50 2 Expressions, formulae and equations 11 Read what Zara says. Show that she is right. 12 Read what Marcus says. Show that he is wrong. Explain the mistake Marcus has made. 13 The diagrams show two rectangles. length 2a length A 3d B The area of rectangle A is 2a2+18a. The perimeter of rectangle B is 14d−10c. Write an expression for the length of each rectangle, in its simplest form. When I expand 5(2x+6)+2(3x−5), then collect like terms and finally factorise the result, I get the expression 4(4x+5) When I expand 6(3y+2)−4(y−2), then collect like terms and finally factorise the result, I get the expression 2(7y+2) Tip You need to factorise the expressions to find the lengths of A and B. You will need an extra step first for rectangle B. Summary checklist I can use the highest common factor (HCF) to factorise an expression. Work with a partner. Take it in turns to define the following terms. a What is a factor? b What is the highest common factor? c What is factorising? How did your answers to a and b help with your answer to c? 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
51 2.5 Constructing and solving equations You already know the difference between a formula, an expression and an equation. Remember Examples A formula is a rule that shows the relationship between two or more quantities (variables). It must have an equals sign. F=ma v=u+10t An expression is a statement that involves one or more variables, but does not have an equals sign. 3x−7 a b 2 + 2 An equation contains an unknown number. It must have an equals sign, and it can be solved to find the value of the unknown number. 3x−7=14 4=6y+22 When you are given a problem to solve, you may need to construct, or write, an equation. 2.5 Constructing and solving equations In this section you will … • write and solve equations. Key word construct Worked example 2.5 a Write if each of the following is a formula, an expression or an equation. i 4c+3e ii P=8h+b iii 9k−2=16 b The diagram shows a rectangle. Work out the values of x and y. Answer a i expression 4c+3e involves two variables but does not have an equals sign. ii formula P=8h+b is a rule showing the relationship between three quantities, P, h and b. iii equation 9k−2=16 contains an unknown number, k, it has an equals sign, and it can be solved to find the value of k. 3(x + 3) cm 24cm 5y − 4cm 3y + 8 cm 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
52 2 Expressions, formulae and equations Continued b 3(x+3)=24 3x+9=24 3x+9−9=24−9 The two lengths must be equal, so construct an equation by writing one length equal to the other. First, multiply out the brackets. Then use inverse operations to solve the equation. Start by subtracting 9 from both sides. 3x=15 x=15 3 =5 Simplify both sides of the equation. Divide 15 by 3 to find the value of x. 5y−4=3y+8 5y−4−3y=3y+8−3y 2y−4=8 2y−4+4=8+4 The two widths must be equal, so write one width equal to the other. Rewrite the equation by subtracting 3y from both sides. Simplify. Use inverse operations to solve the equation. Start by adding 4 to both sides. 2y=12 y=12 2 =6 Simplify both sides of the equation. Divide 12 by 2 to work out the value of y. Exercise 2.5 1 Write if each of the following is a formula, an expression or an equation. a 3y+7=35 b 6(x+5) c T=3a2−8d d 9u−vw 2 Copy and complete the workings to solve these equations. a 3x+5=26 (subtract 5 from both sides) 3x+5−5=26−5 (simplify) 3x = (divide both sides by 3) x = 3 (simplify) x= b 4(x−3)=24 (multiply out the brackets) 4x−12=24 (add 12 to both sides) 4x−12+12=24+12 (simplify) 4x= (divide both sides by 4) x = 4 (simplify) x= 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
53 2.5 Constructing and solving equations c y 4 − = 10 1 (add 10 to both sides) y 4 −10+10=1+10 (simplify) y 4 = (multiply both sides by 4) y= ×4 (simplify) y= d 6y+2=4y+18 (subtract 4y from both sides) 6y−4y+2=4y−4y+18 (simplify) y+2=18 (subtract 2 from both sides) y+2−2=18 (simplify) y= (divide both sides by ) y = (simplify) y= 3 For each learner i write an equation to represent what they say ii solve your equation to find the value of x. The first one has been started for you. a My sister is 15 years old. My Dad is x years old. Half of my Dad’s age minus 3 is the same as my sister’s age. b c My brother is 12 years old. My Mum is x years old. One-third of my Mum’s age plus 1 is the same as my brother’s age. x 2 −3=15 x 2 −3+3=15+3 x 2 = x= × 2 x= My Aunt is 30 years old. My Gran is x years old. One-quarter of my Gran’s age plus 9 is the same as my Aunt’s age. 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
54 2 Expressions, formulae and equations Think like a mathematician 4 Marcus and Sofia are discussing what equation to write to answer this question. The diagram shows an isosceles triangle. All measurements are in centimetres. Work out the value of y. As the triangle is an isosceles triangle, the two sides shown are equal in length. So I would write the equation 2y+7=5y−17 I would write the equation 5y−17=2y+7 because I think this is easier to solve. What do you think? Does it matter which way round you write the equation? Work with a partner to discuss and explain your answers. 2y+ 7 5y − 17 Think like a mathematician 6 Look back at your answers to Question 5. Discuss with a partner how you can check your value for x in each part. 5 Work out the value of x in each isosceles triangle. All measurements are in centimetres. a 6x − 3 9 b +20 x 2 27 c x + 35 5x − 13 Tip Think carefully about which way round you write your equations. 7 Work out the value of y in each shape. All measurements are in centimetres. Show how to check your answers are correct. a 4(y − 3) 2y + 2 b 3(y + 5) 8y − 5 c 2(y + 6) 4(y − 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
55 2.5 Constructing and solving equations 8 Work out the value of x and y in each diagram. All measurements are in centimetres. Show how to check your answers are correct. a 5x − 3 37 2(y + 3) 20 b 4y + 5 2y + 15 3x + 1 2(x + 5) c 3y + 16 3x + 11 8y − 4 5x − 3 d 8(y − 1) 25 16 + 17 x 4 9 Work in a group of three or four. For each part of this question: i Write an equation to represent the problem. ii Compare the equation you have written with the equations written by the other members of your group. Decide who has written the correct equation in the easiest way. iii Solve the equation you chose in part ii. a Emily thinks of a number. She multiplies it by 3 then adds 8. The answer is 23. What number did she think of? b Anders thinks of a number. He divides it by 4 then subtracts 8. The answer is 5. What number did he think of? c Sasha thinks of a number. She multiplies it by 5 then subtracts 4. The answer is the same as 2 times the number plus 20. What number did Sasha think of? d Jake thinks of a number. He adds 5 then multiplies the result by 2. The answer is the same as 5 times the number take away 14. What number did Jake think of? 10 The diagram shows the sizes of the angles in a triangle. a Write an equation to represent the problem. b Solve your equation to find the value of n. c Work out the size of each of the angles in the triangle. Tip Start by writing an expression for the total of the angles in the triangle. Then write an equation. Use the fact that the angles in a triangle add up to … n − 5 ° 6n ° 2n + 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
56 2 Expressions, formulae and equations You can see that instead of multiplying out the bracket, Mo’s first step is to divide both sides of the equation by 4. Use Mo’s method to solve these equations. a 6(3a+4)=12a b 5(4c−9)=25 c 3 4 18 2 5 d + = ( ) 11 The diagram shows the sizes of the two equal angles in an isosceles triangle. 4x − 6 ° 2x + 18 ° a Write an equation to represent the problem. b Solve your equation to find the value of x. c Work out the size of each of the angles in the triangle. 12 Solve these equations. Use the Tip box to help. a 5(2x+3)+2(x−4)=31 b 4(3x−1)−3(5−2x)=35 c 2 3 y = 8 d 3 5 y + = 1 19 13 This is part of Mo’s homework. Tip Start by expanding the brackets and simplifying the left hand side. 2 3 y is the same as 2 3 × y so start by multiplying both sides by 3. Question Solve the equation 4(2b – 3)=−8b Answer Divide both sides by 4 4(2b− 3) 4 = −8b 4 → 2b – 3=−2b Add 3 to both sides 2b – 3+3=−2b+3 → 2b=−2b+3 Add 2b to both sides 2b+2b=−2b+2b+3 → 4b=3 Divide both sides by 4 4b 4 = 3 4 → b= 3 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
57 2.6 Inequalities 14 Art has these cards. 2y+14 8(y−12) y 4 − 18 = 4 −2 −20 He chooses one blue card, the red card and one yellow card to make an equation. Which blue and yellow card should he choose to give him the equation with a the largest value for y b the smallest value for y? Explain your decisions and show that your answers are correct. You have already learned how to use a letter and an inequality sign to represent lots of numbers. Remember the inequality signs: < means ‘is less than’ ⩽ means ‘is less than or equal to’ > means ‘is greater than’ ⩾ means ‘is greater than or equal to’ So if you see the inequality x > 5, this means that x can be any number greater than 5. If you see the inequality y ⩽ 2, this means that y can be any number less than, or equal to, 2. 2.6 Inequalities Key words closed interval inequality integer Summary checklist I can understand equations and solve them. I can write equations and solve them. In this section you will … • use letters and inequalities to represent open and closed intervals. Tip Remember that you use an open circle ( ) for the < and > inequalities and a closed circle ( ) for the ⩽ and ⩾ inequalities. 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
58 2 Expressions, formulae and equations If you see the inequality 3 ⩽ x < 9, this means that x is greater than or equal to 3 and is also less than 9. This inequality represents a closed interval. You can show this on a number line like this. 98765432 10 Worked example 2.6 a i Show the inequality 2 ⩽ x < 6 on a number line. ii List the possible integer values for. b i Show the inequality −5 < y ⩽ −1 on a number line. ii List the possible integer values for y. Answer a i 87654321 Use a closed circle for the ⩽ sign and start the line at 2. Extend the line to 6, where you use an open circle for the < sign. ii 2, 3, 4, 5 x is greater than or equal to 2, so 2 is the smallest integer. x is less than 6, so 5 is the greatest integer. b i −6 −5 −4 −3 −2 −1 0 Use an open circle for the < sign and start the line at −5. Extend the line to −1, where you use a closed circle for the ⩽ sign. ii −4, −3, −2, −1 y is greater than −5, so −4 is the smallest integer. y is less than or equal to −1, so −1 is the greatest integer. Tip Remember that an integer is a whole number. Exercise 2.6 1 Write in words what each of these inequalities means. Part a has been done for you. a 6 1 < < x 1 x is greater than 6 and less than 11 b 12 ⩽ x ⩽ 18 c 0 < x ⩽ 20 d −9 ⩽ x < −1 Tip This closed interval includes the endpoints 3 and 9. 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
59 2.6 Inequalities 2 Write each statement as an inequality. Part a has been done for you. a y is greater than or equal to 3 and less than 17 3< y <17 b y is greater than 15 and less than 25 c y is greater than −2 and less than or equal to 5 d y is greater than or equal to −9 and less than or equal to −3 3 Copy each number line and show each inequality on the number line. a 4 7 < < x b 6 ⩽ x ⩽ 9 3 4 5 6 7 8 5 6 7 8 9 10 c −4 < x ⩽ 1 d −2 ⩽ x < 2 −5 −4 −3 −2 −1 0 1 2 4 Write the inequality that each of these number lines shows. Use the letter x. a 17161514131211 b 543210 c −3 −2 −1 0 1 2 d −3 −2 −1 0 1 2 3 0 2 4 6 8 10 12 Think like a mathematician 5 Sofia and Zara are looking at the inequality x > 5. In pairs or small groups, discuss Sofia’s and Zara’s comments. The inequality x > 5 is equivalent to x−2 > 3. The inequality x > 5 is equivalent to 2x > 10. a How can you show that Sofia and Zara are correct? b Write two different inequalities that are equivalent to x > 5. c Is it possible to say how many different inequalities there are that are equivalent to x > 5? Explain your answer. Tip Remember that ‘equivalent to’ means ‘the same as’ or ‘equal to’. 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
60 2 Expressions, formulae and equations 6 Copy and complete these equivalent inequalities. a x > 8 is equivalent to 3x > b x < 3 is equivalent to x < 15 c y > 7 is equivalent to y+3⩾ d y ⩽ 2 is equivalent to y−4 ⩽ 7 This is part of Ryan’s homework. Question Use the inequality 12 ⩽ x < 18 to write i the smallest integer that x could be ii the largest integer that x could be iii a list of the integer values that x could be. Answer i smallest integer is 13 ii largest integer is 18 iii x could be 13, 14, 15, 16, 17, 18 a Explain the mistakes Ryan has made and write the correct solutions. b Discuss your answers to part a with a partner. Make sure you have corrected all of Ryans’s mistakes. 8 For each of these inequalities, write i the smallest integer that y could be ii the largest integer that y could be iii a list of the integer values that y could be. a 3 8 < < y b 4 < y ⩽ 7 c 0 ⩽ y < 6 d −10 ⩽ y ⩽ −6 Think like a mathematician 9 Arun and Sofia are looking at the inequality 2 < y < 9. In pairs or small groups discuss Arun’s and Sofia’s comments. I think the inequality 2 < y < 9 can be written as 9 > y > 2 I think the inequality 2 < y < 9 can be written as 2 > y > 9 What do you think? Explain your answers. 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
61 2.6 Inequalities 10 Write true (T) or false (F) for each statement. Part a has been done for you. a 7 ⩾ y > 3 means the same as 3 < y ⩽ 7 T b 15 > y ⩾ 5 means the same as 5 < y ⩽ 15 c 10 ⩾ y ⩾ −6 means the same as −6 ⩽ y ⩽ 10 d 8 > y ⩾ −8 means the same as −8 < y ⩽ 8 11 Samir combines two inequalities into one. The two inequalities m < 10 and m > 2 can be combined like this: Step 1: m > 2 is the same as 2 < m Step 2: The two inequalities are now 2 < m and m < 10 I can write these as one inequality: 2 < m < 10 I can use a number line to help me: 86420 10 12 m > 2 or 2 < m 86420 10 12 m < 10 86420 10 12 2 < m < 10 a Use Samir’s method to combine each pair of inequalities into one. Use a number line to help if you want to. i m < 15 and m >8 ii m ⩽ 10 and m > 7 iii m > 0 and m < 6 b Is it possible to write m > 14 and m < 8 as one inequality? Give a reason for your answer. Discuss your answer with a partner. 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
62 2 Expressions, formulae and equations 12 This is part of Sandeep’s classwork. Question a Show the inequality 2 1 2 ⩽ m < 7 1 4 on a number line. b Write i the smallest integer that m could be ii the greatest integer that m could be iii a list of the integer values that m could be. Solution a 54321 6 7 8 b i smallest integer is 2 ii largest integer is 7 iii m could be 2, 3, 4, 5, 6, 7 a Sandeep has made two mistakes. What are they? b For each of these inequalities, write i the smallest integer that m could be ii the greatest integer that m could be ii a list of the integer values that m could be. A 5 3 4 ⩽ m < 91 3 B −6 1 2 ⩽ m < −2 1 8 13 Zara is looking at this question. Draw a line linking each inequality on the left with: the correct smallest integer; the correct largest integer; and the correct list of integers. The first one has been done for you: a and ii and D and Z Inequality Smallest integer Largest integer List of integers a: 1.5 ⩽ x ⩽ 4 i: 3 A: 5 U: 4, 5, 6 b: 08 59 . . < < x ii: 2 B: 6 V: −5, −4, −3, −2, −1, 0 c: 3 < x ⩽ 6.1 iii: −5 C: 1 W: 1, 2, 3, 4, 5 d: 2.2 ⩽ x < 3.9 iv: 4 D: 4 X: −4, −3, −2, −1, 0, 1 e: − < 45 11 . . x < v: 1 E: 0 Y: 3 f: −5.01 < x ⩽ 0 vi: −4 F: 3 Z: 2, 3, 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
2.6 Inequalities Read Zara’s comments. a What do you think of Zara’s method? Can you improve her method, or suggest a better one? b Use what you think is the best method to answer the question. Summary checklist I can understand inequalities. I can draw inequalities. The method I am going to use is to identify the smallest integer for each inequality first. Then I’ll identify the largest integer for each inequality. Then I’ll work out the list of integers for each inequality. 63 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
64 2 Expressions, formulae and equations Check your progress 1 Jin thinks of a number, x. Write an expression for the number Jin gets when he divides the number by 2 then adds 5. 2 a Use the formula K=mg to work out K when m=12 and g=4 . b Rearrange the formula K=mg to make m the subject. c Use your formula in part b to work out m when K=75 and g=10. 3 Expand a x(x+3) b 5y(7y−4w) 4 Factorise a 6x+9 b 2y2−12y 5 Work out the value of x and y in this diagram. All measurements are in centimetres. 6(x + 1) 3x + 21 + 16 20 y 3 6 Write the inequality shown by this number line. Use the letter x. 2520151050 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
65 Project 1 Algebra chains An algebra chain is a sequence of expressions where an input number is substituted as the value of x in the first expression, and then the output of each expression is substituted as the value of x in the following expression. So, in the algebra chain below: 3 is substituted for x in the expression 4x−10, giving the output 2 Then 2 is substituted for x in the expression 8x+4, giving the output 20 3 4x−10 8x+4 20 Can you find a way to arrange the eight cards below into four algebra chains that take the inputs 1, 2, 3 and 4 and give the outputs 40, 30, 20 and 10? 2x+3 2(x+1) 5x−7 12−2x 3x−4 7x+5 3(x−4) 8x−2 1 40 2 30 3 20 4 10 Now choose any two cards and an input number. Work out what the output of your algebra chain would be. Tell a partner the input you chose and the output you got. Can your partner work out which cards you used? Can they still work it out when you make an algebra chain with three cards? 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
66 Getting started 1 Work out a 4.5×10 b 18×10 c 82×100 d 4.6×100 2 Work out a 70÷10 b 342÷10 c 140÷100 d 3120÷100 3 Write the correct answer, A or B, for each part. Round each number to one decimal place. a 7.23 A 7.2 B 7.3 b 12.45 A 12.4 B 12.5 c 0.793 A 0.7 B 0.8 4 Round each number to the given degree of accuracy. a 4.587 (2 d.p.) b 0.672315 (4 d.p.) c 54.78899 (3 d.p.) d 12.050 2997 (5 d.p.) Today, there are hundreds of different languages in use in the world. However, all over the world people write numbers in the same way. Everyone uses the decimal system to write numbers. • The decimal system was first developed in India. • It was adopted by Persian and Arab mathematicians in the 9th century. • It was introduced to Europe about 1000 years ago. • At first it was banned in some European cities because people did not understand it and thought they were being cheated. 3 Place value and rounding 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
67 3 Place value and rounding One system that was used in the past is Roman numerals. 2000 years ago the Romans used letters to represent numbers. You can still see them on clock faces and carvings. Their use continued in Europe for over 1000 years. Here are some examples. Roman III VII IX XX C MCMXXX S Decimal 3 7 9 20 100 1930 0.5 Here are some calculations, multiplying or dividing by 10 (X) or 100 (C), written using Roman numerals. III×X=XXX V×C=D M÷C=X LV÷X=VS Can you work out what D, M and L represent? You can see that arithmetic with Roman numerals is very difficult. You keep needing new letters. The decimal system uses place value. That is why it only needs ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It makes the arithmetic we do today much easier than the arithmetic the Romans did! 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
68 3 Place value and rounding In this section you will … • multiply numbers by 0.1 and 0.01 • divide numbers by 0.1 and 0.01. 3.1 Multiplying and dividing by 0.1 and 0.01 The decimal number 0.1 is the same as 1 10. So when you multiply a number by 0.1, it has the same effect as dividing the number by 10. Example: 8×0.1=8× 1 10 and 8× 1 10=8÷10 The decimal number 0.01 is the same as 1 100. So when you multiply a number by 0.01, it has the same effect as dividing the number by 100. Example: 8×0.01=8× 1 100 and 8× 1 100=8÷100 When you divide a number by 0.1, it has the same effect as multiplying the number by 10. Example: 8÷0.1=8÷ 1 10 and 8÷ 1 10=8×10 When you divide a number by 0.01, it has the same effect as multiplying the number by 100. Example: 8÷0.01=8÷ 1 100 and 8÷ 1 100=8×100 Key words decimal number equivalent calculations inverse operation Worked example 3.1 Work out a 32×0.1 b 4.2×0.01 c 6÷0.1 d 4.156÷0.01 Answer a 32×0.1=3.2 Multiplying by 0.1 is the same as dividing by 10, and 32÷10=3.2 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
69 3.1 Multiplying and dividing by 0.1 and 0.01 Think like a mathematician 3 Work with a partner or in a small group to discuss this question. Sofia and Arun are discussing the best way to work out 56×0.1 a Can you explain how both of these methods work? Whose method do you prefer? b Describe how you would work out 56×0.01 using Sofia’s method and using Arun’s method. Continued b 4.2×0.01=0.042 Multiplying by 0.01 is the same as dividing by 100, and 4.2÷100=0.042 c 6÷0.1=60 Dividing by 0.1 is the same as multiplying by 10, and 6×10=60 d 4.156÷0.01=415.6 Dividing by 0.01 is the same as multiplying by 100, and 4.156×100=415.6 Exercise 3.1 1 Copy and complete these calculations. All the answers are in the cloud. a 20×0.1=20÷10= b 200×0.1=200÷10= c 2000×0.1=2000÷10= d 2×0.1=2÷10= 2 Copy and complete these calculations. All the answers are in the cloud. a 400×0.01=400÷100= b 40 000×0.01=40 000÷100= c 40×0.01=40÷100= d 4000×0.01=4000÷100= 20 200 0.2 2 0.4 40 400 4 When I multiply 56 by 0.1, I move the digits 5 and 6 one place to the right in the place value table. This gives me an answer of 5.6 When I multiply 56 by 0.1, I move the decimal point one place to the left. This gives me an answer of 5.6 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
70 3 Place value and rounding 4 Work out a 62×0.1 b 55×0.1 c 125×0.1 d 3.2×0.1 e 37×0.01 f 655×0.01 g 750×0.01 h 4×0.01 5 Copy and complete these calculations. All the answers are in the cloud. a 2÷0.1=2×10= b 20÷0.1=20×10= c 200÷0.1=200×10= d 0.2÷0.1=0.2×10= 6 Copy and complete these calculations. All the answers are in the cloud. a 4÷0.01=4×100= b 40÷0.01=40×100= c 400÷0.01=400×100= d 0.4÷0.01=0.4×100= 200 2000 2 20 40 40000 400 4000 Think like a mathematician 7 Work with a partner or in a small group to discuss this question. Win uses equivalent calculations to work out 3.2÷0.1 and 12.8÷0.01 This is what she writes. i 3 2 0 1 3 2 10 0 1 10 32 1 . . . . = = × × So 3.2÷0.1=32÷1=32 ii 12 8 0 01 12 8 10 0 01 10 128 0 1 128 10 0 1 10 1280 1 . . . . . . = = = = × × × × So 12.8÷0.01=1280÷1=1280 a Can you explain how Win’s method works? Do you like her method? Explain your answer. b Work out 0 4. . 5 0 ÷ 1 and 78 ÷ 0 0. 1 using Win’s method. 8 Work out a 7÷0.1 b 4.5÷0.1 c 522÷0.1 d 0.67÷0.1 e 2÷0.01 f 8.5÷0.01 g 0.32÷0.01 h 7.225÷0.01 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
71 3.1 Multiplying and dividing by 0.1 and 0.01 9 Jake works out 23×0.1 and 8.3÷0.01 He checks his answers by using an inverse operation. i 23 × 0.1=23÷10=2.3 ii 8.3÷0.01=8.3 × 100=8300 Check: 2.3 × 10=23 ✓ Check: 8300÷100=83 ✘ Correct answer: 830 Work out the answers to these questions. Check your answers by using inverse operations. a 18×0.1 b 23.6×0.01 c 0.6÷0.1 d 4.5÷0.01 10 Which symbol,×or ÷, goes in each box? a 6.7 0.1=67 b 4.5 0.01=0.045 c 0.9 0.1=0.09 d 550 0.01=5.5 e 0.23 0.1=2.3 f 12 0.01=1200 11 Which of 0.1 or 0.01 goes in each box? a 26× =0.26 b 3.4÷ =34 c 0.06× =0.0006 d 7÷ =70 e 8.99× =0.899 f 52÷ =520 12 A jeweller uses this formula to work out the mass of copper in green gold. C=0.1G where: C is the mass of copper G is the mass of green gold a Work out the mass of copper in 125 g of green gold. The jeweller uses this formula to work out the mass of zinc in yellow gold. Z=0.01Y where: Z is the mass of zinc Y is the mass of yellow gold b Work out the mass of zinc in 80 g of yellow gold. The jeweller says, ‘I think that 10% of green gold is copper.’ c Is the jeweller correct? Explain your answer. d What percentage of yellow gold is zinc? Explain how you worked out your answer. 13 a Sort these expressions into groups of the same value. There should be one expression left over. A 24 0 × .1 B 240 × 0 1. C 2. . 4 0 ÷ 01 D 24 0 ÷ .01 E 2. . 4 0 ÷ 1 F 240 × 0 0. 1 G 24 0 ÷ .1 H 0 2. . 4 0 ÷ 01 I 2400 × 0 1. J 0 2. . 4 0 ÷ 1 b Write two new expressions that have the same value as the expression that is left over. Tip Remember, 0.1G means 0.1×G. Tip Remember, ‘percent’ means ‘out of 100’, so 10 10 100 % = . 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
72 3 Place value and rounding 14 Razi thinks of a number. He multiplies his number by 0.1, and then divides the answer by 0.01. Razi then divides this answer by 0.1 and gets a final answer of 12 500. What number does Razi think of first? Explain how you worked out your answer. 15 This is part of Harsha’s homework. Question Write one example to show that this statement is not true. ‘When you multiply a number with one decimal place by 0.01 you will always get an answer that is smaller than zero.’ Answer 345.8 × 0.01=3.458 and 3.458 is not smaller than zero so the statement is not true. Write down one example to show that each of these statements is not true. a When you multiply a number other than zero by 0.1 you will always get an answer that is greater than zero. b When you divide a number with one decimal place by 0.01 you will always get an answer that is greater than 100. Look at these two questions. a 56×0.1 b 3.2÷0.01 Explain to a partner the methods you would use to work out the answers to these questions. Explain why you would use these methods. Does your partner use the same methods? If they use different methods, do you understand their methods? Summary checklist I can multiply numbers by 0.1 and 0.01. I can divide numbers by 0.1 and 0.01. 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
73 3.2 Rounding You already know how to round decimal numbers to a given number of decimal places (d.p.). You also need to know how to round numbers to a given number of significant figures (s.f.). The first significant figure in a number is the first non-zero digit in the number. For example: • In the number 450, 4 is the first significant figure, 5 is the second significant figure and 0 is the third significant figure. • In the number 0.008 06, 8 is the first significant figure, 0 is the second significant figure and 6 is the third significant figure. To round a number to a given number of significant figures, follow these steps: • Look at the digit in the position of the degree of accuracy. The ‘degree of accuracy’ is the number of significant figures you are working to. So, if you have been asked to round to 3 significant figures, look at the third significant figure in the number. • If the number to the right of this digit is 5 or more, increase the digit by 1. If the number is less than 5, leave the digit as it is. 3.2 Rounding In this section you will … • round numbers to a given number of significant figures. Key words decimal places (d.p.) degree of accuracy round significant figures (s.f.) Worked example 3.2 a Round 4286 to one significant figure. b Round 0.080 69 to three significant figures. c Round 0.7963 to two significant figures. Answer a 4286=4000 (1 s.f.) The first significant figure is 4. The digit to the right of it is 2. 2 is less than 5, so 4 stays the same. Replace the 2, the 8 and the 6 with zeros to keep the place value consistent. In this case, rounding to one significant figure is the same as rounding to the nearest 1000. The letters ‘s.f.’ stand for ‘significant figure’. 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
74 3 Place value and rounding 2 Round each of these numbers to two significant figures (2 s.f.). All the answers are in the cloud. a 243 b 0.235 c 24.15 d 0.002 3801 e 2396 f 2.3699 0.0024 0.24 2.4 24 240 2400 Exercise 3.2 1 Round each of these numbers to one significant figure (1 s.f.). Choose the correct answer: A, B or C. a 352 A 4 B 40 C 400 b 7.291 A 7 B 7.3 C 7.29 c 11540 A 12000 B 10 000 C 11 000 d 0.0087 A 9 B 0.09 C 0.009 Continued b 0.08069=0.0807 (3 s.f.) The first significant figure is 8, the second is 0 and the third is 6. The digit to the right of the 6 is 9. 9 is more than 5 so round the 6 up to 7. You must keep the zeros at the start of the number to keep the place value consistent. In this case, rounding to 3 s.f. is the same as rounding to 4 d.p. c 0.7963=0.80 (2 s.f.) The first significant figure is 7 and the second is 9. The digit to the right of the 9 is 6. 6 is more than 5 so round the 9 up to 10. This has the effect of rounding ‘79’ up to ‘80’. You must keep the zero after the 8 to show that you have rounded to 2 s.f. In this case, rounding to 2 s.f. is the same as rounding to 2 d.p. because the first significant figure is also the first decimal place. 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
75 3.2 Rounding Think like a mathematician 3 Work with a partner to discuss the answers to this question. This is part of Harry’s homework. Question Round these numbers to 2 s.f. a 45 150 b 0.032 84 Answer a 45 b 0.03 Harry has rounded one large number and one small number to two significant figures. Both of his answers are wrong. a Explain the mistakes he has made. Write the correct answers. b What must you remember to do when you round a large number to a given number of significant figures? c What must you remember to do when you round a small number to a given number of significant figures? 4 Round each number to the stated number of significant figures (s.f.). a 135 (1 s.f.) b 45678 (2 s.f.) c 18.654 (3 s.f.) d 0.0931 (1 s.f.) e 0.7872 (2 s.f.) f 1.40948 (4 s.f.) g 985 (1 s.f.) h 0.697 (2 s.f.) i 8.595 (3 s.f.) 5 Which answer is correct: A, B, C or D? a 2569 rounded to 1 s.f. A 2 B 3 C 2000 D 3000 b 47.6821 rounded to 3 s.f. A 47.6 B 47.682 C 47.7 D 48.0 c 0.0882 rounded to 2 s.f. A 0.08 B 0.088 C 0.09 D 0.1 d 3.08962 rounded to 4 s.f. A 3.089 B 3.0896 C 3.09 D 3.090 e 19.963 rounded to 3 s.f. A 2 B 20 C 20.0 D 19.96 6 Round the number 209.095 046 to the stated number of significant figures (s.f.). a 1 s.f. b 2 s.f. c 3 s.f. d 4 s.f. e 5 s.f. f 6 s.f. 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
76 3 Place value and rounding Activity 3.2 You are going to write a question for a partner to answer. On a piece of paper, write a question of your own similar to Question 4. Make sure: • you have parts a to d • you use four different numbers • you ask for the numbers to be rounded to different degrees of accuracy • you write the answers on a different piece of paper. Exchange questions with your partner. Answer their question, then exchange back and mark each other’s work. Discuss any mistakes that have been made. 7 a Use a calculator to work out the answer to 26 58 2 + . Write all the numbers on your calculator display. b Round your answer to part a to the stated number of significant figures (s.f.). i 1 s.f. ii 2 s.f. iii 3 s.f. iv 4 s.f. v 5 s.f. vi 6 s.f. 8 At a football match there were 63 475 Barcelona supporters and 32486 Arsenal supporters. How many supporters were there altogether? Give your answer correct to two significant figures. 9 Ahmad has a bag of peanuts that weighs 150 g. There are 335 peanuts in the bag. Work out the average (mean) mass of one peanut. Give your answer correct to one significant figure. 10 The speed of light is approximately 670 616 629 miles per hour. This formula changes a speed in miles per hour into a speed in metres per second. metres per second = miles per hour 2 25. Work out the speed of light in metres per second. Give your answer correct to three significant figures. 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
77 3.2 Rounding 11 Zara and Sofia are looking at this question. Work out the area of this rectangle. 9.6m 0.87m Give your answer to an appropriate degree of accuracy. Read Zara’s and Sofia’s comments. 9.6 and 0.87 are both written to 2 s.f., so I think we should round our answer to 2 s.f. I think our answer should be 8.4 m2. area=9.6×0.87 =8.352 m2 I think we should give 8.352 m2 as our answer. What do you think? Explain your answers. 12 A rugby club sells, on average, 12 600 tickets to a match each week. The average cost of a ticket is $26.80 How much money does the club get from ticket sales, on average, each week? Round your answer to an appropriate degree of accuracy. 13 This formula is often used in science. F=ma Work out the value of a when F=32 and m=15. Round your answer to an appropriate degree of accuracy. Tip Change the subject of the formula first. Tip Choose a sensible degree of accuracy for the context. Think about how accurate you need your answer to be. 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
78 3 Place value and rounding 14 This is part of Jake’s homework. He works out an estimate by rounding each number to one significant figure. Question a Work out an estimate of 0.238 × 576 39.76 b Work out the accurate value. c Compare your estimate with the accurate value. Answer a 0.238≈0.2, 576≈600, 39.76≈40 0.2 × 600=120 and 120÷40=3 Estimate=3 b 0.238 × 576=137.088 137.088÷39.76=3.45 (3 s.f.) Accurate value=3.45 (3 s.f.) c My estimate is close to the accurate value, so my accurate answer is probably correct. Tip The symbol ≈ means ‘is approximately equal to’. Summary checklist I can round numbers to a given number of significant figures. Follow these steps for each of the calculations below. i Use Jake’s method to work out an estimate of the answer. ii Use a calculator to work out the accurate answer. Give this answer correct to three significant figures. iii Compare your estimate with the accurate answer. Decide if your accurate answer is correct. a 0 3941 196 4 796 . . × b 4732 9176 19 5166 + . c 2 764 84 695 9 687 4 19 . . . . × − d 58 432 0 08 0 2 348 × × . . 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
79 3 Place value and rounding Check your progress 1 Work out a 90×0.1 b 552×0.1 c 135×0.01 d 8×0.01 e 6÷0.1 f 23.5÷0.1 g 5.2÷0.01 h 0.68÷0.01 2 Which calculation, A, B, C or D, gives a different answer from the others? Show your working. A 5.2×0.1 B 5.2÷0.01 C 0.052÷0.1 D 52×0.01 3 Round each of these numbers to the given degree of accuracy. a 78.023 (2 s.f.) b 0.06791 (3 s.f.) c 1.54962 (4 s.f.) d 12452673 (5 s.f.) 4 Use a calculator to work out the answer to 89 482 Give your answer correct to two significant figures. 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
80 4 Decimals Getting started 1 Write the correct symbol, < or >, between each pair of decimal numbers. a 4.5 4.1 b 6.57 6.68 c 10.52 10.59 d 2.784 2.781 2 Here are four decimal number cards. 0.763 0.756 0.761 0.759 Write the numbers in order of size, starting with the smallest. 3 Write true (T) or false (F) for each of these. a 6×0.1=0.6 b 12×0.7=0.84 c 0.03×2500=750 d 0.04×25=1 4 Match each blue question card with the correct yellow answer card. 12×1.8 19×1.2 9×2.5 25×0.87 320×0.07 21.75 22.4 21.6 22.5 22.8 5 Work out. a 12.3÷3 b 44.1÷7 c 152.88÷6 d 28.86÷12 Tip Remember: < means ‘is less than’ and > means ‘is greater than’. 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
4 Decimals There are many situations in everyday life where we have to calculate with decimals. If you go to an airport, you will always see a currency exchange desk. This is where you can change money from one currency to another. The exchange rates are shown on a board. They tell you how much of one currency you can exchange for another. An architect is a person who designs buildings, and in many cases, supervises their construction. They need to measure accurately and also calculate using decimals. An incorrect decimal calculation could result in a disaster for the building! 81 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
82 4 Decimals In this section you will … • compare and order decimals. 4.1 Ordering decimals To order decimal numbers, compare the whole-number part first. When the numbers you are ordering have the same whole-number part, look at the decimal part and compare the tenths, then the hundredths, and so on. Look at the three decimal numbers on the right. 8.56 7.4 8.518 1 Highlight the whole numbers. You can see that 7.4 is the smallest number, so 7.4 goes first. 8.56 7.4 8.518 2 The other two numbers both have 8 units, so highlight the tenths. 7.4 8.56 8.518 3 They both have the same number of tenths, so highlight the hundredths. 7.4 8.56 8.518 4 You can see that 8.518 is smaller than 8.56, so in order of size the numbers are: 7.4 8.518 8.56 When you order decimal measurements, you must make sure they are all in the same units. You need to remember these conversion factors. Length Mass Capacity 10mm=1cm 1000g=1kg 1000ml=1l 100cm=1m 1000kg=1t 1000m=1km Key words compare decimal number order term-to-term rule Tip The number of digits after the decimal point is the number of decimal places (d.p.) in the number. Tip When you compare decimal numbers you can use these symbols. =means ‘is equal to’ ≠ means ‘is not equal to’ > means ‘is bigger than’ < means ‘is smaller than’ 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
83 4.1 Ordering decimals Worked example 4.1 a Write these decimal numbers in order of size. 5.6, 4.95, 5.68, 5.609 b Write the correct symbol,=or ≠, between these measures. 7.5m 75 cm c Write the correct symbol, > or <, between these measures. 4.5kg 450 g Answer a 4.95, 5.6, 5.609, 5.68 The smallest number is 4.95 as it has the smallest whole-number part. The other three numbers have the same whole-number part and the same number of tenths, so compare the hundredths. 5.68 has 8 hundredths compared with 5.6 and 5.609 which have 0 hundredths, so 5.68 is the biggest number. Now compare the thousandths in 5.6 and 5.609 5.6 has 0 thousandths and 5.609 has 9 thousandths, so 5.6 is smaller. b 7.5 m ≠ 75 cm There are 100 cm in 1m. 7.5m×100=750 cm, so the measures are not equal. Use the ‘≠’ symbol. c 4.5kg > 450 g There are 1000 g in 1kg. 4.5kg×1000=4500 g, so 4.5kg is greater. Use the ‘>’ symbol. Exercise 4.1 1 Write these decimal numbers in order of size, starting with the smallest. They have all been started for you. a 5.49, 2.06, 7.99, 5.91 2.06, , , b 3.09, 2.87, 3.11, 2.55 2.55, , , c 12.1, 11.88, 12.01, 11.82 11.82, , , d 9.09, 8.9, 9.53, 9.4 8.9, , , 2 Write the correct sign, < or >, between each pair of numbers. a 4.23 4.54 b 6.71 6.03 c 0.27 0.03 d 27.9 27.85 e 8.55 8.508 f 5.055 5.505 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
84 4 Decimals 4 Use your preferred method to write these decimal numbers in order of size, starting with the smallest. a 23.66, 23.592, 23.6, 23.605 b 0.107, 0.08, 0.1, 0.009 c 6.725, 6.78, 6.007, 6.71 d 11.02, 11.032, 11.002, 11.1 5 Write the correct sign,=or ≠, between each pair of measurements. a 6.7 l 670ml b 4.05 t 4500kg c 0.85km 850m d 0.985m 985 cm e 14.5 cm 145mm f 2300g 0.23kg 6 Write the correct sign, < or >, between each pair of measurements. a 4.5 l 2700ml b 0.45 t 547kg c 3.5 cm 345mm d 0.06kg 550g e 7800m 0.8km f 0.065m 6.7 cm Tip Start by converting one of the measurements so that both measurements are in the same units. Think like a mathematician 3 Maya uses this method to order decimals. Question Write these numbers in order of size, starting with the smallest. 26.5 26.41 26.09 26.001 26.92 Answer The greatest number of decimal places in the numbers is 3. Step 1: Write all the numbers with 3 decimal places. 26.500 26.410 26.090 26.001 26.920 Step 2: Compare only the numbers after the decimal point. 500 410 090 001 920 Step 3: Write these numbers in order of size. 001 090 410 500 920 Step 4: Now write the decimal numbers in order. 26.001 26.09 26.41 26.5 26.92 Discuss the answers to these questions with a partner. a Do you understand how Maya’s method works? b Do you like Maya’s method? c Do you prefer Maya’s method to the method shown in Worked example 4.1? Explain your answer. 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
85 4.1 Ordering decimals 7 Write these measurements in order of size, starting with the smallest. a 2.3kg, 780g, 2.18kg, 1950g b 5.4cm, 12mm, 0.8 cm, 9mm c 12m, 650cm, 0.5m, 53cm d 0.55l, 95ml, 0.9 l, 450ml e 6.55km, 780m, 6.4km, 1450m f 0.08t, 920kg, 0.15t, 50kg Tip Make sure all the measurements are in the same units before you start to order them. Tip Draw a number line to help if you want to. Think like a mathematician 8 Look at Arun’s solution to this question. Write these decimal numbers in order of size, starting with the smallest. −4.52 −4.31 −4.05 −4.38 All the numbers start with −4 so I will just compare the decimal parts: 52, 31, 05 and 38. In order, they are 05, 31, 38, 52. So the order is −4.05, −4.31, −4.38, −4.52 Discuss the answers to these questions with a partner. a Is Arun correct? Explain your answer. b What do you think is the best method to use to order negative decimal numbers? 9 Write the correct sign, < or >, between each pair of numbers. a −4.27 −4.38 b −6.75 −6.25 c −0.2 −0.03 d −8.05 −8.9 10 Write these decimal numbers in order of size, starting with the smallest. a −4.67, −4.05, −4.76, −4.5 b −11.525, −11.91, −11.08, −11.6 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
86 4 Decimals 11 Shen and Mia swim every day. They record the distances they swim each day for 10 days. These are the distances that Shen swims each day. 250m 1.25km 0.5km 2500m 2km 1.75km 750m 1500m 25km 0.75km a Shen has written down one distance incorrectly. Which one do you think it is? Explain your answer. These are the distances that Mia swims each day. 1.2km 240m 0.4km 1.64km 820m 640m 0.2km 1.42km 960m 0.88km b Mia says that the longest distance she swam is more than eight times the shortest distance she swam. Is Mia correct? Explain your answer. Shen and Mia swim in different swimming pools. One of the swimming pools is 25m long. The other swimming pool is 20m long. Shen and Mia always swim a whole number of lengths. c Who do you think swims in the 25m swimming pool? Explain how you made your decision. 12 Each of the cards describes a sequence of decimal numbers. A First term: 0.5 Term-to-term rule: ‘add 0.5’ B First term: 0.15 Term-to-term rule: ‘multiply by 2’ C First term: −1.7 Term-to-term rule: ‘add 1’ D First term: 33.6 Term-to-term rule: ‘divide by 2’ E First term: 1.25 Term-to-term rule: ‘add 0.25’ F First term: 10.45 Term-to-term rule: ‘subtract 2’ a Work out the fifth term of each sequence. b Write the numbers from part a in order of size, starting with the smallest. 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
87 4.2 Multiplying decimals 13 Zara is looking at this inequality: 3.27 ⩽x<3.34 If x is a number with two decimal places, there are 8 possible numbers that x could be. Is Zara correct? Explain your answer. 14 y is a number with three decimal places, and –0.274<y ⩽–0.27 Write all the possible numbers that y could be. In this exercise you have written positive decimal numbers, negative decimal numbers and decimal measurements in order of size. a Which have you found the easiest? Explain why. b Which have you found the hardest? Explain why. Summary checklist I can compare and order positive decimal numbers. I can compare and order negative decimal numbers. I can compare and order decimal measurements. Follow these steps when you multiply a decimal by a whole number or a decimal. • First, work out the multiplication without the decimal points. • Finally, put the decimal point in the answer. There must be the same number of digits after the decimal point in the answer as there were in the question. 4.2 Multiplying decimals In this section you will … • multiply decimals by whole numbers and decimals. Key words estimation mentally place value written method 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
88 4 Decimals Exercise 4.2 1 Use a mental method to work out a 0.1×−8 b 0.2×3 c 0.3×−7 d 0.7×8 e 0.9×−4 Worked example 4.2 a Work out mentally i 0.02×−12 ii 3.2×0.04 b Use a written method to work out 5.96×0.35 Check your answer using estimation. Answer a i 2×−12=−24 0.02×−12 =−0.24 Work out the multiplication without the decimal points. Put the decimal point back in the answer. There are 2 digits after the decimal point in the question, so there must be 2 digits after the decimal point in the answer. ii 32×4=128 3.2×0.04=0.128 Work out the multiplication without the decimal points. Put the decimal point back in the answer. There are 3 digits in total after the decimal points in the question, so there must be 3 digits after the decimal point in the answer. b Work out the multiplication without the decimal points. (Use your preferred method for multiplication.) Work out 596×5 Work out 596×30 Add together the two lines above. 5.96×0.35=2.0860 5.96×0.35=2.086 Put the decimal point back in the answer. There are 4 digits in total after the decimal points in the question, so there must be 4 digits after the decimal point in the answer. The zero at the end of the number can now be ignored. Check 6×0.4=2.4 ✓ Round both of the numbers in the question to one significant figure. 2.4 is close to 2.086, so the answer is probably correct. 5 9 6 × 3 5 2 9 8 0 + 1 7 8 8 0 2 0 8 6 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
89 4.2 Multiplying decimals −18 −0.18 −1.8 −0.018 2 Use a mental method to work out a −6×0.03 b −9×0.2 c −18×0.001 d −20×0.9 All the answers are in the cloud. 3 Here are five calculation cards. A 0.6×−12 B 0.039×−180 C 0.85×−9 D 0.44×−16 E 0.04×−182 a Work out the answers to the calculations on the cards. b Write the answers in order of size, starting with the smallest. Think like a mathematician 4 Work with a partner or in a small group to discuss this question. Look at what Arun says. Zara shows Arun this pattern. 2 × 3=6 0.2 × 3=0.6 0.2 × 0.3=0.06 0.2 × 0.03=0.006 0.2 × 0.003=0.0006 How can you use the place value of the digits in 0.2 and 0.3 to explain to Arun why the answer is 0.06 and not 0.6? I don’t understand why 0.2×0.3 is 0.06 and not 0.6 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
90 4 Decimals 5 a Copy and complete these patterns. i 2×4=8 ii 3×5=15 0.2×4= 0.3×5= 0.2×0.4= 0.3×0.5= 0.2×0.04= 0.3×0.05= 0.2×0.004= 0.3×0.005= b Work out. i 0.1×0.09 ii 0.6×0.8 iii 0.07×0.4 iv 0.03×0.05 v 0.12×0.3 vi 0.06×0.11 6 Fill in the missing spaces in this spider diagram. All the calculations give the answer in the middle. All the answers are in the yellow rectangle on the right. 63 0.630.6 43 0.93 39 30.12 31.2 0.36 3 0.4 0.3 36 0.04 0.06 0.09 30.01 Think like a mathematician 7 Work with a partner or in a small group to discuss this question. Jan works out that 42×87=3654 a Use this information to write the answers to these multiplications. i 42×8.7 ii 42×0.87 iii 4.2×87 iv 4.2×8.7 b Explain why your answers to ai and aiii are the same, and why your answers to aii and aiv are the same. c Use generalising to describe a method that someone could follow to work out the answer to any change in 42×87=3654 that involves decimals. Tip An example calculation in part c could be 0.42 × 0.87 or 0.042×8.7 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
91 4.2 Multiplying decimals 8 a Work out 158×46 b Use your answer to part a to write the answers to these multiplications i 15.8×46 ii 158×4.6 iii 15.8×4.6 iv 1.58×4.6 v 15.8×0.46 vi 1.58×0.046 Activity 4.2 a On a piece of paper, write a question of your own similar to Question 8. On a different piece of paper, write the answers to your question. b Exchange questions with a partner and work out the answers to their question. c Exchange back and mark each other’s work. Discuss any mistakes. 9 Sam uses this method to work out and check her answer. Question Work out 0.67 × 4.28 Answer First work out 67 × 428 × 400 20 8 26 800 60 24 000 1200 480 1 340 7 2800 140 56 + 536 Totals 26 800 1340 536 28 676 So 0.67 × 4.28=2.8676 Check Round 0.67 to 0.7 Round 4.28 to 4 0.7 × 4=2.8, which is close to 2.8676 a Write the advantages and disadvantages of Sam’s method. b Can you improve her method? c Which method do you prefer to use to multiply decimals? Write why you prefer this method. 10 Work out these multiplications. Show how to check your answers. a 6.7×9.4 b 0.56×8.3 c 0.23×8.15 d 0.69×0.254 Tip Use estimation to check your answers by rounding all the numbers in the question to one significant figure. 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
92 4 Decimals Use estimation to check if Syra’s answers could be correct. If you don’t think they are correct, explain why. 12 A vet needs to work out how much medicine to give to a cat. The instructions for the medicine say: Give 7.3mg (medicine) per kg (mass of cat) The cat has a mass of 5.8kilograms. a Work out an estimate of the number of milligrams (mg) of medicine that the cat needs. b Calculate the accurate number of milligrams (mg) of medicine that the cat needs. 13 A coin is made from silver and copper. The mass of the coin is 4.2 g. You can use this formula to work out the mass of silver in the coin: mass of silver=0.775×mass of coin a Work out an estimate of the mass of the silver in this coin. b Calculate the accurate mass of the silver in this coin. 11 This is part of Syra’s homework. Summary checklist I can multiply decimals by whole numbers. I can multiply decimals by decimals. Question Work out a 0.45 × 2.8 b 7.8 × 0.0093 c 0.065 × 0.043 Answer a 12.6 b 0.072 54 c 0.027 95 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
93 4.3 Dividing by decimals In this section you will … • divide decimals by numbers with one decimal place. 4.3 Dividing by decimals When you divide a number by a decimal, you can use the place value of the decimal to work out an easier equivalent calculation. An easier equivalent calculation is to divide by a whole number instead of a decimal. For example, you can write 5 67 . . ÷ 0 7 as 5 67 0 7 . . Multiplying the numerator and denominator of the fraction by 10 gives 5 67 10 0 7 10 56 7 7 . . × . × = This makes an equivalent calculation that is much easier to do because dividing by 7 is much easier than dividing by 0.7. Key words equivalent calculation reverse calculation short division Worked example 4.3 Work out a −32÷0.2 b 3.468÷0.8 Answer a − ÷ = − 32 0 2 32 0 2 . . First of all write − ÷ 32 0 2. as a fraction. − × × − = 32 10 0 2 10 320 . 2 Multiply numerator and denominator by 10. −320÷2 = −160 Finally work out the division. b 3 468 0 8 3 468 0 8 . . . . ÷ = First, write 3.468÷0.8 as a fraction. 3 468 10 0 8 10 34 68 8 . . × . × = Multiply the numerator and denominator by 10. Now use short division to work out 34 6. 8 8 ÷ 4 . 3 3 5 8 3 4 . 2 6 2 8 4 0 So 3 468 . . ÷ = 0 8 4 335 . 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
94 4 Decimals Exercise 4.3 1 Copy and complete these divisions. a 2 4 0 4 2 4 0 4 . . . . ÷ = 2 4 10 0 4 10 . . × × = = b 7 2 0 9 7 2 0 9 . . . . ÷ = 7 2 10 0 9 10 . . × × = = c − ÷ = − 42 0 6 42 0 6 . . − × × = = 42 10 0 6 10. d − ÷ = − 45 0 5 45 0 5 . . − × × = = 45 10 0 5 10. 2 Which of these calculation cards is the odd one out? Explain why. A 6.3÷0.9 B 1.4÷0.2 C 4.9÷0.7 D 4.8÷0.6 E 5.6÷0.8 Think like a mathematician 3 Discuss this question with a partner or in a small group. Arun is working out 3÷0.6. This is what he says. What explanation could you give to Arun to show that he is wrong? I understand that I must do 3 0 6 3 10 0 6 10 30 6 5 . . = = = × × because that makes the calculation a lot easier. But I don’t understand why we don’t divide the answer at the end by 10 as we multiplied the numbers at the start by 10. I think the answer should be 5÷10 = 0.5. 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
95 4.3 Dividing by decimals 4 Work out a 0 9. . 2 0 ÷ 4 b 5 74 . . ÷ 0 7 c −774÷0.9 d −288÷0.3 5 Artur pays $1.08 for a piece of string 0.8m long. Artur uses this formula to work out the cost of the string per metre. cost per metre = price of piece length of piece What is the cost of the string per metre? Tip Follow these steps 1 Write the division as a fraction. 2 Multiply the numerator and denominator by 10. 3 Use short division to work out the answer. Tip Remember, the symbol≈means ‘is approximately equal to’. Think like a mathematician 6 Discuss in pairs or small groups the answer to this question. What calculations can you do to check that the answer to a division is probably correct? For example, how can you check that 20 504 . . ÷ = 0 8 25 63 . is probably correct? 7 This is part of Jamal’s homework. Question i Estimate an answer to 498÷0.6 ii Work out the answer to 498÷0.6 Answer 498 × 10 0.6 × 10 =4980 6 i Round 4980 to 1 s.f. to give 5000 5000÷6≈800 because 6 × 800=4800 Estimate=800 ii 8 3 0 6 4 1 9 1 8 0 Answer=830 Use Jamal’s method to work out each of these divisions. i First, estimate the answer. ii Then calculate the accurate answer. a 276 03 . . ÷ b −232 ÷ 0 4. c 306 ÷ 0 9. d −483 ÷ 0 7. e 43 76 . . ÷ 0 8 f −33972 ÷ 0 6. 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
96 4 Decimals 8 Isla works out 50 4. . 6 1 ÷ 2. This is what she writes. 50.46 × 10 1.2 × 10 =504.6 12 4 2 . 5 12 5 0 2 4 . 6 6 0 So, 50.46÷1.2=42.5 a Explain the mistake that Isla has made. b Write the correct answer. 9 Raffa works out 461. . 7 1 ÷ 8. This is what he writes. 461.7 × 10 1.8 × 10 =4617 18 This is my 18 times table: 1 2 3 4 5 6 7 8 9 18 36 54 72 90 108 126 144 162 I can use the table to work out the division like this. 2 5 6 r 9 18 4 6 101 117 So, 461.7÷1.8=256 remainder 9 a What should Raffa have done, instead of stopping the division and writing ‘remainder 9’? b Work out the correct answer. 10 a Copy and complete the table below showing the 19 times table. 1 2 3 4 5 6 7 8 9 19 38 57 b Use the table to help you work out 59 375 . . ÷1 9 c Show how to check your answer to part b is correct. Use estimation with a reverse calculation. 11 a Complete the table below showing the 25 times table. 1 2 3 4 5 6 7 8 9 25 50 75 Tip In part c, your rounded answer to part b×2 should equal about 60. 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
97 4.3 Dividing by decimals b Helen buys a piece of wood for $58.90 The piece of wood is 2.5m long. Work out the cost per metre of the wood. c Show how to check your answer to part b is correct. Use estimation with a reverse calculation. 12 The diagram shows a rectangle with an area of 50.15m2 . The width of the rectangle is 3.4m. length 3.4m Area = 50.15m2 Work out the length of the rectangle. 13 Work with a partner to answer this question. a Balsem works out that 425×27=11475 Use this information to work out i 11475÷27 ii 11 475÷425 iii 11 475÷2.7 iv 11 475÷42.5 b Explain the method you used to work out the answers to part a. c Work out i 1147.5÷2.7 ii 114.75÷2.7 iii 11.475÷2.7 iv 1.1475÷2.7 d Explain the method you used to work out the answers to part c. e Check your answers with some other learners in your class to see if you agree. If you disagree about any of the answers, discuss where mistakes have been made. 14 This is part B of Marcus’s homework. Question Work out 1.798÷0.7 Give your answer to 1 d.p. Answer 1.798 × 10 0.7 × 10 = 17.98 7 2 . 5 6 … 7 1 1 7 . 3 9 4 8 6 0 1.798÷0.7=2.56…=2.6 (1 d.p.) Tip You can use the same formula as in Question 5. I must give my answer to 1 d.p. so I only need to work out the division to 2 d.p. and then I can round. 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
98 4 Decimals Use Marcus’s method to work out these calculations. Round each of your answers to the given degree of accuracy. a 3 79 . . ÷ 0 6 (1 d.p.) b 82 3. . 5 1 ÷ 1 (2 d.p.) c −5689 ÷ 2 3. (1 d.p.) Tip You only need to work out the division to one decimal place more than the degree of accuracy you need. Summary checklist I can divide decimals by numbers with one decimal place. When you are calculating using decimals, there are several methods you can use to make a calculation easier, such as • using the place value of the decimal • breaking a decimal into parts using factors • using the correct order of operations. 4.4 Making decimal calculations easier In this section you will … • simplify calculations containing decimals. Key word factor 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
99 4.4 Making decimal calculations easier Worked example 4.4 Work out mentally a (0.6−0.2)×0.8 b 22×0.9 c 4×12.6×2.5 d 12×0.15 Answer a 0.6−0.2=0.4 0.4×0.8=0.32 Work out the brackets first. 4×8=32 so 0.4×0.8=0.32 b 22×0.9=22×(1−0.1) =22×1−22×0.1 =22−2.2 =22−2−0.2 =20−0.2 =19.8 Replace 0.9 with (1−0.1) Work out 22×1 and 22×0.1 Now subtract 2.2 from 22 by subtracting the 2 first and then subtracting the 0.2 This gives an answer of 19.8 c 4×12.6×2.5=4×2.5×12.6 =10×12.6 =126 You can do the multiplications in any order. Notice that 4×2.5=10, so rearrange the question. Replace 4×2.5 with 10 and work out 10×12.6 d 12×0.15=12×0.5×0.3 =6×0.3 =6×3÷10 =18÷ 10 =1.8 Use factors of 15 to make the calculation easier: 0.15=0.5×0.3 12×0.5 is the easier calculation so do this first: (12×0.5=6) Then work out 6×0.3 Replace 0.3 with 3÷10, then do 3×6=18 Finally 18÷10=1.8 Exercise 4.4 1 Copy and complete the workings for these questions. a (0.3+0.4)×0.2= ×0.2 b (1.1−0.8)×0.4= ×0.4 = = 2 Use the same method as in Question 1 to work out a (0.7+0.1)×0.6 b (0.3+0.9)×0.5 c (0.6−0.4)×1.2 d (1.8−1.5)×1.1 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