Contents. Useful formulae. iv Glossary

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1 Contents Useful formulae iv Glossar v Unit 1 Histograms with equal class widths Get started 1 1 Drawing histograms with equal class widths Estimating the mean from a histogram 4 Practise the methods 6 Problem-solve! 7 Unit Indices and recurring decimals Get started 8 1 Converting recurring decimals to fractions 9 Calculating using fractional and negative indices 10 Practise the methods 1 Problem-solve! 1 Unit Surds Get started 14 1 Simplifing surds 15 Epanding brackets involving surds 16 Rationalising denominators 17 4 Simplifing algebraic epressions involving surds 18 Practise the methods 19 Problem-solve! 0 Unit 4 Cumulative frequenc Get started 1 1 Constructing cumulative frequenc graphs Drawing and interpreting bo plots 4 Comparing distributions 6 Practise the methods 8 Problem-solve! 9 Unit 5 Probabilit Get started 1 1 Combining events Tree diagrams Venn diagrams for conditional probabilit 5 Practise the methods 7 Problem-solve! 8 Unit 6 Direct and inverse proportion Get started 40 1 Direct proportion 41 Inverse proportion 4 Direct and inverse proportion graphs 4 Practise the methods 44 Problem-solve! 45 Unit 7 Accurac and bounds Get started 46 1 Upper and lower bounds of measurements 47 Upper and lower bounds of compound measures 48 Practise the methods 49 Problem-solve! 50 Answers 51 F01_EDNS_G7_WB_GCSE_86_PRE_i-iii.indd 05/04/18 5:05 pm

2 Get started 6 Direct and inverse proportion This unit will help ou to solve problems where is directl or inversel proportional to the square of and to use and recognise graphs showing direct and inverse proportion. A1 Fluenc check 1 a = b p = 10 q i Work out when = 5. i Work out p when q =. ii Work out when = 18. ii Work out q when p = 6.5. Make the subject of each equation. a 4 = b = c = 5 Number sense d a = b Circle the odd one out Ke points When is directl proportional to the square of, write When is inversel proportional to the square 1 of, write These skills boosts will help ou to solve problems involving direct and inverse proportion and to use and recognise graphs showing proportion. 1 Direct proportion Inverse proportion Direct and inverse proportion graphs You might have alread done some work on proportion. Before starting the first skills boost, rate our confidence using these questions. 1 is directl proportional to the square of. When =, = 8. Work out when = 00. is inversel proportional to the square of. When = 0.16, = 5. Work out when = 100. In this graph, is directl proportional to? Eplain our answer. How confident are ou? 40 Unit 6 Direct and inverse proportion M06_EDNS_G7_WB_GCSE_86_U06_ indd 40 05/04/18 5:1 pm

3 Skills boost 1 Direct proportion means is directl proportional to the square of. So = k, where k is a number/constant. Guided practice is proportional to. When = 5, = 75. a Write a formula for in terms of. b Work out the value of when = 8. a So = k 75 = k k = The formula is = b When = 8, = Substitute = 5 and = 75 into the equation = k Simplif and rearrange to work out the value of k. Substitute our value of k into the equation = k Substitute = 8 into our formula to work out. 1 is proportional to the square of. When =, = 45. a Write a formula for in terms of. b Work out the value of when = 6. is proportional to. When =, = 16. Hint To work out when = 100, a Write a formula for in terms of. substitute = 100 b Work out the value of when = 7. into our formula and rearrange it to c Work out the value of when = 100. work out. q is proportional to the square of p. When p = 10, q = 0. a Write a formula for q in terms of p. b Work out the value of q when p = 4. c Work out the value of p when q = b is proportional to a. When a = 4, b = 8. a Write a formula for b in terms of a. b Work out the value of b when a = 1. c Work out the value of a when b = Eam-stle question 5 is proportional to the square of. When = 10, = 5. Work out the value of when = 8.6. ( marks) Reflect Eplain what means and then use what ou know to eplain what means. Unit 6 Direct and inverse proportion 41 M06_EDNS_G7_WB_GCSE_86_U06_ indd 41 05/04/18 5:1 pm

4 Skills boost Inverse proportion 1 means is inversel proportional to the square of. k So = where k is a number/constant. Guided practice is inversel proportional to the square of. When = 10, = a Write a formula for in terms of. b Work out the value of when = 5. Worked eam question 1 a So = k k = k = 10 The formula is = b When = 5, = Substitute, = 10 and = 0.04 into the equation = k Simplif and rearrange to work out the value of k. Substitute our value of k into the equation = k Substitute = 5 into our formula to work out. 1 is inversel proportional to the square of. When = 0.5, = 0. a Write a formula for in terms of. b Work out the value of when =. is inversel proportional to. When =, = a Write a formula for in terms of. b Work out the value of when = 0.4. c Work out the value of when = 150. q is inversel proportional to the square of p. When p = 0.1, q = 80. a Write a formula for q in terms of p. b Work out the value of q when p = 5. c Work out the value of p when q = 5. Hint To work out when = 150, substitute = 150 into our formula and rearrange it to work out. Eam-stle question 4 is inversel proportional to the square of. When = 0., = 6.5. Work out the value of when = 0.1. ( marks) Reflect 1 Eplain what means and then use what ou know to eplain what 1 means. 4 Unit 6 Direct and inverse proportion M06_EDNS_G7_WB_GCSE_86_U06_ indd 4 05/04/18 5:1 pm

5 Skills boost Direct and inverse proportion graphs In this graph, is directl proportional to. In this graph, is directl proportional to the square of. In this graph, is inversel proportional to. In this graph, is inversel proportional to the square of. Guided practice In which of these graphs is directl proportional to? A B C D In graphs A and proportional to. is directl If is directl proportional to, the graph will be a straight line that passes through the origin. Graph B does not pass through the origin. 1 In this graph is directl proportional to. In this graph is inversel proportional to. Work out the values of a and b. Work out the values of p and q. (, 7) (5, b) (a, 14) (, 5) (4, p) (q, 1) Eam-stle question Each graph shows a proportionalit relationship between and. Graph A Graph B Graph C Graph D Match each graph with a statement in the table below. Proportionalit statement is directl proportional to. is inversel proportional to. Graph is directl proportional to the square of. is inversel proportional to the square of. ( marks) Reflect Describe the features of the graphs for direct and inverse proportion. Unit 6 Direct and inverse proportion 4 M06_EDNS_G7_WB_GCSE_86_U06_ indd 4 05/04/18 5:1 pm

6 Get back on track Practise the methods Answer this question to check where to start. Check up Tick the equation that represents the relationship is inversel proportional to the square of. A B C D = k = k k = = k If ou ticked C, work out the value of k given that = when =.5. Then go to Q4. If ou ticked A, B or D go to Q1 for more practice. 1 Match each proportionalit statement with an equation. A is directl proportional to. i = k B is inversel proportional to. ii = k C is directl proportional to the square of. D is inversel proportional to the square of. iii = k iv = k is directl proportional to the square of. When = 1.5, = 18. Write a formula for in terms of. is inversel proportional to the square of. When = 0., = 50. Write a formula for in terms of. 4 is directl proportional to the square of. When = 4, = 0. a Write a formula for in terms of. b Work out the value of when = Circle the graph in which is inversel proportional to. A B C Eam-stle question 6 T is inversel proportional to d. T = 180 when d = 6. Work out the value of T when d = 0.5. ( marks) 44 Unit 6 Direct and inverse proportion M06_EDNS_G7_WB_GCSE_86_U06_ indd 44 05/04/18 5:1 pm

7 Get back on track Problem-solve! Eam-stle question 1 D is directl proportional to the square of n. Carl sas that when n is doubled, the value of D is doubled. Carl is wrong. Eplain wh. (1 mark) In an eperiment, measurements of a and b were taken. a 6 10 b Which of these relationships fits the result? b a b a b a b a Eam-stle questions is inversel proportional to the square of. When = 4, = Work out the value of when = A shape has an area A cm. The shape has a width w cm. A is directl proportional to w. When w = 0, A = 40. Work out the area of the shape when the width is cm. 5 d is inversel proportional to the square of c. When c = 5, d = 00. Work out the value of d when c =.5. ( marks) (4 marks) ( marks) 6 In this graph is inversel proportional to. Work out the values of a and b. (a, 10) (,.5) (4, b) Now that ou have completed this unit, how confident do ou feel? 1 Direct proportion Inverse proportion Direct and inverse proportion graphs Unit 6 Direct and inverse proportion 45 M06_EDNS_G7_WB_GCSE_86_U06_ indd 45 05/04/18 5:1 pm