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1 PLC Papers Created For:

2 Compound Units 2 Grade 5 Objective: Convert standard compound units in numerical and algebraic contexts. Question 1. Ben goes on holiday to Hong Kong. In Hong Kong, Ben sees a camera costing HK$ In London, an identical camera costs 285. The exchange rate is 1 = HK$ Ben buys the camera in Hong Kong. How much cheaper is the camera in Hong Kong than in London? (Total 3 marks)

3 Question 2. There are gallons of fuel in a fuel tank. The fuel is pumped from the fuel tank into a plane at a rate of 1700 litres per minute. 1 gallon = 4.5 litres. How many minutes will it take to empty the fuel tank completely? Give your answer to the nearest minute.... minutes (Total 3 marks)

4 Question 3. Debbie drove from Junction 12 to Junction 13 on a motorway. The travel graph shows Debbie s journey. Ian also drove from Junction 12 to Junction 13 on the same motorway. He drove at an average speed of 66 km/hour. Who had the faster average speed, Debbie or Ian? You must explain your answer. (Total 4 marks) Total /10

5 Direct and inverse proportion 2 Grade 6 Objective: Write and interpret equations that describe direct and inverse proportion Question 1. A is directly proportional to B. A = 100 when B=25. Work out the value of A when B = (Total 2 marks) Question 2. J is directly proportional to M 3. J = 50 when M = 5. (a) Find J when M = 8.5 (b) Find M when J = (3)... (1) (Total 4 marks)

6 Question 3. Y is inversely proportional to X. When Y is 9, X is 4. (a) Find Y when X is (3) (b) Find X when Y is (1) (Total 4 marks) Total /10

7 Gradient and the Rate of Change 2 Grade 5 Objective: Interpret the gradient of a straight-line graph as a rate of change (including linking to graphs that show direct or inverse proportion). Question 1. Simon went for a cycle ride. He left home at 2 pm. The travel graph represents part of Simon s cycle ride. At 3 pm Simon stopped for a rest. (a) How many minutes did he rest?... minutes (1) (b) How far was Simon from home at 5 pm? At 5 pm Simon stopped for 30 minutes. Then he cycled home at a steady speed. It took him 1 hour 30 minutes to get home. (c) Complete the travel graph.... km (1) (3) (Total 5 marks)

8 Question 2. A coach travels from Dronston to Luscoe. The travel graph for this journey is shown below. (a) Work out the average speed of the coach, in kilometres per hour, for the first 10 minutes of the journey. The coach stops in Luscoe for 15 minutes. The coach then returns to Dronston at a constant speed of 42 km/h. (b) Show this information on the travel graph.... km/h (2) (3) (Total 5 marks) Total /10

9 Growth and decay 2 Grade 5 Objective: Set up, solve and interpret the answers in growth and decay problems, including compound interest Question 1 John invests 4,500 in a high interest account at 4.3% interest per annum (compound interest). a) What number is 4,500 multiplied by when invested for 7 years.. (1) b) How many years does it take before 6,000 is in the bank. c) How much money is in the bank after 14 years.. (3). (2) (Total 6 marks)

10 Question 2 Stavros buys a computer for 1,230. The computer depreciates at a rate of 13% per year. a) What number is 1,230 multiplied by to find the value of the car after 3 years.... b) If C is the value of the computer and t is the time in years after buying the computer, write a formula to link C and t (1). c) Stavros wants to know when the computer will have at least halved in value. How many years will this be. Show your working. (1)... (2) (Total 4 marks) Total /10

11 Interpret Proportion 2 Grade 5 Objective: Understand that x is inversely proportional to y is equivalent to x is proportional to 1/y, and interpret equations that describe direct and inverse proportion. Question 1. a is directly proportional to c. When c = 49, a = 35. (a) Find a formula for a in terms of c w is inversely proportional to a³. When a = 2, w = 16. (2) (b) Find a formula for w in terms of a (c) Find the value of w when a = 4. (3) (2) (Total 7 marks)

12 Question 2. (a) y = for values of x from 1 to 5 Which sketch graph is correct? Circle the correct letter. y A B C y y x 1 5 x 1 5 x (1) (b) y = When x = 4, y = Work out the value of y when x = 2 (2) (Total 3 marks) Total /10

13 Percentage change 2 Grade 4 Objective: Express percentages and percentage changes as a fraction or a decimal and interpret these multiplicatively. Question 1. (a) Work out 65 % of (1) (b) Work out 165 % of (2) (Total 3 marks) Question 2 Calculate the decrease as a fraction in the following: (a) CD was 15, now costs 10 (b) Coat was 65, now costs 52 (Total 2 marks)

14 Question 3. In a sale, the normal prices are reduced. A coat is reduced from 258 to What is the percentage decrease? (Total 2 marks) Question 4. Catherine takes the train to work. The cost of a weekly train ticket increases by 15% to 207 Work out the cost of her weekly ticket before the increase. (Total 3 marks) Total /10

15 Problems with compound units 2 Grade 5 Objective: Solve problems involving compound units such as speed, rates of pay, unit pricing, density and pressure. Question 1. A tortoise moves a distance of 120 m in 4 minutes 48 seconds. Work out its speed in m/s.... (Total 2 mark) Question 2. Brass has a density of 8.33 g/cm 3. The mass of a sample of brass is 2kg. Work out its volume. Question (Total 3 marks) A cardboard box exerts a force of 160 Newtons on a kitchen work top. The pressure on the worktop is 98 N/m 2. Calculate the area of the cardboard box that is in contact with the worktop. =...

16 (Total 2 marks) Question 4. Amelia works a basic 35-hour week. Her hourly rate of pay is She is paid time and a quarter for each hour she works on a Saturday and time and a half for each hour she works on a Sunday. How much is Amelia paid for a week when she works a basic 35-hour week plus 5 hours on Saturday and 2 hours on Sunday? (Total 3 marks) Total /10

17 Scale factors and similarity 2 Grade 5 Objective: Use scale factors as a link to similarity. Question 1 (a) Work out the size of x x =.(2) (b) Work out the size of y y =.(2) (total 4 marks)

18 Question 2 AB =.. PS =.. (total 3 marks)

19 Question 3 VU =..cm (total 3 marks) Total marks / 10

20 Simple Interest and Financial Maths 2 Grade 5 Objective: Solve problems involving simple interest and financial maths. Question 1. Greg sells car insurance and home insurance. The table shows the cost of these insurances. Insurance car insurance home insurance Cost Each month Greg earns 530 basic pay 5% of the cost of all the car insurance he sells and 10% of the cost of all the home insurance he sells In May Greg sold 6 car insurances and 4 home insurances Work out the total amount of money Greg earned in May. (Total 5 marks)

21 Question 2. * Here are two schemes for investing 2500 for 2 years. Scheme A gives 4% simple interest each year. Scheme B gives 3.9% compound interest each year. Which scheme gives the most total interest over 2 years? You must show all your working. (Total 5 marks) Total /10

22 Solve Proportion Problems 2 Grade 5 Objective: Solve problems involving direct and inverse proportion. Question 1. The distance, D, travelled by a particle is directly proportional to the square of the time t, taken. When t = 40, D = 30 (a) Find a formula for D in terms of t. (b) Calculate the value of D when t = 64 D =... (3) (c) Calculate the value of t when D = 12 Give your answer correct to 3 significant figures... (1)... (2) (Total 6 marks)

23 Question 2. q is inversely proportional to the square of t. When t = 4, q = 8.5 (a) Find a formula for q in terms of t. (b) Calculate the value of q when t = 5 q =... (3)... (1) (Total 4 marks) Total /10

24 PLC Papers Created For:

25 Compound Units 2 Grade 5 Solutions Objective: Convert standard compound units in numerical and algebraic contexts. Question 1. Ben goes on holiday to Hong Kong. In Hong Kong, Ben sees a camera costing HK$ In London, an identical camera costs 285. The exchange rate is 1 = HK$ Ben buys the camera in Hong Kong. How much cheaper is the camera in Hong Kong than in London? (=258.5) (M1) '258.5' (M1 dep) (correctly stated with currency) (A1) OR (=3505.5) (M1) '3505.5' (=325.95) (M1) HK$ (correctly stated with currency) (A1) (Total 3 marks)

26 Question 2. There are gallons of fuel in a fuel tank. The fuel is pumped from the fuel tank into a plane at a rate of 1700 litres per minute. 1 gallon = 4.5 litres. How many minutes will it take to empty the fuel tank completely? Give your answer to the nearest minute = (M1) = 48.9 (M1 & A1) 49 minutes (Total 3 marks)

27 Question 3. Debbie drove from Junction 12 to Junction 13 on a motorway. The travel graph shows Debbie s journey. Ian also drove from Junction 12 to Junction 13 on the same motorway. He drove at an average speed of 66 km/hour. Who had the faster average speed, Debbie or Ian? You must explain your answer. 24 (mins) and 30 (km) or a pair of other values for Debbie (M1) = 0.4 (M1) = 75 (A1) Debbie is fastest from comparison of with 66 (kph) C1 (dep on M2) (Total 4 marks) Total /10

28 Direct and inverse proportion 2 Grade 6 Solutions Objective: Write and interpret equations that describe direct and inverse proportion Question 1. A is directly proportional to B. A = 100 when B=25. Work out the value of A when B = 8. A B so A = kb 100 = k x 25 k = 4 (M1) A = 4 x 8 = 32 (A1)... (Total 2 marks) Question 2. J is directly proportional to M 3. J = 50 when M = 5. (a) Find J when M = 8.5 J M 3 so J = km 3 (M1) 50 = k x 5 3 k = = 0.4 (M1) J = 0.4 x = (A1)... (3) (b) Find M when J = = 0.4M 3 M 3 = 375 M = 7.2 (A1)... (1) (Total 4 marks)

29 Question 3. Y is inversely proportional to X. When Y is 9, X is 4. (a) Find Y when X is 36. Y 1 so Y = (M1) 9 = 4 k = 18 (M1) Y = = 3 (A1)... (3) (b) Find X when Y is = 18 = 0.5 X = 0.25 (A1)... (1) (Total 4 marks) Total /10

30 Gradient and the Rate of Change 2 Grade 5 Solutions Objective: Interpret the gradient of a straight-line graph as a rate of change (including linking to graphs that show direct or inverse proportion). Question 1. Simon went for a cycle ride. He left home at 2 pm. The travel graph represents part of Simon s cycle ride. At 3 pm Simon stopped for a rest. (a) How many minutes did he rest? (b) How far was Simon from home at 5 pm? At 5 pm Simon stopped for 30 minutes. Then he cycled home at a steady speed. It took him 1 hour 30 minutes to get home. (b) Complete the travel graph. Line for 30 minutes stopped (M1) Straight Line (M1) from 7pm to 5.30 at 20km (A1 dep) 30 minutes (M1) 20km (M1) (3) (Total 5 marks)

31 Question 2. A coach travels from Dronston to Luscoe. The travel graph for this journey is shown below. (a) Work out the average speed of the coach, in kilometres per hour, for the first 10 minutes of the journey = (M1) = km/h (A1) 54km/h (2) The coach stops in Luscoe for 15 minutes. The coach then returns to Dronston at a constant speed of 42 km/h. (b) Show this information on the travel graph = 0.5 (M1) 30 min Correct line drawn on graph (B1) (3) (Total 5 marks) Total /10

32 Growth and decay 2 Grade 5 SOLUTIONS Objective: Set up, solve and interpret the answers in growth and decay problems, including compound interest Question 1 John invests 4,500 in a high interest account at 4.3% interest per annum (compound interest). a) What number is 4,500 multiplied by when invested for 7 years b) How many years does it take before 6,000 is in the bank x = 4, x = 4, x = 5, x = 5, x = 5, x = 5, M x = 6, A1. (1) 7 years B1. (3) c) How much money is in the bank after 14 years. 4,500 x M A1. (2) (Total 6 marks)

33 Question 2 Stavros buys a computer for 1,230. The computer depreciates at a rate of 13% per year. a) What number is 1,230 multiplied by to find the value of the car after 3 years A1. b) If C is the value of the computer and t is the time in years after buying the computer, write a formula to link C and t (1) C=1,230 x 0.87 t A1. c) Stavros wants to know when the computer will have at least halved in value. How many years will this be. Show your working. 1,230 2 = 615 1,230 x = 1, M1 1,230 x = ,230 x = ,230 x = ,230 x = (1) 5 years A1. (2) (Total 4 marks) Total /10

34 Interpret Proportion 2 Grade 5 Solutions Objective: Understand that x is inversely proportional to y is equivalent to x is proportional to 1/y, and interpret equations that describe direct and inverse proportion. Question 1. a is directly proportional to c. When c = 49, a = 35. a α k c (a) Find a formula for a in terms of c 35 = k 49 (M1) 35 = k 7 K = 5 a = 5 c (A1) w is inversely proportional to a³. When a = 2, w = 16. (2) (b) Find a formula for w in terms of a w α k (M1) a 3 16 = k x 8 = k k = 128 (A1) w = 128 (A1) a 3 (c) Find the value of w when a = 4. (3) w = k (M1) a 3 w = w = 2 (A1) (2) (Total 7 marks)

35 Question 2. (a) y = for values of x from 1 to 5 Which sketch graph is correct? Circle the correct letter. y A B C y y B (B1) 1 5 x 1 5 x 1 5 x (1) (b) y = When x = 4, y = Work out the value of y when x = = k (M1) x 16 = 8 (A1) (2) (Total 3 marks) Total /10

36 Percentage change 2 Grade 4 SOLUTIONS Objective: Express percentages and percentage changes as a fraction or a decimal and interpret these multiplicatively. Question 1. (a) Work out 65 % of x 180 = 117 (A1)... (1) (b) Work out 165 % of x 180 = 297 (A2)... (2) (Total 3 marks) Question 2 Calculate the decrease as a fraction in the following: (a) CD was 15, now costs 10 h = 5 15 = 1 3 (A1) (b) Coat was 65, now costs 52 h = 13 = (A1) (Total 2 marks) Question 3. In a sale, the normal prices are reduced. A coat is reduced from 258 to What is the percentage decrease? h = = ( 1) = 14% (A1) (Total 2 marks)

37 Question 4. Catherine takes the train to work. The cost of a weekly train ticket increases by 15% to 207 Work out the cost of her weekly ticket before the increase. The original price was 100% (M1) So 207 represents 115% (M1) = 180 (A1) (Total 3 marks) Total /10

38 Problems with compound units 2 Grade 5 Solutions Objective: Solve problems involving compound units such as speed, rates of pay, unit pricing, density and pressure. Question 1. A tortoise moves a distance of 120 m in 4 minutes 48 seconds. Work out its speed in m/s. Question 2. 4 minutes 48 seconds = (4 60 ) + 48 = 288 seconds (M1) Speed = = = 5 12 or m/s (A1)... Brass has a density of 8.33 g/cm 3. The mass of a sample of brass is 2kg. Work out its volume. Change the mass of 2kg to grams = = 2000g (A1) (Total 2 mark) Density = so volume = Volume = (M1) = 240.1cm 3 (A1)... (Total 3 marks)

39 Question 3. A cardboard box exerts a force of 160 Newtons on a kitchen work top. The pressure on the worktop is 98 N/m 2. Calculate the area of the cardboard box that is in contact with the worktop. = Pressure = so area = Area = (M1) = 1.63 m 2 (A1)... (Total 2 marks) Question 4. Amelia works a basic 35-hour week. Her hourly rate of pay is She is paid time and a quarter for each hour she works on a Saturday and time and a half for each hour she works on a Sunday. How much is Amelia paid for a week when she works a basic 35-hour week plus 5 hours on Saturday and 2 hours on Sunday? 35 hours basic pay = = hours on Saturday at time and a quarter = = hours on Sunday at time and a half = = (Either of the above gets M1) Total pay = (M1) = (A1) (Total 3 marks) Total /10

40 Scale factors and similarity 2 Grade 5 Solutions Objective: Use scale factors as a link to similarity. Question 1 (a) Work out the size of x Scale factor = 43.2/ 8 = 5.4 (M1) 5 x 5.4 = 27 cm (A1) x =.(2) (b) Work out the size of y 37.8 / 5.4 (M1) = 7 cm (A1) y =.(2) (total 4 marks)

41 Question 2 Scale factor 14/4 = 3.5 (M1) 8 x 3.5 = 28 cm (A1) 21 / 3.5 = 6 cm (A1) AB =.. PS =.. (total 3 marks)

42 Question 3 Scale factor = 4 3 RU = 6 x 4 3 = 8 VU = 2 cm (M1) (M1) (A1) VU =..cm (total 3 marks) Total marks / 10

43 Simple Interest and Financial Maths 2 Grade 5 Solutions Objective: Solve problems involving simple interest and financial maths. Question 1. Greg sells car insurance and home insurance. The table shows the cost of these insurances. Insurance car insurance home insurance Cost Each month Greg earns 530 basic pay 5% of the cost of all the car insurance he sells and 10% of the cost of all the home insurance he sells In May Greg sold 6 car insurances and 4 home insurances Work out the total amount of money Greg earned in May. 6 x 200 = 1200 (M1) 4 x 350 = 1400 (M1) 1200 x 5 = 60 (A1) x 10 = 140 (A1) = (A1) (Total 5 marks)

44 Question 2. * Here are two schemes for investing 2500 for 2 years. Scheme A gives 4% simple interest each year. Scheme B gives 3.9% compound interest each year. Which scheme gives the most total interest over 2 years? You must show all your working. Scheme A: 2500 x 4 = 100 (M1) x 2 = 200 (A1) Scheme B: 2500 x = (M1) = (A1) Scheme A gives more interest by 1.20 (C1) (Total 5 marks) Total /10

45 Solve Proportion Problems 2 Grade 5 Solutions Objective: Solve problems involving direct and inverse proportion. Question 1. The distance, D, travelled by a particle is directly proportional to the square of the time t, taken. When t = 40, D = 30 (a) Find a formula for D in terms of t. D α kt 2 (M1) 30 = k x = k 3 = k (M1 dep) 160 D = t 2 x (A1) (3) (b) Calculate the value of D when t = 64 D = t 2 x D = 64 2 x D = 4096 x D = 76.8 (A1) (c) Calculate the value of t when D = 12 Give your answer correct to 3 significant figures D = t 2 3 x = t 2 x 640 = t = t = t 25.3 = t (A1) (M1) (1) (2) (Total 6 marks)

46 Question 2. q is inversely proportional to the square of t. When t = 4, q = 8.5 (a) Find a formula for q in terms of t. q α k (M1) t = x 8.5 = k 136 = k (M1) q = 136 (A1) t 2 (b) Calculate the value of q when t = 5 (3) q = q = 5.44 (A1) (1) (Total 4 marks) Total /10

ABC is a triangle. The point D lies on AC. Angle BDC = 90 BD = 10 cm, AB = 15 cm and DC = 12.5 cm.

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