Algebra Mathematics S. J. Cooper

THOMAS WHITHAM SIXTH FORM Algebra Mathematics S. J. Cooper Year 7 B U R N L E Y C@M P U S, B U R N L E Y, L A N C A S H I R E, B B 1 0 1 J D. T EL. 6 8 2 2 7 2

Algebra (1) Simplif each of the following terms: 1. 3a 4a 2a 2. 8 b 3b b 3. 6p 7 p 4. 3 n 7n 4n n 5. 5 c 9c c 6. 10m m 4m 7. d 3 d 6d d 8. 6e 3e 7e 9. 12 f 15 f 10. 21x 13x 6x 11. 7g g 6g 12. 8 4 10 13. 6 r 3r r 14. 8t 5t 7t 15. 4i 5i 8i 16. 12k 3k 4k 17. 8 w 3w 7w w 18. 9a 5a 4a 19. 3q q 2q 2q 20. 7 5 21. 8a 3b a 2b 22. c 5d 4c 3d 23. 7q p 3q 4p 24. 9x 2 4x 8 25. s 5t 5s 3t 26. 7 e 2 f 6e f Collection of like terms 27. 3g 4h g 2h 28. 6 2 x 7 29. 8m 6n 2m 4n 30. 10i 4 j 6i 3 j 31. 4 x 6 3x 32. 5k 7p k 4p 33. 9b 7c 8b 3c 34. 8x 8 6x 6 35. 4a 9b 7a 7b 36. 6u v u 2v 37. z 6z 8w 4w 38. 5m 3n 2n 4n 3m 39. 7a b 3c 5a b 2c 40. 8e 3 f 5g 5e f 4g 41. h 9i 7 j 6h i 3 j 42. 7m 2n 8p 4m 4n 6p 43. 9t 7r s 7r s 9t 44. 5x 10 8z x 4 3z 45. 15e 17 f g 11e 13 f 9g 46. a 2b a 3b 2a 5b 47. 3 x 2 x 2x 48. 8p 5q 10r 5p 2q 7r 49. 7u 7v 8w u v 5w 50. 7x 3x 4x Find a simplified expression for the perimeters of the following: 51. 52. 53. 54. a 2x 4t 4t a a 3 3 b b 4t 4t a b 4t 2x Thomas Whitham Sixth Form 1

Algebra (2) Simplif each of the following terms: 1. 7 x 2. 3 a 4 b 3. 5 x 3 4. 6 c 2 d 5. 7 4 p q 6. 2 8 x 7. 5 m 7 n 8. 4 e 5 d 9. 8 f 9 a 10. 3 s r 11. 3a 4b 12. 7m 6n 13. 12q 4p 14. 9 k j 15. 7 3x Simplifing Expressions 16. 5u 11v 17. x x 18. p p 19. 2a 9a 20. 5s 6r 21. 4 r r 22. 7h 7i 23. 2 p 3q r 24. 6a 2b 3c 25. e 4 f 7g 26. 4 x 6 z 27. 4t 15t 28. 6q p 3r 29. 5 8 30. 7m 2n Thomas Whitham Sixth Form 2

Algebra (3) Fill in the missing gaps In each of the following questions fill in the missing gaps: 1. What number should be written in the circles in the following: (a) 7 +5 (b) 7 3 (c) 7 X 2 (d) 7 + x (e) 7 (f) (g) (h) (i) (j) a p 3d 5h + b X r + 2 x 2d X 3i Thomas Whitham Sixth Form 3

2. What should be written in the rectangles in the following? (a) + 3 7 (b) 2 9 (c) X 5 20 (d) X 2 2a (e) + a 2 + a (f) (g) b X f x - b ef (h) (i) (j) + 2 x 5d X 3a 7x 2g 5d 9ab Thomas Whitham Sixth Form 4

3. What should be written next to the sign in the following: (a) x 4 12 (b) 4 + 12 (c) 12 4 (d) 3 + 3 + c (e) p X 8p (f) 5s + 5s + r (g) b X 4bc (h) 2 2 3x (i) 9d 2d (j) 5h X 30hg Thomas Whitham Sixth Form 5

4. What should be written in the circles or rectangles for each of the following: (a) 5 Double, and add 3 (b) Double, and subtract 1 11 (c) 3 Multipl b 3 and add x (d) a Multipl b 3 and add x (e) 2 Multipl b a and subtract b (f) d Double, and subtract 1 (g) Double, and add 5 2x + 5 (h) Multipl b 4 and subtract 2 4a 2 (i) Double, and add 2k 8 + 2k (j) 3p Multipl b 4 and subtract 3q Thomas Whitham Sixth Form 6

Algebra (4) Simple Equations Exercise 1 Find the number which must be placed in the box to make both sides equal. 1. + 2 = 5 6. 2 + = 6 11. 7 = 5 16. 6 = 4 2. + 4 = 6 7. 4 + = 7 12. 9 = 6 17. 2 = 9 3. + 6 = 9 8. 3 + = 6 13. 2 = 3 18. + 7 = 13 4. + 3 = 7 9. 8 = 6 14. 3 = 5 19. 7 = 18 5. + 7 = 9 10. 6 = 2 15. 4 = 5 20. + 14 = 27 Exercise 2 Find the number which has to replace the letter to keep both sides equal. 1. x + 3 = 5 6. + 4 = 8 11. a + 3 = 11 16. 9 p = 8 2. + 3 = 9 7. x + 9 = 12 12. p 7 = 2 17. 7 = 2 3. x + 5 = 9 8. d + 4 = 10 13. x 4 = 4 18. 12 r = 8 4. x + 2 = 8 9. 2 + n = 7 14. u 7 = 8 19. 3 = 2 5. x + 3 = 5 10. 1 + p = 8 15. b 5 = 6 20. 6 n = 3 Exercise 3 Find the value of the letter in each of the following: 1. 8x = 24 5. 6c = 48 9. 12m = 36 13. 8x = 72 2. 7x = 35 6. 2n = 40 10. 25q = 50 14. 5d = 40 3. 9a = 54 7. 30r = 90 11. 15t = 60 15. 12f = 84 4. 4b = 28 8. 3 = 27 12. 7e = 28 16. 20q = 100 Exercise 4 Find the number which has to replace the letter to keep both sides equal: 1. 2x 3 9 6. 3b 4 16 11. 4q 3 5 2. 2x 6 8 7. 3m 3 18 12. 4r 8 10 3. 2x 4 12 8. 3n 7 7 13. 6t 2 10 4. 2 1 13 9. 2 4 2 14. 5x 9 14 5. 3a 2 8 10. 2 7 3 15. 5v 6 6 Thomas Whitham Sixth Form 7

Algebra (5) Simple Equations Exercise 5 Solve the following equations: 1. 2x 4 8 2. 2x 3 13 3. 3x 2 10 4. 3x 5 4 5. 4x 1 23 6. 10x 7 5 7. 2x 3 2 8. 2x 1 7 9. 7x 4 38 10. 6x 7 5 11. 5 4x 3 12. 19 7x 2 13. 68 8x 12 14. 7 4x 3 15. 20 6x 7 16. 49 9x 14 17. 6x 1 20 18. 2x 4 10 19. 2x 3 13 20. 3x 1 10 21. 3x 5 11 22. 4x 3 19 23. 10x 7 57 24. 2x 3 10 25. 2x 8 13 26. 5x 7 32 27. 7x 4 25 28. 23 4x 7 29. 7 2x 1 30. 68 6x 8 Exercise 6 Solve each of the following equations: 1. 5x 3 2x 6 2. 4 5 3 26 3. 7q 4 2q 23 4. 5a 7 6a 40 5. 2x 4 3x 31 6. 6m 7 2m 5 7. 10n 1 4n 23 8. 13r 6 4r 60 9. 9p 7 3p 53 10. 4c 12 2c 18 11. 7x 8 5x 14 12. 8t 9 t 63 13. b 6 3b 46 14. 3d 2 3d 34 15. 4g 2g 11 23 16. 5 f 10 2 f 53 17. x 5 2x 44 18. 8 3 2 87 19. 11e 12 4e 78 20. 7h 4 3h 56 Exercise 7 Solve the following: 1. 6x 2 2x 14 6. 3p 15 2p 11. 11e 9 4e 37 16. 2v 3 45 6v 2. 5a 1 2a 8 3. 7 5 3 25 7. 7r 1 89 3r 8. 4t 18 2t 12. 11w 12 8w 6 13. 12 f 9 7 f 14 17. 11s 4 5s 22 18. 15z 8 8z 20 4. 5g 6 7 4g 9. u 8 34 6u 14. 10k 4 9k 7 19. 20 17 8 48 5. 8e 2 3e 32 10. 4 5 41 5 15. h 9 41 4h 20. 2x 9 19 2x Thomas Whitham Sixth Form 8

Algebra (6) Construction of equations 1. Asif is 2 ears older than Kamran. If Kamran is x ears old, how old is Asif? 2. The weight of a bag of bananas is 3 kg lass than a bag of apples. If the bag of apples weigh W kg what weight is the bag of bananas? 3. A taxi travelling between two towns has a charge of 3 plus a further 2 per person. What is the cost for a taxi with (a) one passenger? (b) three passengers? (c) x passengers? 4. An end of ear examination consists of three sheets of A4 paper. How man sheets of paper will be needed for (i) one pupil? (ii) two pupils? (iii) five pupils? (iv) pupils? 5. To cook a turke properl it must spend 40 minutes per pound in the oven on gas mark 7. What is the length of time required to cook a turke of mass (a) 2 pounds? (b) 10 pounds? (c) 15 pounds? (d) m pounds? 6. Garden Gro advertisers their plant food as 5 ml required per litre of water. How man millilitres will ou require for (a) 3 litres? (b) 8 litres? (c) p litres? (d) f litres? 7. A tube of smarties contains, on average, 40 smarties per tube. How man smarties are ou likel to eat in (i) 2 tubes? (ii) 4 tubes? (iii) 10 tubes? (iv) s tubes? 8. A number N is equal to the sum of 8 plus another number a. Write down a formula connecting N and a. 9. A second number S is equal to the sum of two numbers p and q. Write down a formula connecting S, p and q. 10. The number A is equal to twice the number b plus 3. Using algebra express this as a formula. 11. The number x is equal to three times the number minus z. What formula can be written for this statement. Thomas Whitham Sixth Form 9

12. Paul s bag contains p books, q pens and r pencils. If the total number of items in his bag is T write down a formula connecting T, p, q and r. 13. At school pens are sold at x pence each and pencils are sold at pence each. Write down the total cost, C, for a) 2 pens and 1 pencil. b) Three pens and four pencils c) m pens and n pencils. 14. Write down an expression for the perimeter of x this rectangle. x + 3 15. Danielle can run at an average speed of v metres per second. How man metres will she cover in (a) 2 seconds? (b) 5 seconds? (c) 1 minute? (d) g seconds? Thomas Whitham Sixth Form 10

Algebra (7) Construction of equations 1. I think of a number, double it and then add 3. The answer is 15. What was m number? 2. I think of a number, double it and subtract 5. The answer is 7. What was m number? 3. I think of a number, treble it and add 4. The answer is 22. What was m number? 4. I think of a number, multipl it b 3 and subtract 5. The answer is 10. What was m number? 5. I think of a number, multipl it b 4 and subtract 1. The answer is 15. What was m number? 6. I think of a number, multipl it b 7 and add 4. The answer is 39. What was m number? 7. Given the perimeter of the rectangle opposite is 30 cm find its length and width x 8. If the perimeter of the triangle is 45 cm find the sides of the triangle. 2 2 x + 3 9. (a) Work out an expression for the perimeter of the rectangle opposite. (b) Given the perimeter is 54 cm find the value of h. 20 cm h 10. Given the area of the rectangle above is equal to 80 cm2 work out the value of h. 11. I think of a number, double it and add 5. The result is exactl the same as adding 6 to the number. What is the number? 12. I think of a number, multipl it b 3 and subtract 5. The result is equal to the same number plus 9. What is the number? 13. I think of a number, multipl it b 3 and add 1. The answer is equal to the same number doubled add 13. What is the number? Thomas Whitham Sixth Form 11

Algebra (8) Substituting into formulae 1. If a 4 p find a when p 7 2. If m 7 n find m when n 3 3. Given t 6 s work out t when s 9 4. Given a c b what is the value of a when b 14 and c 13? 5. If x z what is the value of x when 34 and z 19? 6. If 2x 1 work out the value of given x 5. 7. If 4x 7 what is the value of when x 2? 8. Given d 6 2 e work out the value of d when e 3. 9. If q 3p 4 what is the value of q when p 12? 10. If g 8 4 h find the value of g when h 5. 11. If m nr 7 work out the value of m when n 8 and r 2. 12. Given mx c what is the valuw of when m 2, x 6 and c 1? 13. Given v u at find v when u 7, a 3 and t 10. 14.Evaluate S 3 T U when T 9 and U 5. 15. If V M work out V when M 144 and D 3. D 16. Given S D work out S for D 240 and T 12. T 17. If h i 6 3 work out h for i 1. Thomas Whitham Sixth Form 12

18. Given m n 4 2 work out the value of m when n 9. 19. Evaluate P q r 5 when q 5 and r 2. 20.Given x 2 what is the value of when x 6? Thomas Whitham Sixth Form 13

Algebra (9) Coordinates 1. Write down the coordinates for each letter in the diagram below. M 10 9 L P 8 J 7 N Q 6 S 5 T R I 4 E C 3 K H F 2 G 1 A D 0 1 B U 2 3 4 5 6 7 8 9 10 x 2. (a) Draw a set of axes labelled from 0 to 8 on both the x and axis. (b) Plot the following coordinates on the set of axes (i) (3, 1) (vi) (6, 3) (ii) (8, 7) (vii) (2, 7) (iii) (0, 4) (viii) (4, 4) (iv) (5, 2) (ix) (8, 0) (v) (1, 0) (x) (7, 5) 3. (a) Draw a set of axes labelled from 0 to 8 on both the x and axis (b) Plot the coordinates for each of the following and join each coordinate with the next coordinate, using a ruler. (i) (2, 1), (7, 1), (5, 6) (ii) (0, 3), (8, 3), (6, 7) (2, 7) (iii) (0, 4), (2, 0), (5, 1), (5, 4) (iv) (8, 8), (3, 1), (4, 6) (v) (5, 0), (2, 7), (0, 1) Thomas Whitham Sixth Form 14

Algebra (10) Coordinates 4. Write down the coordinates for each letter in the diagram below. 6 L P 5 M 4 J N 3 C S 2 A 1 R T E 0 6 5 4 3 2 1 1 2 3 4 5 6 x 1 H K G 2 I 3 4 B D F 5 Q 6 U 5. (a) Draw a set of axes labelled from 8 to 8 on both the x and axis. (c) Plot the following coordinates on the set of axes. (i) (2, 7) (vi) ( 5, 5) (ii) (3, 1) (vii) ( 3, 8) (iii) (4, 2) (viii) (0, 4) (iv) ( 7,1) (ix) (2, 2) (v) (4, 6) (x) ( 1, 0) (xi) (6, 0) (xii) ( 2, 3) (xiii) (8, 4) (xiv) ( 5, 3) (xv) ( 1, 5) 6. (a) Draw a set of axes labelled from 10 to 10 on both the x and axis (c) Plot the coordinates for each of the following and join each coordinate with the next coordinate, using a ruler. (vi) ( 2, 1), (7, 1), ( 5, 6) (vii) (0, 3), (8, 3), (6, 7) (2, 7) (viii) (0, 4), (2, 0), ( 5, 1), ( 5, 4) (ix) (8, 8), ( 3, 1), (4, 6) (x) ( 5, 0), ( 2, 7), (0, 1) Thomas Whitham Sixth Form 15

Algebra (11) Sequences 1. Write down the next two terms for each of the following sequences and a rule in words. (a) 2 5 8 11 (b) 1 8 15 22 (c) 8 11 14 17 (d) 5 11 17 23 (e) 6 10 14 18 (f) 4 12 20 28 (g) 9 11 13 15 (h) 7 12 17 22 (i) 7 18 29 40 (j) 13 21 29 37 2. Write down the next two terms for each of the following and give a rule for continuing the sequence. (a) 20 18 16 14 (b) 37 33 29 25 (c) 99 88 77 66 (d) 47 44 41 38 (e) 56 50 44 38 (f) 80 71 62 53 (g) 63 56 49 42 3. Write down the first five terms for each of the following described sequences. a) Add 2 to the previous term: 5,. b) Add five to the previous term: 9,. c) Subtract 1 from the last term: 10, d) Subtract 8 from the previous term: 65, e) Multipl the previous term b 2: 1,.. f) Multipl the previous term b 3: 3, g) Divide the last term b 2: 64, h) Add the next even number each time: 1, 3, i) Add a number that increases b 2 each time: 3, 4, Thomas Whitham Sixth Form 16

4. Adding together the previous two terms generates the Fibonacci sequence. The first six terms of the Fibonacci sequence starting with 1, 1 are 1, 1, 2, 3, 5, 8. (a) Write down the next three terms in this sequence. (b) Form a new Fibonacci sequence starting with 2, 3 and write down the next five terms. 5. Fill in the missing gaps in the following sequences: a 4,..., 12, 16,...,... b 3,..., 13, 18,...,... c 1,..., 5,..., 9,... d 2, 4,..., 16,...,... e 20,..., 14,..., 8,... f 9, 18,...,..., 45,... 6. For each of the following sequences give the first four terms: a Add 3 to the previous term: 5, 8,... b Divide the previous term b 2: 8, 4,... c Subtract 4 from the previous term: 15, 11,... d Multipl the previous term b 3: 2, 6,... e Add 9 to the previous term: 1, 10,... 7. The coordinates below form a sequence of pairs of numbers (1, 3), (2, 5), (4, 7), (8, 9) a Find the next two sets of coordinates in the sequence. b Find the tenth set of coordinates in the sequence. c Describe the rule for finding (i) the x coordinates (ii) the coordinates. 8. For each of the following sequences below give the next three terms and a rule for continuing the sequence. a 1, 2, 3, 4, 5, 6,...,...,... b 1, 3, 5, 7, 9,...,...,... c 4, 8, 12, 16,...,...,... d 7, 14, 21, 28,...,...,... e 17, 15, 13, 11,...,...,... f 1, 4, 9, 16,...,...,... g 1, 3, 6, 10,...,...,... h 4, 2, 1, 1 2, 1,...,...,... i 2, 6, 12, 20,...,...,... 4 9. For each of the following sequences find the next two terms and a rule for continuing the sequence. a 21, 17, 13,...,... b 8, 16, 24,...,... c 2, 4, 8, 16,...,... d 1, 3, 9, 27,...,... e 1, 8, 27,...,... f 6, 12, 18,...,... Thomas Whitham Sixth Form 17

Algebra (12) Sequences 1. The following collections of dots suggest another sequence of numbers. The sequence is started below. 1, 4, 9, 16, etc. a) Draw the next two patterns in the sequence. b) Write down the next seven terms in this sequence c) Write down the values of (i) the 2 nd term (ii) the 10 th term. d) What sorts of numbers belong to this sequence? 2. Using the squares in our books draw the next two arrangements for each of the following a patterns: b c d Thomas Whitham Sixth Form 18

e f 3. Triangular numbers can be represented b dots arranged as triangles: Draw and write down the next four in the sequence. 1 3 6 10 4. John has constructed a triangle using three pencils, as shown in diagram 1. Elizabeth then added some pencils to John s and constructed diagram 2. Brian finall added some to Elizabeth s and constructed diagram 3. (a) Draw the next two patterns in the sequence. (b) Cop and complete the table below. Diagram number 1 2 3 4 5 6 7 Number of pencils 3 5 (c) State a rule in words for continuing the pattern. 5. Using a new set of pencils a series of diagrams is constructed forming squares. Thomas Whitham Sixth Form 19

(a) Draw the next two patterns in the sequence. (b) Cop and complete the table below. Diagram number 1 2 3 4 5 6 7 Number of pencils 4 12 (c) State a rule in words for continuing the pattern. 6. Ian is going to build a garden path in the shape of a cross. He starts b laing the centre paving stone which is in a dark colour, as shown in figure one. He then places several lighter paving stones around the centre stone as shown in figure 2. Next he continues to alternate the colour of the paving stones as shown in figures 3 and 4. Figure1 Figure2 Figure 3 Figure4 (a) Draw the next two diagrams in the sequence. (b) Cop and complete the table below. Figure number 1 2 3 4 5 6 7 Number of dark paving stones Number of light paving stones (c) State a rule in words for connecting the figure number and (i) the number of dark paving stones. (ii) the number of light paving stones. Thomas Whitham Sixth Form 20

Algebra (13) The nth term 1. Give the first four terms of the sequence for which a) 3n 5 U n e) 8n 5 U n i) 3 U n n b) 5n 1 U n c) U n 4n d) 2n 1 U n f) 5n 4 g) U n 2 U n n h) U n 2 n 2 j) U n 2 n 1 k) U n 15 3n l) 9n 3 U n 2. Find the first 5 terms for the sequence 7n 3 V n 3. Find the first 4 terms for the sequence n n 2 V n 4. Find the first 5 terms for the sequence n V n n 1 5. Find the first 5 terms for the sequence V n 2n 1 3 n 1 6. What nth term (formula) would give ou the sequence 2, 5, 8, 11, 14, 17 Thomas Whitham Sixth Form 21

Algebra(14) Straight line graphs 1. a) Using the formula = 2x + 5 complete the table below for different values of x. x 0 1 2 3 4 b) Plot each of the points found above on a set of axes labeling the x axis from 0 to 5 and the axis from 0 to 15 2. Given = 3x 1, find the different values of for the values of x stated in the table below. x 1 2 3 4 5 Coping the axes drawn below, draw the graph of = 3x 1 labelling our axes carefull. 10 8 6 4 2-4 -2 0 2 4 x -2-4 -6 3. B completing the table below, plot the graph of = 4x + 2 x 0 1 2 3 4 4. (a) B completing the table below, plot the graph of = 3x 4 x 2 3 4 5 (b) Using our graph find the value of when x = 1. Thomas Whitham Sixth Form 22

5. (a) B completing the table below, plot the graph of = 4x 7, labelling our x axis 5 to 5 and the axis from 10 to 15 x 2 3 4 5 (b) Using our graph find the value of when (i) x = 1. (ii) x = 1 6. For each of the following (i) Cop and complete the table (ii) draw a set of axes labeling the x axis from 5 to 5 and the axis from 5 to 16 (iii) draw the graph of the straight line found. a) = x + 7 x 0 1 2 3 4 b) = 2x 1 x 1 2 3 4 c) = 3x + 2 x 0 1 2 3 4 d) = 5x 4 x 1 2 3 4 7. Using our own table, draw each of the following graphs. a) = x + 5 b) = 2x + 8 c) = 3x + 7 d) = 4x 5 e) = 10 2x Thomas Whitham Sixth Form 23

Algebra(15) removal of brackets Remove the brackets for each of the following expressions 1. 3 a + 7 2. 5 b + 8 3. 9 x 3 4. 2 2a + 5 5. 6 4c 7 6. 4 3 1 7. 9 5d + 6 8. 7 2m + 9 9. 10 2e 3 10. 3 3n + 8 11. x x 2 12. a 2a + 7 13. 3b b 3 14. 4m 3m + 2 15. 5h h 4 16. 4 2a + 7b 17. 3 6x 7 18. 4 2p + 3q 1 19. 3x 3 5x 20. 7m 3m 8 Thomas Whitham Sixth Form 24