algebra-1

How much is a single GCSE mark worth? Interesting question, don’t you think? Imagine two GCSE students equally matched in ability in almost every way except that Student A just manages to get the grade they need and Student B just misses by a few marks:

Student A ** Puts in that bit extra and gets the grade necessary to: ** Go to the university of their choice. ** Get the job they want.

Student B ** Thinks they are working hard, but doesn’t pay much attention to detail. Hence: ** Doesn’t get into the university of their choice. ** Doesn’t get the job they want.

Consequence:

Student A gets the job of their dreams.

Student B gets a job they don’t really enjoy doing.

Student A earns at least £2000 per year more than Student B.

Over 35 years that’s £70 000!

If Student B missed the grade they wanted by 5 marks that’s £14 000 per mark! If they missed by just one mark, that’s £70 000 per mark!!!!!!!

And that’s probably a minimum!

How much is a GCSE mark worth to you? Make sure you are Student A by purchasing the answers to these modules and checking every detail of your answers, so you don’t throw away any of those very critical marks in this year’s exams. Probably the best £10 you will ever spend.

Home Study Modules KS4 Foundation Level

Algebra 1 Please enjoy using these free questions. If you would like fully worked answers to the questions in all the GCSE modules (similar to the free Transformations module answers) you may purchase an immediate download for just £10 by following the instructions on the home page of our website:

http://www.gcsemathematics4u.co.uk

Price held until 6th May

GCSEMathematics4U

© Mathematics4u www.gcsemathematics4u.co.uk

Home Study Modules Foundation Level: ALGEBRA 1 ©Mathematics4U www. gcsemathematics4u.co.uk

C1. Write algebraic expressions for the following: a) 6 more than y b) 7 less than w c) 4 more than 2x d) twice r e) three times p f) 6 times d g) t subtract 7 h) t subtract h i) 2m add 4f j) r multiplied by itself s) twice t2 t) f lots of g u) three lots of mn 2. Simplify where possible: a) 2a + 3a b) 6b – 4b c) n + n + n d) 5r + 8r – 7r e) 2j + 6j + 4j – j f) 6t – 3t + 8t g) Half of 6q h) One third of 12b i) 6s + 7s j) 8u – 8u s) 2x + 3y t) –6y + 5y u) 7g – 9g + 5g v) 4x + 8x – 3x + 7x + 4x – 12x + 3x 3. Simplify where possible: a) 6t + 3b – 2t b) 7f – 4f + 5m c) 12c – 5c + 6c d) 3s + 6s + 3d + 6d e) 9p – 4p + 6y f) 2r + 4t – r + 5t g) 5f – 3a + 7a h) 9i – 5i + 5r + 2r i) 9k + 8p – 4k – 9p j) 5h + 4h – 9h s) 100j – 55j + 67j t) 4r + 6t + 8u 4. Find the perimeters of these shapes: a) b) c) d) 5. Multiply out the brackets: a) 2(a + b) b) 5(7m + 4) c) 9(7g + 5) d) 8(y – 5f) e) 7(s + 6t) f) 2(8 + 4r – 6g) g) 4(4f – 6g + 2h) h) 5(3 + 6t – 3d) i) 6.5(2w – 4x) j) 0.5(2m + 6n ) k) ½(9y + 3d) l) 99(–2a + 4f – 6g + 5t)

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Regular Hexagon

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6. In algebra, what is a ‘term’? Say how many terms there are in each of the following expressions: a) 5t + 7y – 6z b) 4m c) –6s – 2w2 d) 6a – 6b) e) 4as2 + 6at3 – 4ae + 2am4 f) –12 – 20r + 36p 7. Multiply out the brackets: a) –4(p + q) b) –9(4u + 5) c) –6(4r – 3) d) –3(–6d + 5e) e) 3(2p + 7r) f) 5(9p – u + 2g) g) –6(3e + 2f + 3g) h) –(4 – 5r + 2d) i) –(2t – 5x) j) –2.5(4q + 12k) 8. Find the value of the following expressions. a) 3m + 5 when m = 25 b) 4d + 6t when d = 3 and t = 4 c) 22 – 6g when g = 3 d) 22 – 6g when g = 6 e) 8y + 4r – 3d when y = 5, r = 2.5 and d = 7 f) 12 – x when x = –3 g) 24 – r when r = –19 h) 30 – 4f when f = –3 i) 6t + 9 when t = 8 2 j) 4p + 4s when p = 5, s = 4 and t = 3 3t k) h + 6 when h = 7 h – 6 l) h + 6 when h = 4 h – 6 m) 12 when t = 1.5 t

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9. Simplify the following: a) d x d x d x d b) a x a + 2 x w x w c) d2 x d3 d) 3a3 x 6a4 e) 4b2 x 8c3 f) c(4c + 4d – 2e) g) 3w(4s – 5t2 + 2w2) h) m2(m + 4n + 2p2) i) 4d3 x 4d2 x 2d3 j) 3s2 x 4s3 x 2s2t4 k) 5h3 x 4h4 l) 6t2 x 5s3t6 x 2s4t2 10. Find the value of the following expressions. a) r2 + 7 when r = 9 b) 2f3 when f = 4 c) 6y2z3 when y = 2 and z = 3 d) 16y2 – 3z when y = 5 and z = 7 e) 4r(2p + 3q2 + 5r3) when p = 3, q = 4 and r = 2 f) 8m2n4 when m = 1.5 and n = 2 g) 6a(2a3 – 4b2 + 3c4) when a = 2, b = 3 and c = 4 h) 4d3(6c3 + 5d2 – 2e3) when c = 4, d = 3 and e = 5 11. Find the value of the following expressions. a) a2 + 3 when a = –4 b) 7g3 when g = –1 c) 8p3q2 + pq when p = 3 and q = –2 d) 7m2 + 6n2 when m = –5 and n = –6 e) 3h3k2 when h = 3 and k = –1.5 f) 2s(2a – 4b + 6c2) when a = 2, b = 3, c = –2 and s = 1.5 12. I have sixteen apples in a barrel. I put in another q apples. How many are in the barrel now? 13. A pencil case contains p pencils. A boy takes out q pencils. How many pencils are now in the pencil case? Write an inequality showing the relationship between p and q. 14. A caretaker divides his keys equally between r key rings. He has k keys on each ring. How many keys does he have altogether? 15. A school has 400 pupils. The pupils are arranged into c classes of equal size. How many

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pupils are there in each class?


16. Peter and Jane each have a bag of sweets. There are 120 sweets in each bag. Peter takes p sweets from his bag and Jane takes three times as many from hers. How many sweets are left in Jane’s bag? 17. A rectangular carpet is x metres long and y metres wide. What is the area of the carpet? 18. A square has a side of length d cm. What is the area of the square? 19. Find the area of this shape.

a cm

b cm c cm

d cm 20. Jenny is t years old. Her brother Ben is 2 years younger than Jenny. Her sister Mary is r years older than Jenny. Her cousin Fred is three times as old as Jenny. Her aunt Helen is q times as old as Mary. How old are Ben, Mary, Fred and Helen? 21. What is the perimeter of this quadrilateral? a + 4b metres

4a – b metres

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22. Solve the following equations: a) 3x + 5 = 11 b) 6x + 8 = 32 c) 5y – 9 = 26 23. Solve the following equations: a) 2(d + 3) = 25 b) 6(4m – 3) = 150 c) t2 + 8 = 24 24. Solve the following equations: a) ½(g + 6) = 14 b) 1/3(m – 7) = 2 c) – 7 = 8 25. Solve 4(2x2 – 5) + 7 = 187 26. Solve these equations: a) 15 – 3y = 6 b) 4(19 – 2t) = 32 c) 5 – = –2

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k 3

27. Three sides of a triangle are 3x – 5 metres, 2x – 4 metres and x + 2 metres. The perimeter of the triangle is 41 metres. What is the value of x? 28. This grass cutter costs £40 to hire plus £6 per day. a) How much does it cost to hire the cutter for 20 days? b) If £c is the cost and n is the number of days of hire, form an equation connecting c and n. c) Use the equation to find the number of days hire if the cost is £328.00

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29. Three angles of a triangle are 3ao, 6ao and 6ao. a) What type of triangle is it? b) Write an equation in a for the sum of the angles of the triangle and hence find a. 30. The cost of a cup is h pence. The cost of a glass is twice as much plus 25p. a) What is the cost of a glass in terms of h ? b) Peter buys 6 cups and 12 glasses and pays £4.50. Find the value of h. 31. A rectangle is 3z – 2 cm long and 12 cm wide.

3z – 2 cm

12 cm

a) Form expressions in z for the perimeter and the area of the rectangle. b) Given that the area is 156 cm2, find the value of z and hence find the length of the perimeter. 32. At a tea party, the organizer knows that the number of cups of tea required is roughly calculated by doubling the number of people attending and adding 20. a) If p is the number of people attending, how many cups of tea should she need to make? b) At the end of the day, the organizer worked out she had made 146 cups of tea. Give an estimate for the number of people attending. 33. A coach company charges £150 plus £3.50 per kilometre to hire a coach. a) If the number of kilometres travelled on a particular journey is k, form an equation connecting k to the cost £p for one journey. b) Mr Brown wants to take his scout group on a return trip to Mount Snowdon. If the total cost is going to be £1025.00, and the journey is pretty straight, how far does Mr Brown and his scout group live from Mount Snowdon? 34. Mary asks Kelly to think of a number. Mary then tells Kelly to multiply her number by four, add six to the answer, multiply the answer by three and subtract sixteen. Kelly’s final answer was eighty. What number did she think of at the beginning?

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35. Soroush draws several rectangles, each of which has an area of 60 square centimetres. a) If the length of a rectangle is m cm, write an expression for the width of a rectangle. b) One of his rectangles is 25 cm long. What is its width? (These last two questions are very difficult. Have a go, but don’t worry too much if you find them particularly stressful.) 36. The grid shows a number square with a Y shape drawn on it.

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37. The grid shows a number square with a Squiggle shape drawn on it.

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The Squiggle shape shown is referred to as S61 because the bottom left number is 61. a) What is the total of all the numbers in the squares on S61 ? b) What is the total of all the numbers in the squares on Sn ? c) If all the numbers on Sp total 562, what is the value of p? d) Draw this particular Squiggle on the grid and show the total of the numbers on this Squiggle is indeed 562.

Home Study Modules KS4

Higher Level

Algebra 1 Please enjoy using these free questions. If you would like fully worked answers to the questions in all the GCSE modules (similar to the free Graphs module answers) you may purchase an immediate download for just £10 by following the instructions on the home page of our website:

http://www.gcsemathematics4u.co.uk

GCSEMathematics4U

© Mathematics4u www.gcsemathematics4u.co.uk

http://www.gcsemathematics4u.co.uk/

Home Study Modules Higher Level: ALGEBRA 1 ©Mathematics4U www. gcsemathematics4u.co.uk

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Revision Topics What you will be practising: Concept

Examples

Working with algebraic expressions Simplify 4ax2 + 6a2x – 3ax2 Find √(t2 + u2) when t = 12, u = 5

Multiplying out brackets 8(2m + 7n) – 5(3m + 5n)

Substituting values for letters Find 5x2 + 3x when x = 9 4x – 4

Solving linear equations Solve 3x + 8 = 35; ⎯5f – 6 = 9 etc

Solving simultaneous equations Solve 4p + 2q = 26 10v – 2w = 12.2 25p – 10q = ⎯40 3v + 9w = 45.9

Re-arranging formulae Express h in terms of the other letters: h2 + g2 + k2 = 1 Make g the subject of: s = ag2

Answering questions based on the above concepts expressed in words

A hexagon has sides a, 3a, 3a, 4a, 6a and 7a metres respectively. The total perimeter of the hexagon is 120q metres. What is the length of each side in terms of q ? If 7 buns and 4 cream cakes cost £3.63 and 3 buns and 5 cream cakes cost £2.87, what is the cost of one bun and the cost one cream cake?


Q1. Simplify the following expressions: a) 2x + 7x + 3y + 5y b) 6t – 5t + 7t c) 4r2 + 2r + 9r2 – 3r + 4r2 d) 4f3 – 6f3 + 7f2 – 5f3 + 8f2 e) 4ax2 + 6a2x – 3ax2 f) 2p2q4r + 6p2q4r – 2pq4r2 2. Multiply out the brackets in the following expressions and simplify where possible: a) 3(g – 2) b) a(r + 5 – 6t2) c) 2p(6p – 8r) d) x2(5x2 + 6x – 8) e) 2p2(8p4 + 4p3 – 5p +6) f) 5pr2(4p + 6r3) g) 6y2(5xy2 + 4x2y) 3. Multiply out the brackets in the following expressions and simplify where possible. a) 2(a + 5) + 6(a – 2) b) 4(x + 2y) + 5(2x – 3y) c) 5(2m – 4n) + 3(3m + 5n) d) 8(5r – 12) + 6(3r – 7) e) 4(7 + 7y) + 6(9 + 8y) f) 8(7g + h) + 9(5g – 3h) g) a(t + 4t2) + 8(at + 3at2) h) b(r + 5m2) + b(2r + 7m2) i) b(m2 – 6m) + m(2bm + 4b) j) 5(r + 2) – 4(r + 6) k) 6(y2 + 7) – 5(y2 + 9) l) 8(2m + 7n) – 5(3m + 5n) m) 5a(2b + b2) – 3b(2a – 4ab) n) 7(r + g2) + 6(r – g2) o) 4r(s + rs2) – 3s(5r – 2rs) 4. Substitute the given values for the letters in the expressions: a) 4r + 7 when r = 10 b) 4r2 – 2r when r = 4 c) 3a + 2b when a = 3, b = 5 d) 2p2 when p = 8 e) 3 – 6r when r = 2.5 f) 6y + 3z2 when y = 5, z = 9 g) 5t2 + 3t – 2 when t = 6 h) 4(2u – 5) when u = 8 i) 1/x when x = 5 j) (3p + 2)(2p – 3) when p = 8 k) 6(p – 7)(p2 + 4) when p = 0 l) 3x + 7 when x = 5 m) 7g2 + 8g when g = 8 n) 5x2 + 3x when x = 9 3x – 4 6g2 – 8g 4x – 4 o) 2r + 5t + 1 when r = 7, t = 9 p) 2u(v2 – q2) when u = 3.5, v = 12, q = 9 3r – t q) rt + r2t2 + r3t3 when r = 3, t = 0.1 r) √(t2 + u2) when t = 12, u = 5 s) 4r + 6t when r = 3q, t = 5p, q = 6, p = 12 t) m2 – n2 when m = 4d, n = 3c, d = s2, c = t2, s = 3, t = 5 u) z when z = 3r, a = ½, r = 0.2 a2

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Q 5. When x = ⎯3, find the value of: a) 5x b) 6 – 5x c) ⎯12 – 5x d) 4x + 6 e) 6x + 13 f) 6x – 9 g) 6x – (⎯5) h) 17 – 3x i) ⎯17 – 3x j) 9x + 12 k) ⎯9x + 12 l) 0 – 4x m) 0.5x n) 8x +9 o) 100x – 25 6. When p = ⎯ 7 and q = ⎯4, find the value of: a) 2p + 5q b) p – q c) q – p d) 6q + 2p e) 5p – 3q f) p + q g) 2q – 3p h) p – ⎯q i) p + ⎯q j) 7p + 9q k) 0.5(2p + 7q) l) q + 2q + 3q m) 10q – 18p n) 45 – q + 2p o) 65 + 3p – 5q 7. When r = 4, s = ⎯6 and t = ⎯2, find the value of: a) 6s b) rs c) st d) rst e) 4s × ⎯6 f) 2st g) s2 h) t2 i) 3st2 j) s2 + t2 k) 4r2 l) 3s2 – 4t2 m) 2s(3t + 4r) n) 6r(t – 4s) o) 4ts(3s – 6t + 2r) p) s3 – t3 q) r2 + 4t2 r) 2rt(3s – 2t) s) 6s2 + 7t t) √(3rst) u) 9t2 – 5rs v) 7s – 12 t – 2r w) 5t(24 – 6s) x) 3s(t – 5r) y) r2 + s3 + t4 8. Solve the following equations: a) 6x + 10 = 22 b) t – 7 = 30 c) 3x + 8 = 35 d) 4r – 8 = 56 e) 8u + 5 = 41 f) 3r – 2 = 19 g) 6p + 9 = ⎯9 h) 7y + 6 = 41 i) 14s – 8 = 27 j) 12 – 5f = 2 k) 16 – 7y = ⎯5 l) 8h – 8 = 56 m) ⎯ 7h + 5 = 8.5 n) 5p + 4 = 10 o) 12e + 6 = 6 p) 25d – 60 = 85 q) ⎯6g – 9 = 18 r) 23 – 6y = 12 s) 9p – 5 = ⎯ 7 t) ⎯5f – 6 = 9 u) 12 + 6t = 18 v) ⎯23 – 6w = 1 w) 7u + 8 = ⎯16.5 x) 5r – 4 = 12

If you are going to be proficient in algebra, it is essential that you can handle negative numbers correctly and confidently, so here is some practice. Please keep trying question 5 to 7 until you can do all the calculations without any problems.


Q9. Solve the following equations: a) 5x + 3 = 2x + 9 b) 4a + 3 = ⎯4a + 19 c) 5y – 2 = ⎯y + 22 d) 3q – 9 = ⎯2q + 16 e) 11r + 5 = 3r + 53 f) ⎯6m + 2 = 2m – 54 g) 7x + 9 = 3x – 8 h) ⎯5x – 9 = 7x – 21 i) 4m + 2 = ⎯6m – 3 j) 2t + 7 = ⎯5t – 28 k) ⎯5p + 9 = 8p – 30 l) 8b + 2 = 2b + 8 10. Solve the following equations: a) 3(2x + 6) = 48 b) 8(5f – 2) = 56 c) 9(8u + 6) = 27 d) 4(r – 2) = ⎯12 e) 5(7g + 3) = 45 f) 9(3h – 6) = 36 g) 3(5k + 3) = ⎯12 h) 7(8d – 3) = 63 11. Solve the following equations: a) 2(4x – 4) + 10(2x – 3) = 6(4x – 5) b) (5y – 7) – 7(y + 1) = 2(3y + 1) c) 10(3b – 4) – 6(2b + 10) + 6(2b – 7) = ⎯22 d) 4(9t – 6) – 3(7t + 8) = 6(t – 5) e) 3(3x – 1) = 3(9x – 5) – 2(21x – 12) f) 8(s – 1) – 5(2s + 3) – (1 – 6s) = 2(2 + 3s) g) 2(18 – 4m) + 2(m + 1) = 8(m – 2) – 5(3m – 7) h) ⎯25 – 35(2p – 3) = 15(p + 1) – 15(5p – 2) – 30 12. Solve the following equations: a) d = 9 b) 5g + 6 = 23 c) 4r – 8 = ⎯6 d) 6h + 4 = 4h – 8 5 2 6 5 6 e) 2x + 5 = 6x – 9 f) 2x – 3 = 4 g) 2u + 1 = 1 h) 5w + 4 = 3 7 3 5 9 3 7 13. Solve the following equations: a) 6x + 3y = 27 b) 8a – 2b = 30 c) 4p + 2q = 26 d) 6r + 3t = 15 8x + 2y = 34 a – 2b = 2 25p – 10q = ⎯40 12r + 4t = 24 e) ⎯5m + 3n = ⎯31 f) 8b – 5c = ⎯3 g) 8q + 5r = 14 h) 8u + 6v = 41 5m – 6n = 13 2b – 5c = ⎯27 ⎯8q + 9r = 14 8u – 6v = ⎯1

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Q 14. Solve the following equations. These are a little harder!! : a) 2p – 5m = 4 b) 2t + 3u = 7 c) 7e + 8f = 60.5 d) 10v – 2w = 12.2 3p – 12m = 15 5t + 4u = 7 2e – 5f = ⎯15.5 3v + 9w = 45.9 e) 4t + 6u = ⎯5 f) 8j + 8k = ⎯36 g) 12q + 6r = ⎯12 h) 6m + 4n = 38 6t – 4u = 25 7j – 6k = 14 4q – 3r = 11 3m – 3n = ⎯28.5 15. Here are two interesting pairs of equations. Solve them and comment on the answers. a) 7p – 5q = 0 b) 8x + 5y = 23 11p + 3q = 0 16x + 10y = 46 16. Make the letter in the bracket the subject of the formula. For example, in part a) make w the subject. a) y = 3x + 2w (w) b) m = r2 (r) c) f = 4m + t2 (t) d) v = ut + ½ at2 (a) e) q = t + 3 (t) f) s = ag2 (g) g) H = r2 + dt2 (t) h) A = LW (W) 7 i) A = πr2 (r) j) A = πr2 (π) k) T = 2π (L) l) a + 6b = 8a – 5b (a) 5 3 17. Express the letter in brackets in brackets in terms of the other letters. a) y = 2b + 7 (b) b) a2 = b2 + c2 (c) c) 2mr = 7yf (f) d) h2 + g2 + k2 = 1 (h) 9 18. I am thinking of a number. If I multiply it by 7 and add 6 the answer is ⎯29. What was the number I first thought of? 19. Five calculators cost £m. What is the cost of n calculators? 20. Harry has twice as much money as Jane. Jane has three times as much money as Francis. The total amount for all three people is £60. How much did they have each? 21. Michelle has twice as much money as Fred. Fred has three times as much money as Jake. The total amount for all three people is £q. How much did they have each?

22. A hexagon has sides a, 3a, 3a, 4a, 6a and 7a metres respectively. The total perimeter of the hexagon is 120q metres. What is the length of each side in terms of q ? 23. n people paid £f each and m people paid £g each to go to a concert. (i) What was the total amount paid? (ii) What was the mean amount paid? 24. A woman is 32 years older than her daughter. Ten years ago the mother was three times as old as the daughter. How old are they both now?

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L g


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Q25. The interior angle of a regular polygon with n sides is given by 180 – degrees. (i) Use this formula to find the interior angle of a regular dodecahedron. (ii) How many sides does a regular polygon have if the interior angle is 140o ? (iii) How many sides does a regular polygon have if the interior angle is 179o ? 26. Two angles of an isosceles triangle are yo and (y – 30)o. What are the two possibilities for the third angle? 27. I am thinking of two numbers. 3 times the first number plus 2 times the second is thirty eight. 5 times the first plus 7 times the second is eighty nine. What are the two numbers? 28. I am thinking of two numbers. Ten times the first number subtract twice the second number is thirteen. Six times the first number add four times the second number is negative thirteen. What are the two numbers? 29. If 7 buns and 4 cream cakes cost £3.63 and 3 buns and 5 cream cakes cost £2.87, what is the cost of one bun and the cost one cream cake? 30. Peter and Jenny were looking at their bank statements and they could not help noticing that one was in credit and the other was overdrawn. Peter said, “If four people with my bank balance were combined with three people with your bank balance, the combined balance would be an overdraft of eighty six pounds!” Jenny said, “If six people with your bank balance were combined with seven people with my bank balance, the combined balance would be an overdraft of seventy four pounds!” Calculate which person is overdrawn with their account and how much their overdrawn balance is. 31. The recommended time to cook a turkey is so many minutes per kilogram plus a fixed time. If T is the total time in minutes, f is the fixed time in minutes, M is the mass of the turkey in Kg and g is the time per kilogram in minutes, write a formula connecting T, f M and g. A turkey with a mass of 4Kg takes 2 hours and 10 minutes to cook and a turkey with a mass of 8Kg takes 3 hours and 30 minutes. Find how long it would take to cook a turkey with a mass of 14 Kg. 32. Electricity from ElecGriddle is priced in the following way: (i) There is a standing charge per month and (ii) a cost per unit of electricity used. Mrs Ghandi used 2100 units and was charged £146.50. Mr Bouhours used 2900 units and was charged £194.50. Write a formula connecting the unit cost, the standing charge, the number of units used and the total cost per month. Find the unit cost and the standing charge. 33. A factory making cups and saucers wastes c pence every time a cup with a fault is made and s pence every time a saucer with a fault is made. On Monday 150 faulty cups and 180 faulty saucers were made and £38.70 was wasted.

360 n

On Tuesday 210 faulty cups 205 faulty saucers were made and £49.95 was wasted. How much was wasted on Wednesday if 180 faulty cups and 260 faulty saucers were made?

algebra-1

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