000mm3 = 1cm3 . 11 Spot the mistakes in these conversions. Correct them. a 5000cm2 = 5m2 b 3500mm2 = 3.5cm2 c 7.5m2 = 750000mm2 12 How many lead cubes of side 2cm can be cast from 4 litres of molten lead? 13 The volume of a cough-medicine bottle is 240cm3 . The prescription reads: Two 5ml spoonfuls to be taken four times a day. How many days will the cough medicine last? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 509 2867_P001_680_Book_MNPB.indb 509 24/03/2020 15:33 4/03/2020 15:33

510 Chapter 36 Solve problems A garden path is made from 350 rectangular paving stones, each 30cm long and 15cm wide. a What is the area of the path? b A company is hired to clean the path. They charge £2.75p/m2 or £50 to clean the whole path. Which method is cheaper? a 30cm = 0.3m, 15cm = 0.15m Area of one stone = 0.3 × 0.15 = 0.045m2. Area of path = 350 × 0.045 = 15.75m2. b Cost of the path = 2.75 × 15.75 = £43.31. It is cheaper to pay per square metre. 14 How many square paving slabs, each with sides of 50cm, are needed to cover a rectangular yard measuring 8m by 5m? 15 A football pitch measures 120m by 90m. It costs £600/month to maintain the pitch. What is the maintenance rate per m²? Give your answer correct to the nearest penny. 16 a A car has a 2.3 litre engine. What is this in cubic centimetres (cc)? b A car engine has a cylinder capacity of 1487cc but when advertised this is rounded to the nearest 100 and converted to litres. What size will it be advertised as? 17 A hockey pitch is 80m long and 45m wide. The annual maintenance costs for the pitch are £1000/hectare. a Work out its area in: i square metres ii hectares. b Work out the annual cost of maintaining the pitch. 36.2 Volume of a prism ● I can calculate the volume of a prism A prism is a three-dimensional (3D) shape that has exactly the same two-dimensional (2D) shape running all the way through it, whenever it is cut across, perpendicular to its length. This 2D shape is called the cross-section of the prism. The shape of the cross-section depends on the type of prism, but it is always the same for a particular prism. You can work out the volume, V, of a prism by multiplying the area, A, of its cross-section by the length, l, of the prism or its height, h, if it stands on one end. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 510 2867_P001_680_Book_MNPB.indb 510 24/03/2020 15:33 4/03/2020 15:33

511 36.2 Develop fluency Calculate the volume of this triangular prism. The cross-section is a right-angled triangle with an area of 6 × 8 2 = 24cm2. So, the volume is given by: V = area of cross-section × length = 24 × 15 = 360cm3 6 cm 8 cm 10 cm 15 cm 1 Calculate the volume of each prism. a 6 cm 4 cm 1 cm b 12 cm 9 cm 15 cm 16 cm c 1.5 m 7 m 2.5 m 8 m 5 m 2 Calculate the volume of each cuboid, in the most appropriate units. 50 cm 30 cm 20 cm 300 cm 400 cm 200 cm 125 cm 6 cm a b c 6 cm 3 Calculate the capacity (in litres or ml as appropriate) of each cuboid in question 2. 4 Calculate the volume and capacity of a cube of side 15cm. 5 A cube has a volume of 8000cm3 . What are the lengths of its sides? 6 The volume of a biscuit tin is 3 litres. The area of the lid is 375cm2 . How deep is the tin? 7 The cross-section of a pencil is a hexagonal prism with an area of 50mm2 . The length of the pencil is 160mm. a Calculate the volume of the pencil, in cubic millimetres. b Write down the volume of the pencil, in cubic centimetres. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 511 2867_P001_680_Book_MNPB.indb 511 24/03/2020 15:33 4/03/2020 15:33

512 Chapter 36 8 A biscuit tin is an octagonal prism with a cross-sectional area of 350cm2 and a height of 9cm. Calculate the volume of the tin. 9 The box of a chocolate bar is an equilateral triangular prism. The area of its cross-section is 15.5cm2 and the length of the prism is 30cm. Calculate the volume of the box. 10 The diagram shows the cross-section of a swimming pool, along its length. The pool is 15m wide. 4 m 6 m 21 m 1 m a Calculate the area of the cross-section of the pool. b Work out the volume of the pool. c How many litres of water does the pool hold when it is full? Reason mathematically A rectangular tank measuring 12cm by 10cm by 14cm contains 1.12 litres of water. a Sasha says that the tank is two-thirds full. Is she correct? b How much more water is needed to fill in the tank completely? Give your answer in millilitres. a Volume of water = 1.12 litres = 1.12 × 1000 = 1120cm³ Hei ght of water level = 1120 14 × 10 = 8cm 8 12 = 2 3, so Sasha is correct. b Capacity of tank = 12cm × 10cm × 14cm = 1680cm3 = 1680ml Amount of water needed = 1680ml – 1120ml = 560ml 12 cm 10 cm 14 cm 11 The cross-section of a block of wood is a trapezium. The volume of the block is 9600cm3 . Tom thinks that the height of the trapezium is 12cm. Is Tom correct? 25 cm 40 cm 15 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 512 2867_P001_680_Book_MNPB.indb 512 24/03/2020 15:33 4/03/2020 15:33

513 36.2 12 Leroy is making a solid concrete ramp for wheelchair access to his house. The dimensions of the ramp are shown on the diagram. a Calculate the volume of the ramp, giving your answer in cubic centimetres. b One cubic metre of cement weighs 2.4 tonnes. What is the mass of concrete that Leroy uses? 13 Krispies are sold in three sizes: mini, medium and giant. The boxes are filled to the top. 36 cm 9 cm 24 cm 24 cm 16 cm 12 cm 8 cm 3 cm 6 cm MINI MEDIUM GIANT a Write the ratio of the heights of the three boxes in its simplest form. b Calculate the volume of each box. c Write the volumes as a ratio in its simplest form. d Can you see a link between the ratios in parts a and c? e How many mini boxes could be fitted into: i a medium box ii a giant box? f The answers in part e are the same as which other answers you have found? Why is  that? g The prices of these packets are 45p, £2.70 and £6.75 respectively. Express these as a simple ratio. h Why do you think the ratio of the prices is different to the ratio of the volumes? 14 An empty regular container has a square base of side 10cm and a height of 20cm. a Ben says that it is filled with 2.1 litres of water. Show that Ben can’t be right. b Jordan says that the container is filled with 1.1 litres of water and the height of the water level is 11cm. Show that Jordan is correct. 140 cm 150 cm 25 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 513 2867_P001_680_Book_MNPB.indb 513 24/03/2020 15:33 4/03/2020 15:33

514 Chapter 36 Solve problems A rectangular tank 30cm by 20cm by 12cm is half full. How much more water is needed to make the tank three-quarters full? Height of water = 1 2 × 20 = 10cm Height of water when 3 4 full = 3 4 × 20cm = 15cm 15cm – 10cm = 5cm Extra volume of water needed = 30 × 12 × 5 = 1800cm3 = 1.8 litres 15 A water tank is in the shape of a trapezium as shown. An overflow pipe is fixed 20cm below the top of the tank. What is the greatest volume of water the tank can hold? 16 The box below is filled with packets of tea. The packets are 5cm square and 12cm high. When full, the box contains 8 dozen packets of tea which are stacked upright in the box. a What volume of tea is in each packet, in cm3 ? b How many packets of tea fit into the base of the box? c Work out how many layers of packets there must be in the box and then find the height of the box. d What is the total volume of tea in the box, in cm3 ? e The packets cost 75p to produce, and are sold at £1.90 each. How much profit is made on the whole box? 17 A rectangular ta nk has a square base of sides 20cm. It contains water to the depth of 5cm. When 8.4 litres of water is added, the tank becomes half-filled with water. What is the height of the tank? 18 A cubical tank of edge 15cm is 1 3 filled with water. The water is then poured into an empty rectangular tank that has a base 25cm by 6cm. The water fills up three-quarters of the rectangular tank. Work out: a the height of the water in the rectangular tank b the height of the tank. 110 cm 90 cm 120 cm 70 cm 30 cm 40 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 514 2867_P001_680_Book_MNPB.indb 514 24/03/2020 15:33 4/03/2020 15:33

515 36.3 36.3 Surface area of a prism ● I can calculate the surface area of a prism You can find the surface area of a prism by calculating the sum of the areas of its faces. For example, in the prism shown here, its total surface area is made up of the two end pentagons plus five rectangles. Develop fluency Calculate the total surface area of this triangular prism. The cross-section is a right-angled triangle with an area of 6 × 8 2 = 24cm2. The total a rea of the two end triangles is 48cm2. The sum of the areas of the three rectangles is: (6 × 15) + (8 × 15) + (10 × 15) = 360cm2 So, the total surface area is: 48 + 360 = 408cm2 6 cm 8 cm 10 cm 15 cm 1 Calculate the total surface area of each prism. a b 1.5 m 7 m 8 m 2.5 m 5 m c 12 cm 9 cm 15 cm 16 cm 12 cm 8 cm 10 cm 20 cm 2 Calculate the surface area of each prism. 3 cm 13 mm 5 mm 12 cm 8 cm 25 mm 12 mm a b 3 Convert your answers to question 2 to square metres. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 515 2867_P001_680_Book_MNPB.indb 515 24/03/2020 15:33 4/03/2020 15:33

516 Chapter 36 4 A fish tank with glass walls has dimensions 60cm wide by 30cm deep by 20cm high with a solid base and no top. How much glass is needed? 5 The cross-section of this prism is a trapezium. 25 cm 17 cm 5 cm 28 cm 34 cm a Calculate the area of the trapezium. b Calculate the total surface area of the prism. 6 The diagram shows a petrol tank in the shape of a prism. a Calculate the area of the cross-section. b Calculate the volume of the tank. c Calculate the capacity of the tank in litres. d Calculate the surface area of the tank in square metres. 50 cm 40 cm 13 cm 12 cm 80 cm 7 This regular octagonal prism has a cross-sectional area of 96cm2 and a length of 32cm. Each edge of the octagon is 6cm long. Calculate the total surface area of the prism. 8 A tent is in the shape of a triangular prism. Its length is 2.4m, its height is 1.6m, the width of the triangular end is 2.4m and the length of the sloping side of the triangular end is 2m. Calculate the surface area of the outside of the tent. 9 This gift box is in the form of an equilateral triangular prism. The cross-sectional area is 27.7cm2 . a Calculate the area of the net needed to make the box. Do not include the area of the tabs. b Calculate the volume of the box. 10 Calculate the surface area of a prism of length 15cm with this isosceles trapezium as its cross section. 32 cm 6 cm 8 cm 25 cm 6 cm 16 cm 13 cm 10 cm 13 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 516 2867_P001_680_Book_MNPB.indb 516 24/03/2020 15:33 4/03/2020 15:33

517 36.3 Reason mathematically Jordan says: ‘To work out the surface area of the prism shown, I need to split the shape into two cuboids, find the surface area of each of them and add them up.’ a Is Jordan correct? b What is the surface area of the shape? a Jordan is not correct. He only needs to calculate the surface areas of the shape’s faces. This shape has two L-shaped faces and six rectangular faces. b Surface area of the L shape = 2 × 6 + 2 × 6 = 24 cm2 Total surface area = 2 × 24 + 8 × 2 + 6 × 2 + 2 × 4 + 6 × 2 + 2 × 2 × 2 = 104 cm2 6 cm 2 cm 2 cm 4 cm 8 cm 11 The cross-sectional area of the prism shown is made from five equal squares, each with side length of 4 cm. The prism is 12cm long. a Calculate the area of the cross-section. b Calculate the total surface area of the prism. c Calculate the volume of the prism. 12 Look at the dimensions of the Krispies packets carefully. 36 cm 9 cm 24 cm 24 cm 16 cm 12 cm 8 cm 3 cm 6 cm MINI MEDIUM GIANT a Work out the surface area of each packet. b Write these areas as a ratio in its simplest form. c There is a quick way to simplify their ratios from their dimensions. Explain the method. d Are these boxes similar shapes or not? Explain your answer. 13 Are these statements true or false? Explain your answers. a When you cut a prism in half, its surface area is halved as well. b Two prisms can have different dimensions, but equal surface areas. 4 cm 4 cm 12 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 517 2867_P001_680_Book_MNPB.indb 517 24/03/2020 15:33 4/03/2020 15:33

518 Chapter 36 14 The surface area of a prism is 348 cm2 . Sketch three different prisms that could have this surface area. Solve problems A cuboid has dimensions 6cm by 10 cm by 2cm. It has the same surface area as the triangular prism shown. Work out the height of the triangle. Surface area of cuboid = 2(6 × 10 + 6 × 2 + 2 × 10) = 184 cm2 Surface area of triangular prism = 184 cm2 2 × 10 × 5 + 6 × 10 + 2 × area of triangle = 184 Area of triangle = 12 cm2 6 × x 2 = 12 x = 4 cm 5 cm x 6 cm 10 cm 15 A cuboid has a surface area of 100 m2 . Its width is 2 m, and its height is 7 m. Calculate its length. 16 The total surface area of this isosceles triangular prism is 896cm2 . Work out the length, l, of the prism. 17 Tom has a small tent which is 1 m wide, 1.2 m high, 2.4m long and has a sewn-in groundsheet. The sloping height of the tent is 1.6 m. a Work out how much material is needed to make this tent. When Tom camps with Scouts they have bigger tents that allow more height before the sloping roof begins. The extra vertical bits of canvas are 50 cm high, joining to a sloping section of 1.5 m. They are 1.8 m wide and 2.8m long but have no groundsheets. The height of the tent is 1.7 m. b How much material is used to make these tents? c How much extra space do you get inside the Scout tent compared to Tom’s own tent? 18 Zahara is finding an expression for the surface area of the prism below. She writes: Area of triangle = 6x Total surface area = 6x + 6x + 6 × 11 + x × 11 + 10 + 11 = 23x + 176 a Is Zahara correct? Explain your answer. b The total surface area of the prism is 312 cm2 . What is the value of x? 10 cm 12 cm 8 cm l 10 cm 11 cm 6 cm x cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 518 2867_P001_680_Book_MNPB.indb 518 24/03/2020 15:33 4/03/2020 15:33

519 36.4 36.4 Volume of a cylinder ● I can calculate the volume of a cylinder A cylinder is a circular prism. The cross-section of a cylinder is a circle of radius r. The area of the cross-section is A. A = πr 2 If the height of the cylinder is h, then the volume, V, of the cylinder is given by: V = πr 2 × h = πr2 h If the length of the cylinder is l, then the volume, V, of the cylinder is given by: V = πr 2 × l = πr 2 l Develop fluency Calculate the volume of each cylinder, giving your answers correct to one decimal place. a 3 cm 2 cm b 15 m 3 m a V = πr 2h = π × 32 × 2 = 56.5cm3 b V = πr 2l = π × 1.52 × 15 = 106.0m3 In this exercise, take π = 3.14 or use the π key on your calculator. 1 Work out the volume of these cylinders: a cross-sectional area 3cm2 , height 15cm b cross-sectional area 1.5m2 , length 4m c cross-sectional area 5.4cm2 , width 8mm 2 Find the volume of these cylinders. Give your answers to one decimal place. a radius 4cm, length 10cm b radius 2.5m, height 6m c diameter 12cm, length 15cm d diameter 8.4m, height 7.5m 3 Calculate the volume of each cylinder. Give your answers correct to one decimal place. 10 cm a 6 cm 3 cm 8 cm b 1.5 m 4 m 12 m c 2 m d 0.5 cm 5 cm e 4 The diameter of a 2p coin is 26mm and its thickness is 2mm. Calculate its volume. Give your answer correct to the nearest cubic millimetre. r h l r © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 519 2867_P001_680_Book_MNPB.indb 519 24/03/2020 15:33 4/03/2020 15:33

520 Chapter 36 5 The diagram shows the internal measurements of a cylindrical paddling pool. a Calculate the volume of the pool. Give your answer in cubic metres, giving your answer to two decimal places. b How many litres of water are there in the pool when it is three-quarters full? Give your answer to the nearest litre. 6 A tea urn is a cylinder. The inner diameter of the cylinder is 30cm and the inner height is 50cm. Calculate the volume of the inner cylinder. Give your answer in litres, correct to one decimal place. 7 A winners’ podium is going to be built at an athletics stadium. The diagram shows the dimensions of the podium. a Calculate the volume of each cylinder. Give your answers correct to the nearest cubic centimetre. b What is the total volume of the podium? Give your answer in cubic metres, correct to two decimal places. 8 The cross-section of this plastic bench has an area of 620cm2 . Calculate its volume in: a cubic centimetres b cubic metres. 1.8 m area of cross-section = 620 cm2 9 Work out the volume of each cylinder. Give your answer in terms of π. a radius8cm, length = 10cm b diameter 8cm, length 10cm c diameter 16m, length 10m d radius 16m, length 10m 10 Calculate the volume of each cylinder. a radius 10cm, length 12cm b diameter 60cm, length 1m c diameter 10m, length 10m d radius 80cm, length 4m 2 m 50 cm 90 cm 20 cm 3rd 90 cm 1st 90 cm 2nd 40 cm 60 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 520 2867_P001_680_Book_MNPB.indb 520 24/03/2020 15:33 4/03/2020 15:33

521 36.4 Reason mathematically A cylinder has a height of 5cm. The volume of the cylinder is 180πcm³. Gary thinks that the diameter of the cylinder is 36cm. a Explain the mistake that Gary has made. b What is the diameter of this cylinder? a Volume of cylinder = 180πcm3 area of circle × 5 = 180π area of circle = 36π Gary has mistaken the formula for the circumference of the circle (πd) for the area of the circle and used this to find the diameter. b πr²= 36π r² = 36, radius = 6cm, diameter = 12cm 11 These are three cake tins. 7 cm 24 cm 14 cm 25 cm 13 cm 10 cm 8 cm 7·5 cm 20 cm a b c Which tin has the greatest volume? 12 A circular pool has a radius of 1.5m and a depth of 60cm. a What is the capacity of the pool to the nearest 100 litres? b How much water does it contain when two-thirds full? 13 A cylinder has a volume of 340cm3 and a radius of 3cm. a What is its height to the nearest centimetre? b Another cylinder has a volume of 37m3 and it is 6m high. What is its radius to one decimal place? c A third cylinder has a capacity of 2 litres, with a diameter of 8cm. How high is it to the nearest centimetre? 14 Are these statements always, sometimes or never true? Explain your reasoning. a If you double the height of a cylinder, the volume also doubles. b If you double the radius of a cylinder, the volume also doubles. c If you double the diameter of a cylinder, the volume doesn’t change. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 521 2867_P001_680_Book_MNPB.indb 521 24/03/2020 15:33 4/03/2020 15:33

522 Chapter 36 Solve problems A drinks manufacturer wants a can to hold 250ml of a fizzy drink. The diameter of the can is 6cm. How tall does the can need to be? 250ml = 250cm3 Radius = 6 ÷ 2 = 3cm 250 = π × 3² × height Height = 250 ÷ 9π = 8.85cm (2 dp) 15 A water pipe has an internal diameter of 40cm. a Calculate: i the radius ii the cross-sectional area iii the volume per metre of the pipe iv the quantity of water, to the nearest litre, in every metre of the pipe. b If the water in the pipe moves at a speed of 0.8 metres per second, how many litres, to the nearest 1000, will pass through it in one hour? 16 A component is manufactured by cutting a circular hole of radius 6mm through a metal block. a Find the volume of the block after the hole has been drilled. b What percentage of the block has been removed? What is this to the nearest unit fraction? c The original block has a mass of 252g. What is its mass after the hole is removed? 17 An urn contains hot water and is a vertical cylinder. It has an internal diameter of 24cm and is 40cm high. The serving tap is located 5 cm above the base, and once the water reaches this level, no more will come out. a Calculate the total capacity of the urn to the nearest millilitre. b Calculate the usable amount of water in the urn to the nearest millilitre. c What percentage of the total capacity is usable? d In the canteen where it is used, tea mugs hold about 25cl, but coffee is served in special 20cl cups. If an equal number of both drinks are served, what is the maximum number of cups that can be filled? 18 A paint tin has a capacity of 10 litres and a height of 40cm. a Write down the volume of the tin, in cubic centimetres. b Calculate the area of the base of the tin. c Calculate the diameter of the base. Give your answer correct to one decimal place. 20 mm 20 mm 30 mm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 522 2867_P001_680_Book_MNPB.indb 522 24/03/2020 15:33 4/03/2020 15:33

523 36.5 36.5 Surface area of a cylinder ● I can calculate the curved surface area of a cylinder ● I can calculate the total surface area of a cylinder A cylinder without a top and bottom is called an open cylinder. When an open cylinder is cut and opened out, it forms a rectangle with the same length as the circumference of the base of the cylinder. The curved surface area of the cylinder is the same as the area of the rectangle. The area of the rectangle is 2πrh. The formula for the curved surface of a cylinder is: A = 2πrh The total surface area of the cylinder is the curved surface area plus the area of the circles at each end. The formula for the total surface area of a cylinder is: A = 2πrh + 2πr 2 Develop fluency A cylinder has a height of 1.2m and a radius of 25cm. Calculate the total surface area of the cylinder. Give your answer in square metres, correct to one decimal place. r = 25cm = 0.25m The total surface area is given by: A = 2πrh + 2πr 2 So, A = 2 × π × 0.25 × 1.2 + 2 × π × 0.252 = 2.3 m2 (1 dp) In this exercise take π = 3.14 or use the pi key on your calculator. 1 Find the surface areas of these cylinders. a radius 4cm, length 10cm b radius 2.5m, height 6m c diameter 12cm, length 15cm d diameter 8.4m, height 7.5m 2 Calculate the total surface area of each cylinder. Give your answers correct to one decimal place. 9 cm c 6 cm 5 cm 12 cm b 3 m 6.5 m a e 2 cm 7 cm d 3 m 10 m h r h C = πd = 2πr © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 523 2867_P001_680_Book_MNPB.indb 523 24/03/2020 15:33 4/03/2020 15:33

524 Chapter 36 3 Calculate the curved surface area of each cylinder. Give your answers correct to one decimal place. 3 cm 15 cm 1.8 m 1.2 m 8.5 cm 10 cm a b c 4 Calculate the total surface area of a cylinder that has a radius of 4.5cm and a height of 7.2cm. Give your answer correct to one decimal place. 5 Calculate the curved surface area of a cylinder that has a diameter of 1.2m and a length of 3.5m. 6 Calculate the surface area of each cylinder. Give your answer in terms of π. a radius 4m, height 12.5m b diameter 10m, height 3m c diameter 12m, height 5m 7 Which of these cylinders, A or B, has the larger surface area? A: radius 4cm and height 10cm B: diameter 4cm and height 20cm 8 Calculate the surface area of each cylinder. Give your answer in terms of π. a radius 4cm, height 3cm b diameter 4cm, height6cm c diameter 10cm, height 0.6cm 9 Calculate the surface area of each open cylinder. Give your answer in terms of π. a radius 5cm, height 10cm b diameter 10m, heig ht 5m c diameter 1m, height 50cm d radius 250cm, height 10m Reason mathematically Lily’s pencil pot is in the shape of a cylinder. The diameter of the cylinder is 16cm and the height is 20cm. She paints the surface area of the pot including the base. She uses 1 tube of paint for every 128cm2 . How many tubes of paint will Lily need? Area of the base = π × 8² = 64πcm² Curved surface area = 2 × π × 8 × 20 = 320πcm² Total surface area = 384πcm² = 1206.37cm² Number of tubes of paint = 1206.37 ÷128 = 9.42 tubes Lily will need 10 tubes of paint. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 524 2867_P001_680_Book_MNPB.indb 524 24/03/2020 15:33 4/03/2020 15:33

525 36.5 10 A tin of salmon has a diameter of 8.5cm. A label, with a height of 4 cm, goes around the curved surface of the tin. Calculate the area of the label, if there is an overlap of 1cm to glue the ends together. Give your answer correct to the nearest square centimetre. 11 Tom calculates the area of the cylinder. He writes: Area of the circle = π ×10² = 100πcm² Curved surface area = π × 20 ×10 = 200πcm² Total surface area = 300πcm² Do you agree with Tom? Explain your answer. 12 A trampoline with a diameter of 12m has protective netting all the way round to a height of 2m. What area of netting is required? 13 a Which of these cylinders has the largest surface area? Explain why. Cylinder A: radius R, height 10cm Cylinder B: radius 2R, 5cm b Substitute R for 5cm and check whether your answer to part a is still correct. Solve problems The curved surface area of a cylinder is 300πcm2 . Its height is 5cm. Work out its radius. Let the radius of the cylinder = x cm. Curved surface area = 2 × π × x × 5 = 10πx 10πx = 300π Set up an equation and solve it. 10x = 300 x = 30 The diameter of the cylinder is 60cm. 14 A circular sandpit is to be built in a children’s play area, kept in place by a wooden fence which needs to be wood stained on both sides before the sand goes in. a The sandpit has a radius of 2.4m and the fence is 40cm high. What is the area to be stained? b The fence needs two coats of wood stain. A 2.5 litre tin claims to cover around 30m2 . What proportion of the tin will not be used? 8.5 cm 20 cm 10 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 525 2867_P001_680_Book_MNPB.indb 525 24/03/2020 15:33 4/03/2020 15:33

526 Chapter 36 15 A cylindrical tube has a diameter of 8cm and a height of 26cm. a Calculate the outside surface area of the tube. b The tube has a lid with a diameter of 8.5cm. Calculate the outside surface area of the lid, which is 0.8cm deep. c Calculate the total surface area of the tube with the lid on. Give all of your answers correct to one decimal place. 16 A firm sells butter in the form of a cylinder with radius 3.8cm and length 12cm. In order to wrap it fully they allow an extra 10% of the surface area for packaging. What area of packaging is used? Give you answer to the nearest whole number? 17 Joanne makes a pencil pot in the shape of a cylinder using a sheet of thin cardboard with dimensions 20cm by 40cm. She uses a circular piece of wood for its base. a Find the two possible values that the radius of the base can be. b Calculate the surface area of the pencil pot. 26 cm 8 cm 8.5 cm 0.8 cm 3.8 cm 12 cm Now I can… convert from one metric unit to another calculate the volume of a cylinder calculate the total surface area of a cylinder calculate the surface area of a prism calculate the curved surface area of a cylinder calculate the volume of a prism © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 526 2867_P001_680_Book_MNPB.indb 526 24/03/2020 15:33 4/03/2020 15:33

527 37 Compound units 37.1 Speed ● I can understand and use measures of speed Jasmine is taking part in a race. She is running at a constant speed. She runs 100 metres in 20 seconds. A formula for working out speed is: speed = distance travelled time taken or distance time Jasmine’s speed is 100 20 = 5 metres per second. The units are metres per second (m/s) because the distance is in metres and the time is in seconds. This is an example of a compound unit that involves other units: in this case, metres and seconds. The units of speed depend on the units used to measure the distance and the time. Develop fluency a A car travels 45 km in 30 minutes. Work out its speed. Include units in your answer. b A car is travelling at a constant speed of 30 m/s. How far does the car travel in one minute? a Speed = distance time = 45 30 = 1.5 km/minute The units are km/minute because the time is in minutes. An alternative answer is 45 0.5 = 90 km/h since 30 minutes = 0.5 hours. b speed = distance time In this case the speed is 30 m/s and the time is 60 s. 30 = d 60 The time must be in seconds. Use d for distance. 30 × 60 = d Multiply by 60 to solve the equation. d = 1800 The distance is 1800 m or 1.8 km. You must include the units. 1 A marathon runner runs 40 km in 2 1 2 hours. Work out his speed, in kilometres per hour (km/h). 2 A train is travelling at a constant speed. It takes 30 minutes to travel 45km. a Work out the speed, in kilometres per minute (km/minute). b Work out the speed in kilometres per hour (km/h). © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 527 2867_P001_680_Book_MNPB.indb 527 24/03/2020 15:33 4/03/2020 15:33

528 Chapter 37 3 Calculate the speed in each case. Put units in each answer. a Peter runs 320 metres in 50 seconds. b A car travels 15 km in 10 minutes. c An aeroplane flies 400 km in half an hour. d A cyclist travels 1500 m in 4 minutes. 4 Matthew is cycling at 18 km/h. Calculate how far he travels in: a 1 hour b 4 hours c 1.5 hours d 2 hours and 30 minutes. 5 Paul can swim at a constant speed of 3 km/h. Calculate how far he can swim in: a 1.5 hours b 1 2 hour c 1 4 hour d 1 minute. 6 Calculate the distance travelled in each case. a Sharon walks at 3 m/s for two minutes. b Nathan drives at 80 km/h for 15 minutes. c A plane flies at 700 km/h for 4.5 hours. d A snail moves at 0.2 m/minute for 150 seconds. 7 The top speed of a sprinter is 8 m/s. Calculate the time it takes at that speed to sprint: a 40 m b 80 m c 50 m d 200 m. 8 Calculate the time taken to travel: a 40 km at 120 km/h b 22 km at 4 km/h c 200 m at 5 m/s d 5 km at 4 m/s. 9 Anita is walking in the countryside. She is travelling at 6 km/h. a Copy and complete this table. Time (t hours) 0.5 1 1.5 2 2.5 Distance (d km) b Work out how long it takes Anita to travel 8 km. 10 An aeroplane is flying at 850 km/h. a Calculate how far the aeroplane flies in 3 1 2 hours. b Calculate the time the plane takes to fly 5000 km. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 528 2867_P001_680_Book_MNPB.indb 528 24/03/2020 15:33 4/03/2020 15:33

529 37.1 Reason mathematically The graph shows the journey that Kate and Charlie made from their homes to the park. Kate forgets her phone and has to go back home to pick it up. Charlie meets a friend on the way to the park. a Which line represents Kate’s journey? Which line represents Charlie’s journey? b Who lives the closest to the park? How much closer? c Who walks at a faster pace? Give a reason for your answer. d Charlie says his average speed is about 1.87 km/h. Is he correct? a The green line represents Kate’s journey and the brown line represents Charlie’s journey. b Charlie lives 1800 – 1400 = 400 m closer to the park. c Kate walks at a faster pace. Her graph is steeper. She walks 400 m in 5 minutes. Charlie walks 600 m in 10 minutes or 300 m in 5 minutes. d Charlie travels 1.4 km in 40 minutes = 2 3 hours. Average speed = 1.4 ÷ 2 3 = 2.1 km/h. Charlie is not correct. 0 400 800 1200 1600 200 600 1000 1400 1800 2000 0 5 10 15 20 25 30 35 40 45 50 Time (minutes) Distance (m) Key: Charlie Kate 11 This graph shows the journey of a car. a How far does the car travel in 20 minutes? b Explain how you know the car is travelling at a constant speed. c Work out the speed of the car, in kilometres/minute. d How long does it take the car to travel 50 km? 12 A plant grows 3.6 cm in 2 days. What is the rate of growth in millimetres per hour (mm/h)? 0 20 10 30 15 35 5 25 0 5 10 15 20 Time (minutes), m Distance (km), d © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 529 2867_P001_680_Book_MNPB.indb 529 24/03/2020 15:33 4/03/2020 15:33

530 Chapter 37 13 Lewis and Neil ran a 2000-metre race. The distance–time graph below shows the race. a What was Neil’s average speed for the race in i m/s ii km/h? b After 800 m Lewis managed to sprint for about 15 seconds in order to catch up with Neil. What was Lewis’ speed for that part of the race in m/s and km/h? c Use the graph to help you fill in the gaps in the report of this race: ‘Lewis started well but soon tired and Neil took the lead after about_____ seconds. Lewis made an effort to draw level at _____ metres, but could only hold the lead for _____ minutes. Neil won the race in a time of _____ minutes, and Lewis finished _______ seconds later.’ 14 Michael draws a graph to show the journey from his home to his grandparents’ house. Explain which of these graphs can’t be his graph. 9 km Time (minutes) Time (minutes) Time (minutes) Distance (km) a 9 kmDistance (km) b 9 kmDistance (km) c Solve problems Matilda took 10 minutes to cycle one-third of the distance to her home. She cycled at an average speed of 9 km/h. She walked th e rest of the distance at an average speed of 4.5 km/h. What is Matilda’s average speed for the journey? 10 minutes = 10 60 of an hour = 1 6 hour Distance cycled = 1 6 hour × 9 km/h = 1.5 km Second part of the journey = 1.5 km × 2= 3 km Time taken for the second part = 3 km 4.5 km/h = 3 4.5 = 30 45 = 2 3 hour = 40 minutes Total distance = 1.5 km × 3 = 4.5 km Total time = 50 minutes = 50 60 of an hour = 5 6 hour Average speed = distance time = 4.5 km 5 6 hour = 5.4 km/h 0 1000 800 600 400 200 1200 1400 1600 1800 2000 0 1 2 3 4 5 6 Time (minutes) Distance (m) Lewis Neil © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 530 2867_P001_680_Book_MNPB.indb 530 24/03/2020 15:33 4/03/2020 15:33

531 37.2 15 Some friends took part in a K1 kayak slalom competition. Over the 1 km course, there were 12 down gates and 6 up gates. It is reckoned that every down gate adds 1 second to a competitor’s time and each up gate adds 3 seconds. Any missed gate adds a 15 second time penalty. a Lucy gets through every gate successfully and finishes in 2 minutes 30 seconds. What was her average speed? b Gemma’s usual average speed without any gates is 33 km/h. Adding extra time for the gates, how long did she take if she got through every gate successfully? c Sarah paddles at an average 36 km/h but misses one gate. In what order did these three girls finish? 16 The speed of sound is 340m/s. There is an explosion 2km away from Sam. Calculate how many seconds will pass before Sam hears the explosion. 17 On a journey lasting 23 minutes, this bicycle wheel rotated at a rate of 95 revolutions per minute. a Calculate the circumference of the wheel. b Calculate the length of the journey. Give your answer in km, correct to the nearest 100 m. 18 Amy takes 2 hours to travel a third of her journey at a speed of 70 km/h. At what speed must she travel the rest of the journey so that she can complete the whole journey in 7 hours? 37.2 More about proportion ● I can understand and use density and other compound units Another example of compound units is the rate of flow, which is a measure of how quickly a liquid is flowing. Another example of a compound unit is density. This is calculated as ‘mass per unit volume’: density = mass volume Examples of possible units are: • grams per cubic centimetre (g/cm³) • grams per litre (g/litre). If you compare equal volumes of different substances, the denser one will be heavier. 42 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 531 2867_P001_680_Book_MNPB.indb 531 24/03/2020 15:33 4/03/2020 15:33

532 Chapter 37 Develop fluency Water is flowing out of a tap. In 5 minutes, 24 litres flow out of the tap. a Work out the rate of flow, in litres per minute. b How long does it take to fill a 7.5 litre bucket? a The rate of flow is measured as litres minutes . This is the number of litres ÷ number of minutes and the unit is litres/minute. Rate of flow = 24 5 = 4.8 litres/minute. b 4.8 = 7.5 m Call the number of minutes m. 4.8m = 7.5 Multiply by m. m = 7.5 4.8 = 1.5625 Divide by 4.8. It takes 1.56 minutes. Round the answer. It is just over 1 1 2 minutes. A piece of iron has a volume of 20 cm3 and a mass of 158 g. a Calculate the density of iron. b Calculate the mass of 36 cm3 of iron. a Density = mass volume = 158 20 = 7.9 g/cm3 The units must involve grams and cm3. b Density = mass volume 7.9 = m 36 Use the answer from part a. Use m for the mass. m = 7.9 × 36 = 284.4 Multiply by 36. The mass is 284 g to three significant figures. 1 A driver takes 20 seconds to put 56 litres of petrol in the fuel tank of her car. What is the rate of flow of the petrol? Give units in your answer. 2 A tap is dripping. In 20 minutes, 0.3 litres drip from the tap. What is the rate of flow, in litres per hour (litres/h)? 3 Water flows down a stream at a rate of 6 litres/s. a Calculate how much water flows in one minute. b Calculate how much water flows in half an hour. c Work out how long it takes for 1000 litres to flow past a particular point. 4 A shower has a rate of flow of 12 litres/minute. a Work out how much water is used when Sam has a shower that lasts four minutes. b Work out how long it takes to use 1 litre of water. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 532 2867_P001_680_Book_MNPB.indb 532 24/03/2020 15:33 4/03/2020 15:33

533 37.2 5 Balsa wood is used to make model aeroplanes because it has a low density of 0.16 g/cm3 . a A piece of balsa wood has a volume of 25 cm3 . Calculate its mass. b Calculate the volume of 1 g of balsa wood. 6 Oxygen has a density of 1.43 g/litre. Work out the mass of 200 litres of oxygen. 7 Graphite is used to make pencils. It has a density of 2.3 g/cm3 . a Work out the volume of 50 g of graphite. b A pencil contains 1.2 cm3 of graphite. Work out the mass of graphite in 10 pencils. c Work out the mass of 100 cm3 of graphite. 8 A piece of copper has a mass of 45 g and a volume of 5 cm3 . a Work out the density of copper. b Work out the volume of 120 g of copper. c Work out the mass of a cube of copper with a side of 5 cm. Reason mathematically The graph shows the relationship between the mass (m kg) of metal A and its volume (V cm3 ). Georgina says that she used 300 g of metal A to make a cube with a side length 3 cm. Explain why Georgina is incorrect. Density of metal A = 6.25 500 = 0.0125 kg/cm3 Volume of cube A = 33 = 27 cm3 Mass of cube A = 0.0125 × 27 = 0.3375 kg 1 kg = 1000 g Mass of cube A = 337.5 g. It is not 300 g. m v (500, 6.25) 0 9 This graph shows the connection between mass and volume for a type of steel. a Use the graph to find the mass of 2 m3 of steel. b Work out the density of the steel. Give your answer in kg/m3 . c The mass of a particular type of steel beam is 400 kg. Show that the volume of the beam is 0.05 m3 . d Explain how you know that the mass of the beam is proportional to its volume. Hint: 1m3 = 1000000cm3 0 8000 4000 12 000 6000 14 000 16 000 18 000 2000 10 000 0 0.5 1 1.5 2 Volume (m3 ) Mass (kg) © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 533 2867_P001_680_Book_MNPB.indb 533 24/03/2020 15:33 4/03/2020 15:33

534 Chapter 37 10 A hosepipe is able to fill this can at a rate of 15 litres/minute. a Calculate: i the volume of the can in cubic centimetres ii the capacity of the can in litres. b How long does it take to fill the can? Give your answer to the nearest second. 11 The ratio between the mass (m, g) and volume (V, cm3 ) of substance P, is 3 : 10. a Write a formula connecting the mass and volume of substance P. b What is the density of substance P? c Which of the graphs below can represent substance P? Explain why. m v (40, 12) a 0 m v (12, 40) b 0 m v 12 40 c 0 12 40 m v d 0 Solve problems The density of copper is 8.96 g/cm3 . The mass of 50 cm3 of zinc is 356.50 g. 1344 g of copper and 356.50 g of zinc were mixed together. Work out the density of the mixture. Give your answer to two decimal places. density = mass volume Mass of mixture = mass of copper + mass of zinc = 1344 g + 356.50 g = 1700.50 g Volume of copper = 1344 ÷ 8.96 = 150 cm3 Volume of mixture = volume of copper + volume of zinc = 150 + 50 = 200 cm3 Density of mixture = 1700.50 ÷ 200 = 8.50 g/cm3 12 a A hosepipe is accidentally left on for 15 minutes. The water flows at 9 litres per minute. How much water is wasted? b The charge for water at this house is 80p per cubic metre. How much has this wastage cost? 20 cm 30 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 534 2867_P001_680_Book_MNPB.indb 534 24/03/2020 15:33 4/03/2020 15:33

535 37.3 13 a Car A takes 35 seconds to fill its fuel tank with 70 litres of diesel. What is the rate of flow of the fuel at this pump? b At the same pump, Car B takes 22 seconds to fill up. How much fuel does it hold? c Mary’s car holds 16 gallons of petrol. How long does it take to fill up, to the nearest second? d If diesel costs £1.39 per litre and petrol costs £1.32, which of the three cars costs most to fill up? 14 The density of gold is 19.3 g/cm3 . a Work out the mass of: i 3.5 cm3 of gold ii 3.5 g of gold.. b The largest gold bar in the world has a mass of 250 kg. Work out its volume. 15 An alloy is made from 3.2 kg of copper, 1.5 kg of lead and 600 g of tin. Calculate the density of the alloy. Metal Density (g/cm3 ) copper 8.96 lead 11.40 tin 7.30 37.3 Unit costs ● I can understand and use unit pricing Packets and containers of food and other items are sold in different sizes. If you want to compare the prices of the same item, in different sized containers, it is helpful to be able to work out a unit price. This is the price of one gram or one litre, or any other suitable unit. Develop fluency Compare the prices of the rice in these packets. • You could find the cost/gram of rice from each packet. For the smaller packet, 89p ÷ 200 = 0.445 p/g (pence per gram). For the larger packet, 209p ÷ 500 = 0.418 p/g. The larger packet is better value, as it has the lower cost per gram. • You could find how much you can buy for 1p. For the smaller packet, the number of grams/p is 200 ÷ 89 = 2.247... g/p. For the larger packet, it is 500 ÷ 209 = 2.392... g/p. The larger packet is better value because you can buy more for 1p. 89p £2.09 1 1 kg of margarine costs £2.50. a Calculate the cost: i per 100 g ii per gram. b Calculate the number of grams bought for: i 1p ii £1. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 535 2867_P001_680_Book_MNPB.indb 535 24/03/2020 15:33 4/03/2020 15:33

536 Chapter 37 2 A bag of pasta has a mass of 250 g and costs 87p. a Calculate the cost: i per 100 g ii per gram. b Calculate the number of grams bought for: i 1p ii £1. 3 A 160 g can of tuna costs £1.85. A pack of four cans costs £6.20. a Work out the cost per 100 g for one can. b Work out the cost per 100 g if you buy the pack of four cans. c Which is better value? Give a reason for your answer. 4 You can buy tomato puree in tubes or jars. A shop sells 140 g tubes for 69p and 200 g jars for £1.09. a Work out the cost per 100 g in a tube of tomato puree. b Work out which is better value for money. Justify your answer. 5 Aziz buys 700 g of apples for £1.68. Calculate the cost per kilogram. 6 Miriam downloaded 928 kB of data from the Internet in 18 seconds. a How much data did she download each second, to the nearest kB? b How much data could she download at the same rate in: i 45 seconds ii 7 minutes? Reason mathematically Tom and Ben travel to work. Tom’s car uses 10.6 litres of petrol on a 160 km journey. The price of petrol is £1.26/litre. Ben travels 256 km by train. His train ticket costs £25.50. Which is better value, to drive or get the train? Explain why. Cost of driving per km = 10.6 1.26 160 = £0.083475 per km Cost of travelling by train per km = 25.50 256 = £0.0996 per km £0.083475 per km is less than £0.0996 per km. It is cheaper to drive. 7 A shop sells cans of pineapple in two sizes. The prices are shown in this table. Mass 225 g 435 g Price 83p £1.45 Which is better value? Justify your answer. 8 A 600 g box of muesli costs £2.65. An 850 g box costs £3.95. Which is better value? Justify your answer. 9 Toothpaste comes in tubes of different sizes. A 125 ml tube costs £2.99 and a 75 ml tube of the same brand costs £1.89. Which is better value for money? Justify your answer. 10 A pack of four 120 g pots of yogurt costs £2.00. A 450 g pot of the same yogurt costs £1.79. Show that the large pot is better value. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 536 2867_P001_680_Book_MNPB.indb 536 24/03/2020 15:33 4/03/2020 15:33

537 37.3 Solve problems Milk is sold in litres or pints. One pint is 568 ml. A supermarket sells 1 litre of milk for 95p or 1 pint for 49p. Which is better value? Work out the cost per litre. 1 pint = 568 ml = 0.568 litre Cost per litre = 49p ÷ 0.568 =86.3p per litre The pint of milk is better value, because it is cheaper per litre. 11 a Eighteen eggs cost a shopkeeper £3.60. How much does each egg cost him? b He puts them in boxes of six and charges £1.80 per box. How much profit does he make on each box? Express this as a percentage of the cost. c He does an offer of three boxes for five pounds. What is his percentage profit here? 12 A 400 g packet of beef costs £4.60. What price would you expect to pay for a 750 g pack? Justify your answer. 13 The diagram shows how two businesses charge for the hire of a canoe. a For which business is the charge directly proportional to the hire period? Give a reason for your answer. b For this business, what is the hire charge for one hour? c For this business write an equation connecting the hire charge, H, with the hire period, p. d i What would this company charge to hire the canoe for 4.5 hours? ii What is the hire period corresponding to a hire charge of £18 with this company? Give your answer in hours and minutes. 5 0 10 15 20 25 0 1 2 3 4 5 6 7 8 9 10 Hire period, p (hours) Hire charge, H (£) Rapids White Water Canoe hire charges Now I can… understand measures of speed use measures of speed use unit pricing understand density and other compound units use density and other compound units understand unit pricing © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 537 2867_P001_680_Book_MNPB.indb 537 24/03/2020 15:33 4/03/2020 15:33

538 38 Solving equations graphically 38.1 Graphs from equations of the form ay ± bx = c ● I can draw any linear graph from any linear equation ● I can solve a linear equation from a graph You have already met the linear equation y = mx + c, which generates a straight-line graph. It is this equation that is seen here as ay ± bx = c. When its graph is plotted it will, of course, still produce a straight line. To construct the graph of ay ± bx = c, follow these steps: • Substitute x = 0 into the equation so that it becomes ay = c. • Solve for y, i.e. y = c a so that one point on the graph is 0, c a . • Substitute y = 0 into the equation so that it becomes bx = c. • Solve for x, i.e. x = c b , so that one point on the graph is c b , 0 . • Plot the two points on the axes and join them up. Develop fluency Draw the graph of 4y – 5x = 20. Substitute x = 0: 4y = 20 y = 5 The graph passes through (0, 5). Substitute y = 0: –5x = 20 x = –4 The graph passes through (–4, 0). Plot the points and join them up. Note that this method is sometimes called the ‘cover-up’ method as all you have to do to solve x or y is cover up the other term. 6 4 2 0 –2 –4 –1 2 3 x y –5 –4 –3 –2 0 1 1 Draw the graph for each equation. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. a y = x + 3 b y = x – 4 c y = 2x d y = 3x + 1 2 Draw the graph for each equation. Use a grid that is numbered from 0 to +10 on both the x-axis and the y-axis. a 2y + 3x = 6 b 4y + 3x = 12 c y + 2x = 8 d 3y + 2x = 6 e 5y + 2x = 10 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 538 2867_P001_680_Book_MNPB.indb 538 24/03/2020 15:33 4/03/2020 15:33

539 38.1 3 Draw the graph for each equation. Use a grid that is numbered from –10 to +10 on both the x-axis and the y-axis. a y – 5x = 10 b 2y – 3x = 12 c 2y – 5x = 10 d 3x – 4y = 12 e 5y – 2x = 10 4 For each graph, find the coordinates of the two points where the graph intersects the x-axis and the y-axis. a 3x + 2y = 18 b 5y − x = 15 c 4y − 7x = −28 5 Draw the graph for each equation. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. a 3y + 2x = 12 b 4x + 5y = 40 c 3y + 7x = 21 d 6x − y = 6 e 3y − 4x = 24 6 a Using a grid with axes numbered from –2 to 10, draw the graph of y = 4x + 3. b Use the graph to solve these equations. i 4x + 3 = 7 ii 4x + 3 = 9 iii 4x + 3 = 5 7 a Using a grid with axes numbered from –5 to 8, draw the graph of y = 3x – 1. b Use the graph to find the value of y when i x = – 1 ii x = 0 iii x = 3. c Use the graph to find the value of x when i y = – 1 ii y = 1 iii y = 5. 8 a Using a grid with axes numbered from –2 to 10, draw the graph of y = 3 – 2x. b Write down the coordinates of the point where your graph crosses the i x-axis ii y-axis. c Use your graph to find the value of x when y = 6. 9 a Using a grid with axes numbered from –3 to 4, draw the graph of y = 3 – x. b The point with coordinates (p, 4.5) lies on the graph of y = 3 – 3x. Use your graph to find the value of p. Reason mathematically Vicky says that the graph shown has the equation 3x + 4y = 12. Is she correct? Explain your answer. Find the x and y intercept of 3x + 4y = 12. When y = 0, 3x = 12 so x = 4. When x = 0, 4y = 12 so y = 3. Vicky is incorrect as the line intercepts the axes at x = 3 and y = 4. –1 0 1 2 3 4 –4 –3 –2 –1 –5 5 –4 –3 –2 1 2 3 4 x y © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 539 2867_P001_680_Book_MNPB.indb 539 24/03/2020 15:33 4/03/2020 15:33

540 Chapter 38 10 a Draw the graphs of all these equations on the same grid. Use a grid that is numbered from −10 to +10 on both the x-axis and the y-axis. i y = 1 2 x + 4 ii y = 1 2 x − 2 iii y = 1 2 x b What do you notice about all these graphs? c Explain how you could now draw the graph of y = 1 2 x + 6. 11 Draw the graphs for all these equations on the same grid. Use a grid that is numbered from −10 to +10 on the x-axis and from −2 to +10 on the y-axis. a i 3y + x = 6 ii 4x − 5y = −10 iii x + y = 2 iv 2x − 9y = −18 b What do you notice about all these graphs? 12 a Draw the graphs of all these equations on the same grid. Number both axes from –2 to 10. i x + y = 6 ii x + y = 8 iii x + y = 10 iv x + y = 2 v x + y = 1 vi x + y = 7 b What do you notice about all these graphs? c Explain how you could now draw the graph of x + y = –5.3. 13 a Copy and complete the table for the graph y = 6 x . b Why can you not find the value of y when x = 0? c Using a grid with axes numbered from 0 to 6, draw the graph of y = 6 x . Solve problems At 0 hours, there are 42 litres of fuel in the fuel tank of a car. After 7 hours, there is no fuel left in the fuel tank. Assume that the fuel is used at a constant rate. a Draw a graph to show the amount of fuel left in the tank against time. b Use your graph to estimate: i the amount of fuel used per hour of driving ii the volume of fuel in the tank after 3 hours iii the number of hours it takes for the tank to be half-full. a b i After 1 hour, the volume of the tank is 36 litres. The amount of fuel used per hour is 6 litres. ii When t = 3 hours, V = 24 litres. iii When V = 21 litres, t = 3 1 2 hours 0 1 2 3 Time (hours) Volume (litres) 4 5 6 7 0 10 20 30 40 50 y x x 123456 y © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 540 2867_P001_680_Book_MNPB.indb 540 24/03/2020 15:33 4/03/2020 15:33

541 38.2 14 Angela works on Saturdays at the local cafe. One day, Angela receives £64 for 8 hours work. a Draw a graph to show the monthly pay, P, that Angela receives based on the time, T, that she works. b What is Angela’s hourly rate? c Use the graph to calculate the number of hours Angela needs to work to earn £100. d Why is it not realistic to find a value for P when T = 100? 15 A car is slowing down. The speed of the car, s metres per second, after time t seconds is given by the formula s = 30 – 4t a Calculate the initial speed of the car. b Sketch the graph showing the relationship between speed (s) and time (t). c Use the graph to find the time that it will take for the car to stop. 16 A bamboo shoot is 10cm high. It then grows 4cm every day. a Draw a graph to show the relationship between the height of the bamboo and the number of days. b Use your graph to calculate the number of days it takes for the bamboo to reach the height of 0.9m. c Ben thinks that he can’t use the same graph to find the height of the bamboo after two years. Explain why. 17 For each graph sketch, think of a real-life representation it could represent. State the quantities that x and y can represent and create a problem based on the graph. a b 38.2 Graphs from quadratic equations ● I can draw graphs from quadratic equations A quadratic equation involves two variables where the highest power of one of the variables is a square. Some examples of quadratic equations are y = x2 , y = x2 + 3x, y = 2x2 + x – 1, y = (x + 2)(x + 1). 0 1 2 3 4 5 0 1 2 3 4 y x 0 1 2 3 4 5 0 1 2 3 4 y x © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 541 2867_P001_680_Book_MNPB.indb 541 24/03/2020 15:33 4/03/2020 15:33

542 Chapter 38 Develop fluency Draw the graph of the equation y = x2 + 3x. First, draw up a table of values for x, then substitute each value of x into x2 and 3x to determine the y-value. x –4 –3 –2 –1 0 1 2 x2 16 9 4 1 0 1 4 3x –12 –9 –6 –3 0 3 6 y = x2 + 3x 4 0 –2 –2 0 4 10 Now take the pairs of (x, y) coordinates from the table, plot each point on a grid, and join up all the points. Note that the shape is a smooth curve. It is important always to try to draw a quadratic graph as smoothly as possible, especially at the bottom of the graph where it needs to be a smooth curve. 2 0 4 6 8 10 –2 –4 –3 –2 –1 0 1 3 2 x y 1 a Copy and complete this table of values for y = x2 + 2x. x –3 –2 –1 0 1 2 x2 941014 2x –2 0 y = x2 + 2x 0 b Draw a grid with the x-axis numbered from –3 to 2 and the y-axis from –2 to 10. c Use the table to help you draw, on the grid, the graph of y = x2 + 2x. 2 a Copy and complete this table of values for y = x2 + 4x. x –5 –4 –3 –2 –1 0 1 x2 25 4x –20 y = x2 + 4x 5 b Draw a grid with the x-axis numbered from –5 to 1 and the y-axis from –5 to 6. c Use your table to help draw, on the grid, the graph of y = x2 + 4x. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 542 2867_P001_680_Book_MNPB.indb 542 24/03/2020 15:33 4/03/2020 15:33

543 38.2 3 a Copy and complete this table of values for y = x2 + 3x + 2. x –4 –3 –2 –1 0 1 x2 9 0 3x –9 0 2 222222 y = x2 + 3x + 2 2 2 b Draw a grid with the x-axis numbered from –4 to 2 and the y-axis from –1 to 8. c Use your table to help you draw, on the grid, the graph of y = x2 + 3x – 2. 4 a Copy and complete this table of values for y = x2 + 2x – 3. x –4 –3 –2 –1 0 1 2 x2 2x –3 y = x2 + 2x – 3 b Draw a grid with the x-axis numbered from –4 to 2 and the y-axis from –5 to 6. c Use your table to help you draw, on the grid, the graph of y = x2 + 2x – 3. 5 Copy and complete this table of values for y = x2 + x. x −1 0 1 2 3 x2 x y 6 a Copy and complete this table of values for y = x2 + 6x. x −7 −6 −5 −4 −3 −2 −1 0 1 x2 6x y b Draw a grid with the x-axis numbered from −7 to 1 and the y-axis from −10 to 10. c Use the table to help you draw, on the grid, the graph of y = x2 + 6x. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 543 2867_P001_680_Book_MNPB.indb 543 24/03/2020 15:33 4/03/2020 15:33

544 Chapter 38 7 a Copy and complete this table of values for y = 2x2 + 1. b Draw a grid with the x-axis numbered from −2 to 2 and the y-axis from 0 to 10. c Use the table to help you draw, on the grid, the graph of y = 2x2 + 1. 8 Construct a table of values for each equation. Using suitable scales, draw the graphs. a y = x2 + 5x b y = x2 + 3x + 1 c y = x2 + 4x – 3 9 a Copy and complete the table of values for y = 3x2 – 5. x –2 –1 0 1 2 x2 4 0 3x2 12 0 y = 3x2 – 5 7 –5 b Draw a grid with the x-axis numbered from –2 to 2 and the y-axis numbered from –7 to 10. c Use the table to help you draw, on the grid, the graph of y = 3x2 – 5. Reason mathematically a Construct a table of values for each equation. i y = x2 ii y = x2 + 1 iii y = x2 + 4 b Plot all the graphs on the same pair of axes. Number the x-axis from −3 to 3 and the y-axis from 0 to 13. c Comment on your graphs. d Sketch onto your diagram the graph with the equation y = x2 + 3. a x −3 −2 −1 0 1 2 3 y = x2 9410149 y = x2 + 1 10 5 2 1 2 5 10 y = x2 + 4 13 8 5 4 5 8 13 b See the graph. c The graphs are the same shape, however they have been translated in the positive y direction. y = x2 translated by 1 unit in the y-direction is y = x2+ 1. y = x2 + 1 translated by 3 units in the positive y direction is y = x2 + 4. –3 –2 –1 0 1 2 3 4 5 6 7 8 1 2 3 y = x2 y = x2 + 1 y = x2 y = x + 3 2 + 4 y x x −2 −1 0 1 2 x2 2x2 y © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 544 2867_P001_680_Book_MNPB.indb 544 24/03/2020 15:33 4/03/2020 15:33

545 38.2 10 a Construct a table of values for each equation, then plot all their gra phs on the same pair of axes. Number the x-axis from –4 to 4 and the y-axis from –10 to 55. i y = x2 – 10 ii y = 2x2 – 10 iii y = 3x2 – 10 iv y = 4x2 – 10 b Comment on your graphs. c Sketch onto your diagram the graphs with these equations. i y = 1 2 x2 – 10 ii y = 2 1 2 x2 – 10 iii y = 5x2 – 10 11 a Construct a table of values for each equation. Then plot all their graphs on the same pair of axes. Number the x-axis from −2 to 2 and the y-axis from −3 to 16. i y = 3x2 – 2 ii y = 3x2 iii y = 3x2 + 1 iv y = 3x2 + 3 b Comment on your graphs. c Sketch onto your diagram the graph with the equation y = 3x2 + 2. 12 a Copy and complete this table of values for y = 8 − 2x − x2 . x −5 −4 −3 −2 −1 0 1 2 3 8 −2x − x2 y b Draw a grid with the x-axis numbered from −5 to 3 and the y-axis from −8 to 10. c Use the table to help you draw, on the grid, the graph of y = 8 − 2x − x2 . d Comment on the shape of the graph. 13 a Copy and complete this table for y = (x + 1)2 . x −5 −4 −3 −2 −1 0 1 2 3 x + 1 (x + 1)² b Using the method in part a, construct a table for each equation. Plot all the graphs on the same pair of axes. Number the x-axis from −5 to 3 and the y-axis from 0 to 50. i y = x2 ii y = (x – 1)2 iii y = (x – 2)2 c Comment on your graphs. What do you notice? d Sketch onto your diagram the graph with the equation y = (x – 4)2 . © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 545 2867_P001_680_Book_MNPB.indb 545 24/03/2020 15:33 4/03/2020 15:33

546 Chapter 38 Solve problems Bob sows grass seeds in the garden. This chart shows the height of the grass over the next 30 days. Days 4 5 10 15 20 25 30 Height (mm) 0 0.5 2.0 6.0 7.5 9.0 10.5 a Use this information to draw a graph. b How many days is it before the grass shows above the ground? c When is the grass growing at its slowest pace? d What is the height of the grass after 12 days? e How many days is it before the grass is 10mm long? a 0 5 Days Height (mm) 10 15 20 25 30 0 5 10 15 20 25 y x b 5 days c From day 4 to day 5 d 4cm e 29 days 14 Andrew makes gold rings. This chart shows his charges for rings of different diameters. Diameter (mm) 5 10 15 20 25 Cost (£) 45 61 87 124 170 a Use this information to draw a graph. b Use your graph to estimate the cost of a ring with diameter: i 12mm ii 24mm. c Brian bought a gold ring from Andrew for £100. What was the diameter of the ring? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 546 2867_P001_680_Book_MNPB.indb 546 24/03/2020 15:33 4/03/2020 15:33

547 38.2 15 The length (L) of a pendulum is related to the period of its swing cycle (T). The table shows the lengths of five pendulums and their periods. T (seconds) 2 4 6 8 10 L (metres) 1.0 4.0 8.9 15.9 24.8 a Draw a graph with T on the x-axis and L on the y-axis. b Use your graph to estimate the length of a pendulum with a period of 7.5 seconds. c Use your graph to estimate the period of a pendulum with a length of 5m. 16 The time (T) it takes to complete a 5km run is related to the speed (S) that a person is running. The table shows the time it took to complete a 5km run. The measurements were taken every kilometre. Length, L (km) 12345 Time, T (minutes) 5.8 11.4 18.1 23 29.2 a Draw a graph with L on the x-axis and T on the y-axis. b Use your graph to estimate the time it took to run 2.5km. c Use your graph to estimate how far can a person run in 20 minutes. d Ashley uses the graph to estimate the time she will take to run 10km. Explain why this may not be accurate. 17 A ball is kicked from the ground up in the air. The table shows the height of the ball 10 seconds after it was kicked in the air. Time, T (seconds) 0 1 2 3 4 5 6 7 8 9 10 Height, H (m) 0 9 16 21 24 25 24 21 16 9 0 a Draw a graph with time, T, on the x-axis and height, H, on the y-axis. b Use your graph to estimate the length of the ball’s journey. c Use your graph to estimate the maximum height the ball reached. d Use your graph estimate the time the ball was 20m or higher above the ground. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 547 2867_P001_680_Book_MNPB.indb 547 24/03/2020 15:33 4/03/2020 15:33

548 Chapter 38 38.3 Solving quadratic equations by drawing graphs ● I can solve a quadratic equation by drawing a graph You can find out a lot of information from a quadratic graph, when you know how to do it. Develop fluency a Draw the graph of y = x2 + 4x – 5. b Use your graph to find the value of y when x = –3.5. c Find the solution to the equation x2 + 4x – 5 = 3.2. d What are the solutions to the equation x2 + 4x – 5 = 0? a Draw the graph, as shown. b Draw a dotted line from x = –3.5 to the graph. Following this across to the y-axis, you can see that when x = –3.5, y = –6.75. c To find the solution of the equation x2 + 4x – 5 = 3.2, draw a dotted line across the graph of y = x2 + 4x – 5 where y = 3.2. The dotted line for y = 3.2 cuts the graph in two places. So there are two solutions to this equation. Drawing dotted lines down from the graph to the x-axis, you can see that the solutions are x = –5.5 and x = 1.5. d You can find the solution of the equation x2 + 4x – 5 = 0 on the graph where y = 0. This is where the graph cuts the x-axis. You can see that this will be where x = –5 and x = 1. Note that the solution of a quadratic equation will often give two answers, but not always! 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 2 3 4 5 6 7 8 –6 –5 –4 –3 –2 –1 0 1 2 x y For all graphs in this exercise, use a scale of 2cm to 1 unit on each axis. 1 a Draw the graph of y = x2 from x = –3 to 3. b Write down the value of y when x = 2.1. c Use the graph to find the solutions to these equations. i x2 = 3 ii x2 = 6 iii x2 = 7.5 2 a Draw the graph of y = x2 + 2x from x = –3 to 3. b Write down the value of y when x = 0.7. c Use the graph to find the solutions to these equations. i x2 + 2x = 2 ii x2 + 2x = 1 iii x2 + 2x = 0 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 548 2867_P001_680_Book_MNPB.indb 548 24/03/2020 15:33 4/03/2020 15:33

549 38.3 3 a Draw the graph of y = x2 – x from x = –2 to 3. b Write down the value of y when x = –0.9. c Use the graph to find the solutions to these equations. i x2 – x = 3 ii x2 – x = 1.5 iii x2 – x = 0.5 4 a Draw the graph of y = x2 + 3x – 2 from x = –4 to 2. b Write down the value of y when x = 1.6. c Use the graph to find the solutions to the following equations. i x2 + 3x – 2 = 0 ii x2 + 3x – 2 = –1 iii x2 + 3x = 3 5 a Draw the graph of y = x2 + 2x – 3 from x = –4 to 4. b Write down the value of y when x = –1.7. c Use the graph to find the solutions to the following equations. i x2 + 2x – 3 = 1 ii x2 + 2x – 3 = 0 iii x2 + 2x – 3 = –1 6 Draw a graph to find the solutions of x2 + x – 7 = 0. 7 Draw a graph to find the solutions of x2 – 4x + 1 = 0. 8 a Draw the graph of y = x2 − 2x − 4 from x = −3 to 5. b Use the graph to find solutions to these equations. i x2 − 2x − 4 = 0 ii x2 − 2x = 1 iii x2 − 2x = 10 9 Draw graphs to find the solutions of: a x2 − 3x − 1 = 0 b x2 + 4x − 6 = 2 c x2 − x − 4 = 5. 10 Draw a graph to find the solutions of 2x2 − 3x − 1 = 0. Reason mathematically a Draw the graph of y = x2 + 2x + 4 from x = −4 to 2. b Use the graph to explain why x2 + 2x + 4 = 0 has no solution. a –4 –3 –2 –1 0 1 2 0 2 4 6 8 10 12 14 x y b The graph does not intersect the x-axis, so there are no solutions. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 549 2867_P001_680_Book_MNPB.indb 549 24/03/2020 15:33 4/03/2020 15:33

550 Chapter 38 11 Lo ok at the graph of y = x2 − 2x − 4 from question 8. Explain why there is no solution for values of y that are less than –5. 12 Look at the graph of y = x2 − 3x − 1 from question 9a. a Explain why there is only one solution for y = − 13 4 . b Explain why there are no solutions for y < − 13 4 . 13 Look at the graph of y = 2x2 − 3x – 1 from question 10. Explain whether each statement is true or false. a The minimum point of y = 2x2 − 3x – 1 is (3 4, –3). b When x < 3 4, 2x2 − 3x – 1 = 0 has no solutions. c 2x2 − 3x – 1 = –3 has no solutions. Solve problems A rectangle is (x + 3.5) cm long and x cm high. a Show that the area of the rectangle is x2 + 3.5x. b Sketch a graph for the area for 0 < x < 5. c The area of the rectangle is 30cm2 . Use your graph to find the length and height of the rectangle. a Area of rectangle = x × (x + 3.5) = x2 + 3.5x b 0 2 4 6 0 5 10 15 20 25 30 x y c x = 4cm, so height is 4cm and length is 7.5cm. 14 a Draw the graph of y = 2x2 + x − 3 from x = −3 to 3. b Write down the value of y when x = −1.7 c Use the graph to find the solutions to the equations: i 2x2 + x − 3 = 0 ii 2x2 + x = 7 iii 2x2 + x − 2 = 0 d Draw a straight line on your graph to solve the equations: i 2x2 + x − 3 = x + 5 ii 2x2 + 2x − 7 = 0 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 550 2867_P001_680_Book_MNPB.indb 550 24/03/2020 15:33 4/03/2020 15:33