Algebra: Equations and formulae

Transcription

1 Algebra: Equations and formulae. Basic algebra HOMEWORK A FM AU Find the value of x + when a x = b x = 6 c x = Find the value of k when a k = b k = c k = 0 Find the value of + t when a t = b t = 8 c t = Evaluate f when a f = b f = 6 c f = 0 d 7 Evaluate when a d = b d = c d = Find the value of x + when a x = b x = c x =. Evaluate w when a w = b w = c w =. Evaluate 0 x when a x = b x = 6 c x =.6 Find the value of t when a t =. b t =.6 c t = 0.0 Evaluate t when a t =. b t =.8 c t = 0. Two of the first recorded units of measurement were the cubit and the palm. The cubit is the distance from the fingertip to the elbow and the palm is the distance across the hand. A cubit is four and a half palms. The actual length of a cubit varied throughout history, but it is now accepted to be cm. Noah s Ark is recorded as being 00 cubits long by 0 cubits wide by 0 cubits high. What are the dimensions of the Ark in metres? In this algebraic magic square, every row, column and diagonal should add up and simplify to a + 6b + c. D a b + c a + 8b + c a + b c a + b + c a b + 7c a + b + c a + 7b c a Copy and complete the magic square. b Calculate the value of the magic number if a =, b = and c =. FM Functional Maths AU (AO) Assessing Understanding PS (AO) Problem Solving RECALL

2 CHAPTER : Algebra: Equations and formulae D AU FM 6 FM 7 AU 8 The rule for converting degrees Fahrenheit into degrees Celsius is: C = (F ) a b Use this rule to convert 68 F into degrees Celsius. Which of the following is the rule for converting degrees Celsius into degrees Fahrenheit? F = (C + ) F = C + F = C + F = C The formula for the cost of water used by a household each quarter is: per litre of water used. A family uses 0 litres of water each day. a How much is their total bill per quarter? (Take a quarter to be days.) b The family pay a direct debit of per month towards their electricity costs. By how much will they be in credit or debit after the quarter? Using x = 7., y = 8. and z = 0.6, work out the value of: y x + y x a x = b c + y z z z Expand these expressions. a ( + m) b 6( + p) c ( y) d (6 + 7k) e ( f ) f ( w) g 7(g + h) h (k + m) i 6(d n) j y(y ) k g(g + ) l h(h ) m y(y + k) n 6m (m p) o h(h + ) p h(h ) q d(d d ) r w(w + t) s a(a b) t h (h + g) An approximate rule for converting degrees Fahrenheit into degrees Celsius is: C = 0.(F 0) a Use this rule to convert F into degrees Celsius. b Which of the following is an approximate rule for converting degrees Celsius into degrees Fahrenheit? F = (C + 0) F = 0.(C + 0) F = (C + ) F = (C ) Copy the diagram below and draw lines to show which algebraic expressions are equivalent. One line has been drawn for you. y y + y y + 6 y y 0 (y ) y y y y + (y + ) 0 RECALL

3 CHAPTER : Algebra: Equations and formulae 0 PS FM AU PS Expand and simplify. a ( + t) + ( + t) b 6( + k) + ( + k) c ( + m) + ( + m) d ( + y) + ( + y) e ( + f) + (6 f ) f 7( + g) + ( g) g ( + h) ( + h) h (g + ) (g + ) i (y + ) (y + ) j (t + ) (t + ) k (k + ) (k ) l (e + ) (e ) m m( + p) + p( + m) n k( + h) + h( + k) o t( + n) + n( + t) p p(q + ) + q(p + ) q h( + j) + j(h + ) r y(t + ) + t( + y) s t(t + ) + t( + t) t y( + y) + y(6y ) u w(w + ) + w( w) v p(p + ) p( p) w m(m ) + m( m) x d( d) + d(d ) y a(b + a) + a(a + c) z y(w + y ) + y(y t) a Laser printer cartridges cost 7 and print approximately 00 pages. Approximately how many pence per page does it cost to run, taking only ink consumption into consideration? b A printing specialist uses a laser printer of this type. He charges a fixed rate of.0 to set up the design and five pence for every page. Explain why his profit on a print run of x pages is, in pounds, x c How much profit will the printing specialist make if he prints 000 race entry forms for a running club? The expansion (x + 8y) = x + y. Write down two other expansions that give an answer of x + y. Adult tickets for a concert cost x and children s tickets cost y. At the afternoon show there were 0 adults and 60 children. At the evening show there were 60 adults and 0 children. a Write down an expression for the total amount of money taken on that day in terms of x and y. b The daily expense for putting on the show is 00. If x = and y =, how much profit did the theatre make that day? Don wrote the following: (x ) + (x + ) = x + 0x + = x Don has made two mistakes in his working. Explain the mistakes that Don has made. An internet site sells CDs. They cost (x + 0.7) each for the first five and then (x + 0.) for any orders over five. a Moe buys eight CDs. Which of the following expressions represents how much Moe will pay? i 8(x + 0.7) ii (x + 0.7) + (x + 0.) iii (x + 0.7) + (x + 0.) iv 8(x + 0.) b If x =, how much will Moe pay? C RECALL

4 CHAPTER : Algebra: Equations and formulae. Solving linear equations HOMEWORK B D FM AU PS 6 Solve the following equations. g m f h a + = 8 b = c + = d = 6 8 h t d x e + = 7 f = 6 g + = 8 h = 8 x + t + w y i = j = k = l = Solve the following equations. Give your answers as fractions or decimals as appropriate. a (x + 6) = b 6(x ) = 0 c (t + ) = 0 d (x + ) = e (y 7) = f (x + ) = 88 g (t + ) = 8 h (t + ) = i (6x + ) = 8 j (y ) = 0 k (k + ) = l (x + 8) = 0 m (y 7) = n (t ) = 7 o 8(x 7) = 6 p 8(x ) = 6 q (x + 7) = 8 r (x ) = s (t + ) = t (x ) = 8 u (t + ) = 0 v (x ) = 6 w (6y 8) = 8 x (x + 7) = Solve each of the following equations. a x + = x + 6 b y + = y + c a = a + d 6t + = t + e 8p = p + f k + = k + A rectangular room is four metres longer than it is wide. The perimeter is 8 metres. It cost to carpet the room. How much is the carpet per square metre? Mike has been asked to solve the equation a(bx + c) = 60 Mike knows that the values of a, b and c are, and, but he doesn t know which is which. He also knows that the answer is an even number. What are the correct values of a, b and c? As the class are coming in for the start of a mathematics lesson, the teacher is writing some equations on the board. So far she has written: (x + ) = (x + ) = Zak says, That s easy both equations have the same solution, x =. Is Zak correct? If not, then what mistake has he made? What are the correct answers? C 7 Solve each of the following equations. a (d + ) = d + b (x ) = (x + ) c (y + ) = (y ) d (b ) + = (b +) e (c + ) 7 = (c + ) c AU 8 The solution to the equation x + = is x = 8. ax Make up two more different equations of the form ± c = d, where a, b, c and d are b positive whole numbers, for which the answer is also 8. RECALL

5 CHAPTER : Algebra: Equations and formulae FM 0 Solve the following equations. x t m + 8p 6 a = b = c = d = x t + x 8 x e = f = g = h = A party of eight friends went for a meal in a restaurant. The bill was x. They received a per person reduction on the bill. They split the bill between them. Each person paid.. a Set this problem up as an equation. b Solve the equation to work out the bill before the reduction. C. Setting up equations HOMEWORK C PS PS Set up an equation to represent each situation described below. Then solve the equation. Do not forget to check each answer. A girl is Y years old. Her father is years older than she is. The sum of their ages is 7. How old is the girl? A boy is X years old. His sister is twice as old as he is. The sum of their ages is. How old is the boy? D PS The diagram shows a rectangle. Find x if the perimeter is cm. cm (x ) cm PS Find the length of each side of the pentagon, if it has a perimeter of cm. x x x x x PS PS 6 PS 7 FM 8 FM On a bookshelf there are b crime novels, b science fiction novels and b + 7 romance novels. Find how many of each type of book there is, if there are 6 books altogether. Maureen thought of a number. She multiplied it by and then added 6 to get an answer of 6. What number did she start with? Declan also thought of a number. He took from the number and then multiplied by to get an answer of. What number did he start with? Books cost twice as much as magazines. Kerry buys the same number of books and magazines and pays.0. Derek buys one book and two magazines and pays 6. How many magazines did Kerry buy? Sandeep s money box contains 0p coins, coins and coins. In the box there are twice as many coins as 0p coins and more coins than 0p coins. There are coins in the box. a Find how many of each coin there are in the box. b How much money does Sandeep have in her money box? RECALL C

6 CHAPTER : Algebra: Equations and formulae. Trial and improvement HOMEWORK D C FM Find two consecutive whole numbers between which the solution to each of the following equations lies. a x + x = 7 b x + x = c x + x = 0 d x + x = 8 Find a solution to each of the following equations to decimal place. a x x = 0 b x x = c x x = 0 d x x = Show that x + x = has a solution between x = and x =, and find the solution to decimal place. Show that x x = has a solution between x = and x =, and find the solution to decimal place. A rectangle has an area of 00 cm. Its length is 8 cm longer than its width. Find the dimensions of the rectangle, correct to decimal place. A gardener wants his rectangular lawn to be m longer than the width, and the area of the lawn to be 800 m. What are the dimensions he should make his lawn? (Give your solution to decimal place.) A triangle has a vertical height cm longer than its base length. Its area is 0 cm. What are the dimensions of the triangle? (Give your solution to decimal place.) A rectangular picture has a height cm shorter than its length. Its area is 0 cm. What are the dimensions of the picture? (Give your solution to decimal place.) This cuboid has a volume of 000 cm. x x a b x Write down an expression for the volume. Use trial and improvement to find the value of x to decimal place. AU 0 Darius is using trial and improvement to find a solution to the equation x x =. The table shows his first trial. x x x Comment 8 Too low Continue the table to find a solution to the equation. Give your answer to decimal place. RECALL

7 CHAPTER : Algebra: Equations and formulae PS Two numbers a and b are such that ab = 0 and a b =. Use trial and improvement to find the two numbers to one decimal place. You can use a table like the one below. The first two lines have been done for you. a b = (0 a) a b Comment Too low 0 8 Too high C. Solving simultaneous equations HOMEWORK E 7 Solve the following simultaneous equations. x + y = 7 x + y = x y = x y = x y = 7 x + y = 6 x y = x + y = 6 x + y = 7 x + y = x + y = 7x y = x + y = 8 x y = 6 x y = 7 x y = B.6 Solving problems with simultaneous equations HOMEWORK F PS PS FM FM Read each situation carefully, then make a pair of simultaneous equations in order to solve the problem. A book and a CD cost.00 together. The CD costs 7 more than the book. How much does each cost? Ten second-class and six first-class stamps cost.6. Eight second-class and 0 first-class stamps cost.8. How much do I pay for three second-class and four first-class stamps? At the shop, Henri pays.7 for six cans of cola and five chocolate bars. On his next visit to the shop he pays for three cans of cola and two chocolate bars. A few days later, he wants to buy two cans of cola and a chocolate bar. How much will they cost him? In her storeroom, Chef Mischa has bags of sugar and rice. The bags are not individually marked, but three bags of sugar and four bags of rice weigh kg. Five bags of sugar and two bags of rice weigh kg. Help Chef Mischa to work out the weight of two bags of sugar and five bags of rice. Ina wants to buy some snacks for her friends. She works out from the labelling that two cakes and three bags of peanuts contain 6 g of fat; one cake and four bags of peanuts contain 6 g of fat. Help her to work out how many grams of fat there are in each item. B RECALL

8 CHAPTER : Algebra: Equations and formulae B A PS 6 FM 7 FM 8 FM FM 0 The difference between my son s age and my age is 8 years. Five years ago my age was double that of my son. Let my age now be x and my son s age now be y. a Explain why x = (y ). b Find the values of x and y. In a record shop, three CDs and five DVDs cost In the same shop, three CDs and three DVDs cost.0. a Using c to represent the cost of a CD and d to represent the cost of a DVD set up the above information as a pair of simultaneous equations. b Solve the equations. c Work out the cost of four CDs and six DVDs. Four apples and two oranges cost.0. Five apples and one orange costs.7. Baz buys four apples and eight oranges. How much change will he get from a 0 note? Wath School buys basic scientific calculators and graphical calculators to sell to students. An order for 0 basic scientific calculators and graphical calculators came to a total of 0. Another order for basic scientific calculators and 0 graphical calculators came to a total of.. Using x to represent the cost of basic scientific calculators and y to represent the cost of graphical calculators, set up and solve a pair of simultaneous equations to find the cost of the next order, for basic scientific calculators and graphical calculators. Five bags of compost and four bags of pebbles weigh 0 kg. Three bags of compost and five bags of pebbles weigh kg. Carol wants six bags of compost and eight bags of pebbles. Her trailer has a safe working load of 00 kg. Can Carol carry all the bags safely on her trailer?.7 Solving quadratic equations HOMEWORK G B 7 Solve the following quadratic equations. x + 6x 7 = 0 x 8x 0 = 0 x 7x + = 0 x = 0 8x = 0 6 x + x = 0 x 6x + = 0 8 x + 8x = 0 x x + = 0 0 x + x = 0 6 RECALL

9 CHAPTER : Algebra: Equations and formulae.8 The quadratic formula HOMEWORK H PS AU PS Solve the following equations using the quadratic formula. Give your answers to decimal places. a x + x = 0 b x + x + = 0 c x x 7 = 0 d x + x = 0 e x + 7x + = 0 f x + x + = 0 g x + x + = 0 h x + x = 0 i x + x 6 = 0 Solve the equation x = x + 7, giving your answers correct to significant figures. A rectangular lawn is m longer than it is wide. The area of the lawn is 60 m. How long is the lawn? Give your answer to the nearest cm. Gerard is solving a quadratic equation using the formula method. He correctly substitutes values for a, b and c to get x = What is the equation that Gerard is trying to solve? Eric uses the quadratic formula to solve: x x + = 0 June uses factorisation to solve: x x + = 0 They both find something unusual in their solutions. Explain what this is, and why. ± 6 A. Solving problems with quadratic equations HOMEWORK I Work out the discriminant b ac of the following equations. In each case say how many solutions the equation has. a x + 6x + = 0 b x + x = 0 c x + x + = 0 d 8x + x = 0 e x + x + = 0 f x + x + = 0 A * PS Bill works out the discriminant of the quadratic equation x + bx c = 0 as: b ac = There are six possible equations that could lead to this discriminant, where a, b and c are integers. What are they? HOMEWORK J PS PS The sides of a right-angled triangle are x, (x + ) and (x ). Find the actual dimensions of the triangle. The length of a rectangle is m more than its width. Its area is 0 m. Find the actual dimensions of the rectangle. Solve the equation: x + = x Give your answers correct to decimal places. A * RECALL 7

10 CHAPTER : Algebra: Equations and formulae A * PS FM 6 FM 7 8 FM Solve the equation: x + = 7 x The area of a triangle is cm. The base is 8 cm longer than the height. Use this information to set up a quadratic equation. Solve the equation to find the length of the base. On a journey of 0 km, the driver of a train calculates that if he were to increase his average speed by 0 km/h, he would take 0 minutes less. Find his average speed. After a p per kilogram increase in the price of bananas, I can buy kilograms less for 6 than I could last week. How much do bananas cost this week? Gareth took part in a 6-mile race. a He ran the first miles at an average speed of x mph. He ran the last miles at an average speed of (x ) mph. Write down an expression, in terms of x, for the time he took to complete the 6-mile race. b Gareth took four hours to complete the race. Using your answer to part a, form an equation in terms of x. c i Simplify your equation and show that it can be written as: x 7x + = 0 ii Solve the equation and obtain Gareth s average speed over the first miles. Ana, an interior decorator, is told that a rectangular room is m longer than it is wide. She is also told that it cost to carpet the room. The cost of the carpet was per square metre. Help her to work out the width of the room. Functional Maths Activity Picture framing Sam has a photograph that measures 8 cm by 0 cm. She puts it into a frame that is the same width all the way round. She notices that the area of the rectangle formed by the frame is twice the area of the original photograph. How wide is the frame? Explain how you found the answer. 8 RECALL

11 Answers: New GCSE Maths AQA Modular Homework Book Higher Algebra: Equations and formulae. Basic algebra HOMEWORK A a b 7 c 7 a b c a b c a b c 6 a 0. b 6. c 6. 6 a 8 b c 0. 7 a b c 7 8 a b 6 c. a b c a. b. c m by 7 m by 6. m a a + 6b c and a b + c b 8 a 0 C b F = C + a. b 0. in credit a 6. b 76 c 7. 6 a + m b 8 + 6p c 6 y d 8 + k e 0f f 8 6w g 7g + 7h h 8k + 6m i d 6n j y y k 8g + 6g l 8h h m y + ky n 8m 6m p o h + h p h 0h q 8d d r w + wt s a ab t h + gh 7 a C b F = (C + ) 8 y + y = y, y + 6 = (y + ), y 0 = (y ) a 8 + 7t b + k c + m d 7 + y e 8 + f f 0 + g g + h h g + i 6y + j 7t k 7k + 6 l 6e + 0 m m + p + mp n k + h + hk o t + n + 7nt p p + q + 8pq q 6h + j + hj r y + t +0ty s t + t t y + 7y u w + w v 7p + 6p w m + 8m x d d y a + 0a + ab + ac z y + y + yw ty 0 a p b Basic charge of.0 plus p profit ( ) per page c.0 Correct answers such as: (6x + y), (x + y), 6(x + y) a 00x + 00y b 700 HarperCollins Publishers Ltd

12 Answers: New GCSE Maths AQA Modular Homework Book Higher He has worked out as instead of 6 And he has worked out + as, not + Answer should be 6x + a ii, (x + 0.7) + (x + 0.) b.0. Solving linear equations HOMEWORK B a 8 b 8 c d 6 e 6 f g. h i j k 8 l a b 0 c d. e f g h i 0. j k l m 7 n 7 o. p q r s 0 t u 0. v w x a x = b y = c a = d t = e p = f k = Length is m; width is m; area is m. Carpet costs.0 per square metre. a =, b =, c = 6 Zak is wrong, as he has not multiplied the bracket correctly to get 0x + = in both cases. First equation x = 0., second equation x = a d = 7 b x = c y = 6 d b = e c = 8 Any valid equations a 8 b c 6 d e f g. h. x 6 0 a =. 8 b 06. Setting up equations HOMEWORK C Y + = 7, years X =, 8 years (x + 7) =, x = x + =, x = 6 6b + = 6, b = 0, 0 crime novels, 8 science fiction and 7 romance 6 x + 6 = 6, so x = 0, x = 7 (x ) =, so x = 8, x = HarperCollins Publishers Ltd

13 Answers: New GCSE Maths AQA Modular Homework Book Higher 8 If a magazine costs m pence, then a book costs m pence. Then, Derek will have spent m on a book and m on magazines, so m = 600, m = 0p or.0. Assuming Kerry bought x books and x magazines, he will have spent x (m + m) which is mx or 0x pence. But he paid.0, which is 0p, so 0x = 0 and x =. Kerry bought magazines. a Suppose there are x 0p coins. Then, totalling the numbers of coins, x [ coins] + x [0p coins] + (x + ) ( coins] = x + coins. Now x + =, so x = 0. Therefore, there are 0 coins, 0 0p coins and coins. b (0 ) + (0 0p) + ( ) = =. Trial and improvement HOMEWORK D a and b and c and d and a. b.6 c. d and 8.7 cm 6.8 and 6.8 m 7. and 7. cm 8.6 and.6 cm a x + x = 000 b. cm and.6. Solving simultaneous equations HOMEWORK E x =, y = x = 6, y = x =, y = x =, y = x =, y = 6 x =, y = 7 x =, y = 8 x =, y =.6 Solving problems with simultaneous equations HOMEWORK F CD 0.0, book kg HarperCollins Publishers Ltd

14 Answers: New GCSE Maths AQA Modular Homework Book Higher g in cakes and g in peanuts 6 a My age minus equals x (my son s age minus ) b x = 6 and y = 7 a c + d = 770, c + d = 0 b c = 7.0, d = c a + n = 0, a + n = 7 gives 0a + n =, 6a = 8, p =, n = 6. Total cost for Baz is.0, so he will get.60 change c + p = 0 kg, c + p = kg, c = kg, p = kg The bags of compost weigh kg, so Carol cannot carry the bags safely on her trailer..7 Solving quadratic equations HOMEWORK G x =, x = 7 x =, x = 7 x =, x = x =, x = or x = ± x = ± 6 x = 0, x = 7 x = 8 x =, x = x =, x = 0 x =, x = 6.8 The quadratic formula HOMEWORK H a x =., x =.7 b x = 0., x =.7 c x =., x =. d x = 0., x = 0.77 e x = 0.7, x =.77 f x = 0.0, x =. g x = 0., x =.8 h x =.6, x =.6 i x =.6, x =.6 6.,. x + x 60 = 0, x =.6 and 0.6, so lawn is m 6 cm long. x x 8 = 0 ± 0 Eric gets x = 8 and June gets (x ) = 0, which only gives one solution: x =. The quadratic curve only touches the x-axis once, so it is a tangent to the curve. HarperCollins Publishers Ltd 6

15 Answers: New GCSE Maths AQA Modular Homework Book Higher. Solving problems with quadratic equations HOMEWORK I a 0, b, c, 0 d 7, e, 0 f 0, From the given equation, a = x ± x = 0; x ± x 6 = 0; x ± x 8 = 0 HOMEWORK J,, 0 m and m.6 and 0. and cm 6 60 km/h 7 per kilogram 8 a + x x b + = x x c i Multiply through by x(x ) and simplify ii 7. mph = 6., x + x 6. = 0, x + 6x 6 = 0, (x + )(x ) = 0, x =. metres Functional Maths Activity Picture framing.8 cm to dp or.8 cm to dp Possible explanation: If the width of the frame is x cm, then the area is (x+8)(x+0) cm. As the photograph is 80 cm, we must have (x+8)(x+0) = 60 The area of the frame is therefore 60 cm, (i.e. twice the area). The border between the edge of the frame and the photograph is the same width all the way around, so let its width be x cm. Then the width of the frame is (8 + x) cm and the height of the frame is (0 + x) cm. Then the area of the frame is (8 + x)(0 + x) = x + x. Using all the information given, and rearranging, gives: x + 6x + 80 = 60 x + 6x 80 = 0 x + x 0 = 0 Using the quadratic formula, x =.8 cm or x = 0.8 cm. Ignore the negative value, so the width of the frame is =.68 cm HarperCollins Publishers Ltd 7