GCSE Homework Unit 2 Foundation Tier Exercise Pack New AQA Syllabus

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1 GCSE Homework Unit 2 Foundation Tier Exercise Pack New AQA Syllabus

2 The more negative a number, the smaller it is. The order of operations is Brackets, Indices, Division, Multiplication, Addition and Subtraction. When rounding, find the appropriate column if the column to 1 Number the right is less than 5, this column stays the same. If it is 5 or more, this column increases by 1. A multiple of a number is in the number s times table. The lowest common multiple of two numbers is the lowest number that is a multiple of both numbers. A factor of a number goes into that number. The highest common factor of two numbers is the highest number that is a factor of both numbers. 1. Write these sets of numbers in order from smallest to largest: 2. Using your preferred method, complete these multiplication and division problems: a) 47 x 6 b) 78 x 4 c) 73 x 5 d) 96 x 6 e) 583 x 4 f) 389 x 7 g) 639 x 8 h) 804 x 67 i) 487 x 62 j) 942 x 58 k) 81 3 l) 72 4 m) n) o) p) Using your preferred method, complete these multiplication and division problems: a) 1.5 x 6 b) 7.6 x 3 c) 3.7 x 8 d) 7.82 x 3 e) 9.5 x 7.2 f) 6.08 x 3.4 g) 13.6 x 2.4 h) x 5.42 i) j) k) l) Use the correct order of operations to calculate the following:

3 5. Solve the following word problems: a) Bottles of lemonade are 64p each. John needs 18 for a party. How much does he spend? b) A roadside florist sells only tulips. They cost 32p each. He takes 8.96 at the end of the day. How many tulips has he sold? c) A train carriage can hold 118 people. If a train has 14 carriages, how many people can it transport? d) Drawing pins comes in boxes of 48. The mathematics department orders 17 boxes. How many drawing pins have they ordered? e) The new intake in a large school is 468. They are to be put in classes of 26 pupils each. i) How many classes are there to be? ii) Each pupil from the new intake pays 54p towards the School Fund. How much does the new intake contribute to the School Fund in total? 6. Round these numbers to nearest integer: a) b) c) d) Round these numbers to the nearest 10: a) 48 b) 621 c) 23.5 d) Round these numbers to the nearest 100: a) 1866 b) 219 c) 308 d) Round these numbers to the nearest 1000: a) 1452 b) c) d) Round these numbers to one decimal place: a) b) c) d) Round these numbers to two decimal places: a) b) c) d) Round these numbers to one significant figure: a) 15,672 b) c) d) Write the first ten multiples of: a) 4 b) 7 c) 6 d) Write the lowest common multiple of: e) 4 and 7 f) 3 and 7 g) 4 and 6 h) 3 and Write all of the factors of: a) 28 b) 32 c) 42 d) Write the highest common factor of: a) 28 and 32 b) 28 and 42 c) 16 and 42 d) 16 and 28

4 2 Algebra We simplify expressions by collecting like terms. To expand brackets, multiply everything inside the brackets by what it outside. When factorising expressions, find a factor of all terms, and place this outside the brackets divide by this to leave terms within the brackets. 1. Simplify these expressions: a) a + a + a + a b) a + b + 2a + 3b - 4 c) 4d 3e 2d + e + 5 d) 4gh + 3g 2hg + 2h e) 5m² + 6m - m² - 4n f) 3pq + 3q qp + 2qp + 4q Expand (Multiply out) these brackets: a) 2(b + 7) b) 5(2d 3) c) -3(2f + 8) d) k(k + 4) e) 3m(2n 1) f) 2m(2m + 3n r) 3. Factorise each of these expressions: a) 2n + 6 b) 10g + 25h c) 35k 14 d) n² + 4n e) 6p² - 15p f) 2p³ - 6p² + 10p 4. Expand and simplify each of these brackets: a) 2(3p 5) + 3(p + 3) b) 3(4t + 3) 5(2t 1) c) 2x(2x + 1) + 3(5x 4) d) 4(3z 1) 5(z + 2)

5 3 Sequences The term-to-term rule of a sequence describes what happens between the terms in a sequence. The n th term of a sequence describes the sequence using each position. 1. Find the first five terms of the sequences described below: a) First term: 5. Term-to-term rule: add 7. c) First term: 3. Term-to-term rule: multiply by 2. e) First term: 10. Term-to-term rule: divide by 2. b) First term: 35. Term-to-term rule: subtract 8. d) First term: 2. Term-to-term rule: multiply by 2, subtract 1. f) First term: 0. Second term: 1. Term-to-term rule: add together the previous two terms. 2. Write the first term, and the term-to-term rule of each of these sequences: a) 4, 7, 10, 13, 16, b) 5, 11, 17, 23, 29, c) 19, 14, 9, 4, -1, d) 7, 21, 63, 189, 567, e) 1, 4, 9, 16, 25, (What do we call these numbers?) f) 1, 3, 6, 10, 15, (What do we call these numbers?) 3. Write the first five terms, and the extra term, for each of these sequences: a) 4n 2 b) 2n + 3 Write the 17 th term. Write the 60 th term. c) 7n - 5 Write the 23 rd term. d) 50 4n Write the 10 th term. e) n² - 1 Write the 10 th term. f) n³ + 5 Write the 8 th term. Write the n th term of these sequences: a) 1, 6, 11, 16, 21, b) 4, 7, 10, 13, 16, c) 7, 13, 19, 25, 31, d) 35, 31, 27, 23, 19, e) 1, 4, 9, 16, 25, f) 49, 46, 41, 34, 25,

6 4 Fractions 1. Complete these equivalent fractions: The top part of a fraction is called the numerator. The bottom part of a fraction is called the denominator. To simplify a fraction, divide numerator and denominator by the same number. To order, add or subtract fractions, you must rewrite the fractions with a common denominator. 2. Write these fractions in their simplest terms: 3. Write these mixed numbers as improper fractions: 4. Write these improper fractions as mixed numbers: 5. Put these fractions in order from smallest to largest, by using decimals or a common denominator. a) 2, 1, 5, 7, 1. b) 7, 2, 2, 4, c) 7, 2, 27, 16, d) 2, 5, 7, 5,

7 6. Complete these addition and subtraction problems by writing the fractions with a common denominator: 7. Calculate these fractions of quantities: 8. Write the first quantity as a fraction of the second in its simplest form. a) Write 15 as a fraction of 60. b) Write 12 as a fraction of 50. c) Write 21 as a fraction of 77. d) Write 2.50 as a fraction of 15. e) Write 150g as a fraction of 1.5kg. f) Write 340m as a fraction of 1km.

8 5 Decimals To order decimals, make each decimal the same length by writing 0s, or compare the decimals using each column. A recurring decimal is a decimal where values in the decimal places repeat. 1. Write these sets of decimals in order from smallest to largest: a) 0.4, 0.37, 0.387, 0.42, b) 1.763, 1.75, 1.735, 2.7, Write these fractions as recurring decimals: a) 1. 3 b) 5. 6 c) 1. 7 d) 3. 7 e) 4. 9 f) Complete the table to show equivalent fractions, decimals and percentages. Fraction (Simplest form) Decimal Percentage % 17.5% 42% 72.5%

9 6 Coordinates and Graphs 1. Plot these coordinates on this pair of axes: a) (1, 2) b) (3, 1) c) (3, -2) d) (-4, 2) e) (-4, -3) Coordinates are in the form (x, y), so we plot horizontally, and then vertically. When drawing a line from an equation, draw a table of values and calculate the coordinates plot the points and join with a line. To find the midpoint of two points (x 1, y 1 ) and (x 2, y 2 ), use the formula m = (½(x 1 + x 2 ), ½(y 1 + y 2 )). Coordinates in 3D come in the form (x, y, z). Be careful with the axes and the way that they are labelled. 2. On the same pair of axes, draw the graphs of: a) x = 3 b) x = -4 c) y = 2 d) y = -1

10 3. On the same pair of axes, draw the graphs of: a) y = 2x - 1 b) y = ½x + 3 c) y = x² (Find y-values for each value of x in this case) 4. Write the equation of each of these lines:

11 5. Water is poured at a constant rate into each of the following beakers: These four graphs show the depth of water against time. Which graph is for which beaker?

12 6. Draw a pair of axes with the x axis going from 0 to 10kg (letting 1 square = 1kg), and the y-axis from 0 to 18lbs (letting 1 square = 1lb). Use the face that 5kg 11lbs to draw a conversion graph. Use your graph to convert: a) 4kg to lbs. b) 6kg to lbs. c) 13lbs to kg. d) 5lbs to kg. e) 40kg to lbs. f) 1300lbs to kg. 7. For each of these distance-time graphs find: a) The length of time spent stationary. b) The speed in the first moving part of the journey. c) The speed in the second moving part of the journey. d) The total distance travelled. e) The total time taken. f) The average speed for the whole journey. 8. Draw these distance-time graphs:

13 7 Percentages Percentages are always out of 100. We can write a percentage as a fraction with 100 as a denominator. To turn a percentage into a decimal, divide by 100. To find 10% of a number, divide that number by 10. To find 1% of a number, divide that number by 100. To apply a percentage increase, find the percentage and add it on. To apply a percentage decrease, find the percentage and subtract it. 1. Use percentages to compare these proportions. a) From a survey, 17 out of 25 people said they liked football. 13 out of 20 people said they liked Rugby. Which was the most liked sport: Football or Rugby? b) 7 out of 10 people said they enjoyed their time at Flamingo Land. 37 out of 50 said that they enjoyed their time at Alton Towers. Which was the most liked theme park? 2. Find these percentages of amounts: a) 25% of 160 b) 10% of 70 c) 30% of 90 d) 15% of 30 e) 10% of 26g f) 5% of 74cm g) 13% of 160 h) 17.5% of i) James spends on his mobile phone bill every month. 70% of this is his line rental, 10% of this is his calls, and the rest is on text messages. How much of his bill is spent on text messages? j) It is recommended that women eat 300g of carbohydrates each day. Jane says that, for her, approximately 42% of this comes from bread. Approximately, how much bread does she eat per day? 3. Solve these percentage increase and decrease problems: a) Increase 30g by 10%. b) Increase 85cm by 20%. c) Decrease 50p by 30%. d) Decrease 150ml by 12%. e) The price of a TV is 350. In a sale, the price is reduced by 25%. How much is the sale price? f) A shop buys stock in at 2 per item. They want to make 47% profit on these items. How much do they sell it for? g) Mr and Mrs Taylor get loan of 15,000 over 3 years. It must be repaid with interest of 5% per annum. What is the total amount repayable? h) Amazon.co.uk is having a 20% off sale. You also have a 10% discount card. An ipod you want is 150. How much does you pay? 4. Write these amounts as percentages. a) 26g out of 200g. b) 7ml out of 40ml. c) 85cm out of 2.5m. d) 275g out of 1kg.

14 5. Work out the percentage increases or decreases in each case. a) After a storm, the amount of water in a pond b) A hi-fi system usually costs 400. In a sale, the increases from 200m³ to 280m³. price is reduced to 260. What is the percentage increase? What is the percentage decrease? c) A box of washing up powder usually contains 2kg. In a promotion, the box contains 2.3kg. What is the percentage increase? d) One week, the attendance at a football match is 30,000. The following week, the attendance is 27,000. What is the percentage decrease?

15 8 Indices To square a number, we multiply it by itself. Each square root has both a positive and negative value. To cube a number, we multiply it by itself and by itself again. a n x a m = a n+m a n a m = a n-m (a n ) m = a nxm Prime numbers have exactly two factors. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 1. Find the value of these square and cube numbers: a) 4² b) 7² c) 13² d) (-3)² e) 1³ f) 5³ g) 3³ h) (-2)³ 2. Find the value of these square and cube roots: a) 36 b) 81 c) 4 d) 25 e) 3 64 f) g) 3 27 h) Simplify these expressions using index notation: a) b x b x b x b x b b) f x f x f c) m² x m³ d) k 7 x k 2 x k 4 e) h 6 h 4 f) p 8 p 3 g) (t 2 ) 4 h) (x 6 ) 3 i) a x b x a x a x b j) c 5 x c 4 k) g 12 g 7 l) (z 4 ) 5 4. Write these numbers as a product of its prime factors. a) 32 b) 24 c) 60 d) 56 e) 36 f) 28 g) 45 h) 49

16 9 Ratio and Proportion Ratio compares one part to other parts. We write it in the form a : b. Proportion compares one part to the total. We write proportions as fractions. 1. There are 30 balls in a bag. 10 are white, 7 are red, 8 are blue and 5 are yellow. Write the ratio of: a) white balls to red balls. b) blue balls to yellow balls. c) yellow balls to red balls. d) red balls to blue balls to yellow balls. e) white balls to yellow balls to blue balls. 2. Write these ratios in their simplest form: a) 4 : 6 b) 10 : 8 c) 6 : 15 d) 15 : 35 e) 14 : 35 f) 14 : 10 : 18 g) 2.5 : 1.5 h) 15 : 45 : Write these ratios in the form 1 : n. a) 5 : 20 b) 20 : 50 c) 10 : 15 d) 6 : Write these ratios in the form n : 1. a) 10 : 4 b) 5 : 2 c) 21 : 4 d) 14 : Share the following amounts in the given ratio: a) 300 in the ratio 2 : 3 b) 500 in the ratio 7 : 3 c) 120 in the ratio 1 : 3 d) 450ml in the ratio 4 : 5 e) 350g in the ratio 2 : 5 f) 40cm in the ratio 3 : 2 6. Find these proportions of the following amounts: a) 2 out of 5 people in a school are male. If there are 450 people in the school, how many in total are male? b) 4 out of 5 fans at a football match are home fans. If there are 7,000 people at the match, how many away fans are there?

17 7. This recipe will make pancakes for 4 people: 110g plain flour 2 eggs 250ml milk 50g butter How much of each ingredient will I need for: a) 2 people?... g plain flour...eggs... ml milk... g butter b) 6 people?... g plain flour...eggs... ml milk... g butter 8. This recipe will make sausage bolognese for 4 people: 6 sausages, skins removed 1 tsp fennel seeds 250g mushrooms, sliced 600g tomato pasta sauce 300g penne How much of each ingredient will I need for: a) 2 people?... sausages, skins removed... tsp fennel seeds... g mushrooms, sliced... g tomato pasta sauce... g penne b) 12 people?... sausages, skins removed... tsp fennel seeds... g mushrooms, sliced... g tomato pasta sauce... g penne 9. Jay has 4.80 in change. He only has 20 pence coins and 1 coins. He has twice as many 20 pence coins as 1 coins. 10. Yasin wants to share 126 between two friends in the ratio 9 : 5. What is the size of the largest share? How many 1 coins does he have?

18 10 Equations and Inequalities When solving equations, you are attempting to find the value of the missing variable use inverse operations to do so. When showing inequalities on a number line, the coloured in circle means it can be equal to the open circle means strictly less than or greater than. 1. Solve these equations: a) 8x = 24 b) p + 7 = 9 c) n 1 = 5 d) 7m + 4 = 39 e) 10t 3 = 47 f) 7y 11 = 24 g) 3c 1 = 14 h) ½p + 7 = 9 i) ½(p + 7) = 7 j) 3(2b 1) = 27 k) 3(3d 2) = 75 l) ¼(a + 3) = 10 m) 2v 3 = v + 1 n) 6q - 3 = 8q - 8 o) 2f 2 = f p) ½p 3 = 9 - p q) ½g 3 = ¼g + 2 r) 4(y + 3) = 2(y + 10) 2. Solve these inequalities, showing your solutions on number lines: a) 2x > 6 b) 3n 4 5 c) p 6 < -4 d) 8m 4 16 e) 2t 7 < -3 f) 5h 5 > 15 g) 11 4x 1 < 19 h) 9 ½y i) z + 4 < 3z + 2 z + 16 For parts g, h and i, write the possible integer values of each variable.

19 11 Formulae and Algebraic Argument 1. Write expressions for the following statements: a) Five times the value of x plus two. b) Six multiplied by the value of y, minus eleven. c) The sum of twice the number d and four times the value of d. d) The sum of w and t, multiplied by a) In the formula D = 3a + 2, find the value of D when a = 3. b) In the formula y = z, find the value of y when z = 36. c) In the formula S = 5x + 7y, find the value of S when x = 4 and y = 8. In the formula P = 2l + 2w, P is the subject. An expression is a mathematical statement written in symbols. e.g. 3x + 1. An equation is a statement showing that two expressions are equal. e.g. 2y 1 = 15 A formula is an equation showing a relationship between two or more variables. e.g. E = mc². A proof generalises a problem for all different cases. It is best to use algebra to write proofs. To disprove a proof, you must find one example that does not fit. 3. The formula for the perimeter of a rectangle of length l and width w is P = 2l + 2w. Find the perimeter of a rectangle: a) 5cm long and 9cm wide. b) 6 yards long and 2.5 yards wide. 4. Use the formula p = m² + 5n to find: a) p when m = 4 and n = -6. b) p when m = -7 and n = -8. c) p when m = -10 and n = The ferry fare to cross a lake is 7 for adults and 4 for children. a) Write down a formula to calculate the total ferry cost, F, for A adults and C children. b) Use your formula to find the total cost for: i) 4 adults and 3 children. ii) 2 adults and 5 children. 6. Make y the subject of the equation 4x + 7y = 8 7. The change in momentum, M, of a body of mass m when it moves from a speed u to a new speed v is given by the formula M = mv mu. Make v the subject. 8. Prove that the sum of four consecutive numbers is always an even number. 9. Prove that the sum of three consecutive even numbers is always a multiple of six. 10. Prove that the mean of three consecutive numbers is always equal to the middle number.