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PLC Papers

Created For:

PiXL PLC 2017 Certification

Compare Fractions, Decimals and Percentages 2 Grade 4

Objective: Compare quantities by calculating equivalent fractions, decimals and percentages.

Question 1

Write down three-quarters as a percentage.

.............................................(1)

Question 2

Complete the blanks to make the statements correct.

(a) 50% = 2 ..............................................(1)

(b) 0.7 = 10 ..............................................(1)

(c) 13 = …......%

..............................................(1)

(Total 3 marks)

Question 3

(a) Write ¼ as a percentage

..............................................(1)

(b) Write 30% as a decimal

..............................................(1)

(c) Write ¼, 30% and 0.2 in order with the smallest first

..............................................(1)

(Total 3 marks)


Question 4

Work out the difference between

10% of 350 and ½ of 76

..............................................(3)

Total /10


Compare lengths, area, volume 2 Grade 4

Objective: Compare lengths, areas and volumes using ratio notation.

Question 1

(a) Share 45 kg in the ratio 2 : 3

..............................................

(1)

(b) Share 5 minutes in the ratio 9 : 6

Give your answer in seconds.

.............................................metres (2)

(Total 3 marks)

Question 2

The parallelogram below has area of 42 cm2

It is to be enlarged with a scale factor of 5.

What will the area of the enlarged parallelogram be?

…………………………………

(Total 3 marks)


Question 3

A tank has a volume of 32m³. It is enlarged with scale factor 3.

(a) What is the height of the enlarged tank?

……………………..(1)

(b) What is the volume of the enlarged tank?

……………………….. (3)

(Total 4 marks)

Total /10


Compare quantities using ratio 2 Grade 4 Express a multiplicative relationship between two quantities

Question 1

Here is a list of ingredients for making a peach dessert for 6 people.

Bob is going to make a peach dessert for 15 people.

Work out the amount of each ingredient he needs.

...........................................................g jelly ...........................................................sponge fingers

...........................................................ml custard ...........................................................g peaches

(Total 3 marks)


Question 2

A barrel of squash contains orange juice and water in the ratio 2:5. If there are 18 litres more water than orange juice, how much liquid is in the barrel?

(Total 3 marks)

Question 3

The side lengths of a rectangle are in the ratio 3:8. If the longer side is 12cm more than the shorter side, what is the perimeter of the rectangle?

(Total 4 marks)

Total /10


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$3179.55. In London, an identical camera costs £285. The exchange rate is £1 = HK$12.30. Ben buys the camera in Hong Kong. How much cheaper is the camera in Hong Kong than in London?

(Total 3 marks)


Question 2.

There are 18500 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)


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


Division of a quantity as a ratio 2 Grade 4 Objective: Express the division of a quantity into two parts as a ratio

Question 1

In Year 11 at the school, the ratio of the number of pupils who study P.E to the number of pupils who study Dance is 2 : 3

105 pupils in Year 11 study Dance. Work out the number of pupils in Year 11 who study P.E.

...........................................................

(Total 2 marks) Question 2

Matt, Camilla and Steve share some money in the ratios 5 : 9 : 6 In total, Matt and Camilla receive £56 Work out the amount of money Steve receives.

£...........................................................

(Total 3 marks)


Question 3

James and Sanjana share some sweets in the ratio 3 : 8

Sanjana gets 32 sweets.

How many sweets does James get?

...........................................................


Pritam, Sarah and Emily share some money in the ratios 3 : 6 : 4 Sarah gets $15 more than Emily.

Work out the amount of money that Pritam gets.

$ ........................................................

(Total 3 marks)

Total /10


Expressing one quantity as a percentage of another 2 Grade 4 Objective: Understand that a percentage means "parts per hundred", and express one quantity as a percentage of another, including percentages greater than 100%

Question 1

Susie brings 500 flapjacks to a local fair. Susie sells 450 of the flapjacks. What percentage of the flapjacks did Susie not sell?

......................%

(Total 2 marks)

Question 2

Samuel gains 13 out of 20 in a Maths test. Work out his Maths test mark as a percentage.

......................%

(Total 2 marks)

Question 3

Bryan buys a car costing £15000. He pays a deposit £6000. Express the amount that he has left to pay as a percentage of the original cost of the car.

......................%

(Total 3 marks)


Question 4

The population of a country is 3.54 x 105. The number of people that own a dog is 1.24 x 104. Calculate the percentage of the population that own a dog. Give your answer correct to 2 significant figures.

......................%

(Total 3 marks)

Total /10


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?

.......................................... km (1)

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.

(3)

(Total 5 marks)


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.

.............................................. km/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.

(3)

(Total 5 marks)

Total /10


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.

………………………….

(3)

c) How much money is in the bank after 14 years.

………………………….

(2)

(Total 6 marks)


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.

………………….. ……….

(1)

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.

………………………..…….

(2)

(Total 4 marks)

Total /10


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

(2) w is inversely proportional to a³. When a = 2, w = 16.

(b) Find a formula for w in terms of a

(3) (c) Find the value of w when a = 4.

(2) (Total 7 marks)


Question 2.

(a) y = for values of x from 1 to 5

Which sketch graph is correct? Circle the correct letter.

(1)

(b) y =

When x = 4, y =

Work out the value of y when x = 2

(2)

(Total 3 marks)

Total /10

x

y

0.4

1 5

10

x

y

0.4

1 5

10

x

y

0.4

1 5

10

A B C


Percentage Change 2 Grade 5 Objective: Solve problems involving percentage change, including original value problems.

Question 1.

During August 2015 the number of people visiting Woodlands Adventure Park fell from 76,800 to 65,280.

What was the percentage decrease?

………………%

(Total 2 marks)

Question 2.

In January 2012 the average price of a 3-bed semi-detached house in Hitchin was £362,500.

In January 2016 the average price had increased to £482,000.

What was the percentage increase, to the nearest one per cent?

………………%

(Total 2 marks)


Question 3.

Toby got 18 questions wrong on a test. He got 70% of the questions correct.

Work out the total number of questions in the test.

………………

(Total 2 marks)

Question 4.

Markus takes his family out for a meal. He pays £184 including a 15% tip.

How much tip did he leave?

£………………

(Total 2 marks)

Question 5.

In New York City a sales tax of 4.5% is added onto all merchandise at the till.

Emily spends $133.76 in one shop.

How much did her goods come to before the sales tax?

$………………

(Total 2 marks)

Total /10


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 180

..............................................

(1)

(b) Work out 165 % of 180

..............................................

(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)


Question 3.

In a sale, the normal prices are reduced.

A coat is reduced from £258 to £221.88

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


Problems involving ratios 2 Grade 4 Objective: Solve problems involving ratios, e.g. conversion, comparison, scaling, mixing, concentrations

Question 1

On a farm

the number of cows and the number of sheep are in the ratio 6 : 5 the number of sheep and the number of pigs are in the ratio 2 : 1

The total number of cows, sheep and pigs on the farm is 189

How many sheep are there on the farm?

........................................................


Build-a-mix makes concrete.

1 cubic metre of concrete has a weight of 2400 kg.

15% of the concrete is water.

The rest of the ingredients of concrete are cement, sand and stone. The weights of these ingredients are in the ratio 1 : 2 : 5

Work out the weight of cement, of sand and of stone in 1 cubic metre of concrete.

cement = . . . . . . . . . . . . . . . . . . . . . . kg

sand = . . . . . . . . . . . . . . . . . . . . . . kg

stone = . . . . . . . . . . . . . . . . . . . . . . kg

(Total 4 marks)


Question 3

Rob is learning about the planets.

Rob makes a model of the Sun. He also makes a model of the planet Jupiter.

Rob is going to hang the two models in the school hall.

Rob wants a distance of 16 m between the two models. The real distance between the planet Jupiter and the Sun is 8 × 108 km.

Work out the scale Rob should use. Give your answer in the form 1 : n

..........................................................................................................................

(Total 3 marks)

Total /10


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/cm3.

The mass of a sample of brass is 2kg.

Work out its volume.

…………………………………………………………………………...............................................

(Total 3 marks)

Question 3.

A cardboard box exerts a force of 160 Newtons on a kitchen work top.

The pressure on the worktop is 98 N/m2.

Calculate the area of the cardboard box that is in contact with the worktop. �� = ��

…………………………………………………………………………...............................................


(Total 2 marks)

Question 4.

Amelia works a basic 35-hour week. Her hourly rate of pay is £6.60.

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


Proportion and ratio 2 Grade 4 Objective: Understand and use proportion as equality of ratios

Question 1

Lewis has a copper pipe with a length of 150 cm and a mass of 800 grams. He cuts a piece of the copper pipe with a length of 90 cm. Work out the mass of this piece of copper pipe.

........................................................... grams

(Total 3 marks)

Question 2

* 225 grams of flour are needed to make 9 cakes.

Marian wants to make 20 of these cakes. She has 475 grams of flour.

Does Marian have enough flour to make 20 cakes? You must show all your working.

(Total 3 marks)


Question 3

5 schools sent some students to a conference.

One of the schools sent both boys and girls. This school sent 16 boys. The ratio of the number of boys it sent to the number of girls it sent was 1 : 2

The other 4 schools sent only girls. Each of the 5 schools sent the same number of students.

Work out the total number of students sent to the conference by these 5 schools.

(Total 4 marks)

Total /10


Ratio Sharing 2 Grade 4

Objective: Divide a quantity in a given ratio.

Question 1.

Ken and Susan share £20 in the ratio 1 : 3 Work out how much money each person gets.

Ken £ .....................

Susan £ ..................

(Total 3 marks)

Question 2.

Amy, Beth and Colin share 36 sweets in the ratio 2 : 3 : 4 Work out the number of sweets that each of them receives.

(Total 3 marks)


Question 3.

Peter won £75 as a prize. He gave 4/5 of the prize money as a present to Roger and Bethan. Roger and Bethan shared the present in the ratio 2:3 Work out how much they each got.

Roger..................................

Bethan..................................

(Total 4 marks)

Total /10


Ratio and fractions 2 Grade 4 Objective: Relate ratios to fractions and linear functions

Question 1

The ratio of girls to boys in a school is 5 : 7 . What fraction of these students are girls?

. . . . . . . . . . . . . . . . . . . . . .

(Total 1 mark)

Question 2

Ann and Bob shared £240 in the ratio 3 : 5

Ann gave a half of her share to Colin.

Bob gave a tenth of his share to Colin.

What fraction of the £240 did Colin receive?

. . . . . . . . . . . . . . . . . . . . . .

(Total 5 marks)


Question 3

Peter won £75 as a prize. He gave 45 of the prize money as a present to Roger and Bethan.

Roger and Bethan shared the present in the ratio 2:3

Work out how much they each got.

........................................................................................................................

(Total 4 marks)

Total /10


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)


Question 2

AB = …………………………..

PS = …………………………..

(total 3 marks)


Question 3

VU = ……………………………………………..cm

(total 3 marks)

Total marks / 10


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 £200 £350

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)


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


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.

D = ..................................... (3)

(b) Calculate the value of D when t = 64

..................................... (1)

(c) Calculate the value of t when D = 12 Give your answer correct to 3 significant figures

..................................... (2)

(Total 6 marks)


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 = ...................................... (3)

(b) Calculate the value of q when t = 5

............................... (1)

(Total 4 marks)

Total /10

PLC Papers

Created For:


Compare Fractions, Decimals and Percentages 2 Grade 4 SOLUTIONS

Objective: Compare quantities by calculating equivalent fractions, decimals and percentages.

Question 1

Write down three-quarters as a percentage. 75% (A1)

.............................................(1)

Question 2

Complete the blanks to make the statements correct.

(a) 50% = 12 (A1) ..............................................(1)

(b) 0.7 = 710 (A1) ..............................................(1)

(c) 13 = 33

13 % (A1)

..............................................(1)

(Total 3 marks)

Question 3

(a) Write ¼ as a percentage 25% (A1)

..............................................(1)

(b) Write 30% as a decimal 0.3 (A1)

..............................................(1)

(c) Write ¼, 30% and 0.2 in order with the smallest first

0.2, ¼ , 30% (A1)

..............................................(1)

(Total 3 marks)


Question 4

Work out the difference between

10% of 350 and ½ of 76

10% of 350 = 35 (M1)

½ of 76 = 38 (M1)

38 – 35 = a difference of 3 (A1)

..............................................(3)

Total /10


Compare lengths, area, volume 2 Grade 4 SOLUTIONS

Objective: Compare lengths, areas and volumes using ratio notation.

Question 1

(a) Share 45 kg in the ratio 2 : 3 45 / 5 = 9 per share 18 : 27 (A1)

..............................................

(1)

(b) Share 5 minutes in the ratio 9 : 6 Give your answer in seconds. 5 minutes = 300 seconds 300/15 = 20 secs per share (M1) 180 : 120 (A1)

.............................................metres (2)

(Total 3 marks)

Question 2

The parallelogram below has area of 42 cm2

It is to be enlarged with a scale factor of 5.

What will the area of the enlarged parallelogram be?

Area factor is 5 x 5 = 25 (M1) 42 x 25 (M1) = 1050 cm2 (A1)

…………………………………

(Total 3 marks)


Question 3

A tank is 4 m tall, and has a volume of 32m³. It is enlarged with scale factor 3.

(a) What is the height of the enlarged tank? 4m x 3 = 12 m (A1)

(b) What is the volume of the enlarged tank?

Volume factor is 3 x 3 x 3 = 27 (M1)

32 x 27 (M1)

864 m³ (A1)

……………………….. (Total 4 marks)

Total /10


Compare quantities using ratio 2 Grade 4 Solutions Express a multiplicative relationship between two quantities

Question 1

Here is a list of ingredients for making a peach dessert for 6 people.

Bob is going to make a peach dessert for 15 people.

Work out the amount of each ingredient he needs.

15 ÷ 6 = 2.5 M1 2.5 × 150 = 375 2.5 × 10 = 25 2.5 × 500= 1250 2.5 × 200 = 500 A2 for all four correct answers

............................ 375......................g jelly ............................ 25..........sponge fingers ........................... 1250...............ml custard

........................... 500...................g peaches

(Total 3 marks)


Question 2

A barrel of squash contains orange juice and water in the ratio 2:5. If there are 18 litres more water than orange juice, how much liquid is in the barrel?

5 - 2 = 3

18 ÷ 3 = 6 M1

2 × 6 = 12 5 × 6 = 30 M1

12 + 30 = 42 A1

(Total 3 marks)

Question 3

The side lengths of a rectangle are in the ratio 3:8. If the longer side is 12cm more than the shorter side, what is the perimeter of the rectangle?

8 - 3 = 5

12 ÷ 5 = 2.4 M1

3.8 × 2.4= 7.2 8 × 2.4 = 19.2 M1

2(7.2 + 19.2) = 52.8 M1

52.8 cm A1

(Total 4 marks)

Total /10


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$3179.55. In London, an identical camera costs £285. The exchange rate is £1 = HK$12.30. Ben buys the camera in Hong Kong. How much cheaper is the camera in Hong Kong than in London? 3179.55 ÷ 12.3 (=258.5) (M1) 285 - '258.5' (M1 dep) £26.50 (correctly stated with currency) (A1) OR 285 × 12.3 (=3505.5) (M1) '3505.5' - 3179.55 (=325.95) (M1) HK$325.95 (correctly stated with currency) (A1)

(Total 3 marks)


Question 2.

There are 18500 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.

1700 ÷ 4.5 = 377.78 (M1) 18500 ÷ 377.78 = 48.9 (M1 & A1)

49 minutes

(Total 3 marks)


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) 24 ÷ 60 = 0.4 (M1) 30 ÷ 0.4 = 75 (A1) Debbie is fastest from comparison of “74 – 76” with 66 (kph) C1 (dep on M2)

(Total 4 marks)

Total /10


Division of a quantity as a ratio 2 Grade 4 Solutions Objective: Express the division of a quantity into two parts as a ratio

Question 1

In Year 11 at the school, the ratio of the number of pupils who study P.E to the number of pupils who study Dance is 2 : 3

105 pupils in Year 11 study Dance. Work out the number of pupils in Year 11 who study P.E. 105÷ 3 = 35 M1 2 × 35 = 70 A1

...........................................................


Matt, Camilla and Steve share some money in the ratios 5 : 9 : 6 In total, Matt and Camilla receive £56 Work out the amount of money Steve receives. 5 + 9 = 14 56 ÷ 14 = 4 M1 4 × 6 M1 £24 A1

£...........................................................

(Total 3 marks)


Question 3

James and Sanjana share some sweets in the ratio 3 : 8

Sanjana gets 32 sweets.

How many sweets does James get?

32 ÷ 8 = 4 M1 4 × 3 = 12 A1

...........................................................


Pritam, Sarah and Emily share some money in the ratios 3 : 6 : 4 Sarah gets $15 more than Emily.

Work out the amount of money that Pritam gets.

6 – 4 = 2 15 ÷ 2 = 7.5 M1 3 × 7.5 M1 $22.50 A1

$ ........................................................

(Total 3 marks)

Total /10


Expressing one quantity as a percentage of another 2 Grade 4 Solutions Objective: Understand that a percentage means "parts per hundred", and express one quantity as a percentage of another, including percentages greater than 100%

Question 1

Susie brings 500 flapjacks to a local fair. Susie sells 450 of the flapjacks. What percentage of the flapjacks did Susie not sell?

500 – 450 = 50 50500 × 100 M1

10 A1

......................%

(Total 2 marks)

Question 2

Samuel gains 13 out of 20 in a Maths test. Work out his Maths test mark as a percentage.

1320 × 100 M1

65% A1

......................%

(Total 2 marks)

Question 3

Bryan buys a car costing £15000. He pays a deposit £6000. Express the amount that he has left to pay as a percentage of the original cost of the car.

15000 – 6000 = 9000 M1 900015000 × 100 M1

60 % A1

......................%


(Total 3 marks)

Question 4

The population of a country is 3.54 x 105. The number of people that own a dog is 1.24 x 104. Calculate the percentage of the population that own a dog. Give your answer correct to 2 significant figures.

1.24 × 104 = 12400 M1 for either answer 3,54 × 105 = 354 000 12400354000 × 100 M1

3.5 % A1

......................%

(Total 3 marks)

Total /10


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?

30 minutes (M1)

(b) How far was Simon from home at 5 pm? 20km (M1)

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)

(3)

(Total 5 marks)


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. 10 ÷ 60 = 0.16666667 (M1) 9 ÷ 0.16666667 = 53.999 54km/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. 21÷ 42 = 0.5 (M1) 30 min Correct line drawn on graph (B1)

(3)

(Total 5 marks)

Total /10


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.

1.0437

……1.342734535………A1…………….

(1)

b) How many years does it take before £6,000 is in the bank. 4500 x 1.0431 = 4,693.50 4500 x 1.0432 = 4,895.32 4500 x 1.0433 = 5,105.82 4500 x 1.0434 = 5,325.37 4500 x 1.0435 = 5,554.36 4500 x 1.0436 = 5,793.20 M2 4500 x 1.0437 = 6,042.31

……7 years………B1 ……….

(3)

c) How much money is in the bank after 14 years. 4,500 x 1.04314 M1

……£8113.21…………A1……….

(2)

(Total 6 marks)


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.

0.873

………0.6585…………A1……….

(1)

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

…………C=1,230 x 0.87t………A1…….

(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.

1,230 ÷ 2 = 615

1,230 x 0.871 = 1,070.10 M1 1,230 x 0.872 = 930.99 1,230 x 0.873 = 809.96 1,230 x 0.874 = 704.66 1,230 x 0.875 = 613.06

………5 years………A1……….

(2)

(Total 4 marks)

Total /10


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) Find a formula for a in terms of c

a α k√c

35 = k√49 (M1)

35 = k 7 K = 5

a = 5√c (A1)

(2) w is inversely proportional to a³. When a = 2, w = 16.

(b) Find a formula for w in terms of a

w α k (M1) a3 16 = k 23 16 x 8 = k

k = 128 (A1)

w = 128 (A1) a3

(3) (c) Find the value of w when a = 4.

w = k (M1) a3

w =128 43 w = 2 (A1)

(2) (Total 7 marks)


Question 2.

(a) y = for values of x from 1 to 5

Which sketch graph is correct? Circle the correct letter.

B (B1)

(1)

(b) y =

When x = 4, y =

Work out the value of y when x = 2

0.5 = k (M1) 42 0.5 x 16 = 8 (A1)

(2)

(Total 3 marks)

Total /10

x

y

0.4

1 5

10

x

y

0.4

1 5

10

x

y

0.4

1 5

10

A B C


Percentage Change 2 Grade 5 SOLUTIONS Objective: Solve problems involving percentage change, including original value problems.

Question 1.

During August 2015 the number of people visiting Woodlands Adventure Park fell from 76,800 to 65,280.

What was the percentage decrease?

76800−6528076800 × 100 (M1)

15 (A1)………………%

(Total 2 marks)

Question 2.

In January 2012 the average price of a 3-bed semi-detached house in Hitchin was £362,500.

In January 2016 the average price had increased to £482,000.

What was the percentage increase, to the nearest one per cent?

482000−362500362500 × 100 (M1)

33 (A1)………………%

(Total 2 marks)


Question 3.

Toby got 18 questions wrong on a test. He got 70% of the questions correct.

Work out the total number of questions in the test.

18 = 30% (M1)

0.6 = 1%

60 = 100%

60 (A1)………………

(Total 2 marks)

Question 4.

Markus takes his family out for a meal. He pays £184 including a 15% tip.

How much tip did he leave?

£184 = 115% (M1)

£1.6 = 1%

£160 = 100%

£160 (A1)………………

(Total 2 marks)

Question 5.

In New York City a sales tax of 4.5% is added onto all merchandise at the till.

Emily spends $133.76 in one shop.

How much did her goods come to before the sales tax?

$133.76 = 104.5% (M1)

$1.28 = 1%

$128 = 100%

$128 (A1)………………

(Total 2 marks)

Total /10


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 180 65100 x 180 = 117 (A1)

..............................................

(1)

(b) Work out 165 % of 180 165100 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�� =

515 = 13 �� (A1)

(b) Coat was £65, now costs £52 �ℎ�� =

1365 = 15 �� (A1)

(Total 2 marks)

Question 3.

In a sale, the normal prices are reduced.

A coat is reduced from £258 to £221.88

What is the percentage decrease? �ℎ�� = 36.12258 =

750 (�1) = 14% �� (A1)

(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.

The original price was 100% (M1)

So £207 represents 115% (M1)

207115 × 100 = £180 �� (A1)

(Total 3 marks)

Total /10


Problems involving ratios 2 Grade 4 Solutions Objective: Solve problems involving ratios, e.g. conversion, comparison, scaling, mixing, concentrations

Question 1

On a farm

the number of cows and the number of sheep are in the ratio 6 : 5 the number of sheep and the number of pigs are in the ratio 2 : 1

The total number of cows, sheep and pigs on the farm is 189

How many sheep are there on the farm?

6 : 5 = 12: 10

2:1 = 10 : 5 M1

C : S : P

12 : 10 : 5 M1

× 189 = 70 A1

........................................................


Build-a-mix makes concrete.

1 cubic metre of concrete has a weight of 2400 kg.

15% of the concrete is water.

The rest of the ingredients of concrete are cement, sand and stone. The weights of these ingredients are in the ratio 1 : 2 : 5

Work out the weight of cement, of sand and of stone in 1 cubic metre of concrete.

× 2400 = 360 M1

2400 – 360 = 2040

2040 ÷ 8 = 255 M1

1 × 255 = 255 kg cement

2 × 255 = 510 kg sand A2 for all three correct answers

5 × 255 = 1275 kg stone

cement = . . . . . . . . . . . . . . . . . . . . . . kg

sand = . . . . . . . . . . . . . . . . . . . . . . kg

stone = . . . . . . . . . . . . . . . . . . . . . . kg

(Total 4 marks)


Question 3

Rob is learning about the planets.

Rob makes a model of the Sun. He also makes a model of the planet Jupiter.

Rob is going to hang the two models in the school hall.

Rob wants a distance of 16 m between the two models. The real distance between the planet Jupiter and the Sun is 8 × 108 km.

Work out the scale Rob should use. Give your answer in the form 1 : n

16 metres : 8 x 108

16 : 8 × 108 × 1000 M1

16 : 8 × 108 ×103

16 : 8 × 1011 M1

1 : 5 × 1010 A1

..........................................................................................................................

(Total 3 marks)

Total /10


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.

4 minutes 48 seconds = (4 × 60 ) + 48 = 288 seconds (M1)

Speed = ��

= 120288 =

512 or 0.416̇ m/s (A1)

…………………………………………………………………………...............................................

(Total 2 mark)

Question 2.

Brass has a density of 8.33 g/cm3.

The mass of a sample of brass is 2kg.

Work out its volume.

Change the mass of 2kg to grams = 2 × 1000 = 2000g (A1)

Density = �� so volume =

��

Volume = 20008.33 (M1)

= 240.1cm3 (A1)

…………………………………………………………………………...............................................

(Total 3 marks)


Question 3.

A cardboard box exerts a force of 160 Newtons on a kitchen work top.

The pressure on the worktop is 98 N/m2.

Calculate the area of the cardboard box that is in contact with the worktop. �� =��

Pressure = �� so area =

�� Area =

16098 (M1)

= 1.63 m2 (A1) …………………………………………………………………………...............................................

(Total 2 marks)

Question 4.

Amelia works a basic 35-hour week. Her hourly rate of pay is £6.60.

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 = 35 × 6.60 = £231 5 hours on Saturday at time and a quarter = 5 × 1.25 × 6.60 = £41.25 2 hours on Sunday at time and a half = 2 × 1.5 × 6.60 = £19.80 (Either of the above gets M1)

Total pay = £231 + £41.25 + £ 19.80 (M1)

= £292.05 (A1)

(Total 3 marks)

Total /10


Proportion and ratio 2 Grade 4 Solutions Objective: Understand and use proportion as equality of ratios

Question 1

Lewis has a copper pipe with a length of 150 cm and a mass of 800 grams. He cuts a piece of the copper pipe with a length of 90 cm. Work out the mass of this piece of copper pipe.

850 ÷ 150= 5.3333… M1

90 × 5.3333… M1

480 A1

........................................................... grams

(Total 3 marks)

Question 2

* 225 grams of flour are needed to make 9 cakes.

Marian wants to make 20 of these cakes. She has 475 grams of flour.

Does Marian have enough flour to make 20 cakes? You must show all your working.

225 ÷ 9= 25 M1

20 × 25 = 500 M1

No, she is 25g short C1

(Total 3 marks)


Question 3

5 schools sent some students to a conference.

One of the schools sent both boys and girls. This school sent 16 boys. The ratio of the number of boys it sent to the number of girls it sent was 1 : 2

The other 4 schools sent only girls. Each of the 5 schools sent the same number of students.

Work out the total number of students sent to the conference by these 5 schools.

16 × 2 = 32 M1

16 +32 = 48 M1

48 × 5 M1

240 A1

(Total 4 marks)

Total /10


Ratio Sharing 2 Grade 4 Solutions

Objective: Divide a quantity in a given ratio.

Question 1.

Ken and Susan share £20 in the ratio 1 : 3 Work out how much money each person gets. 1 + 3 = 4 or 20 ÷ 4 = 5 (M1)

5 x 1 = 5 (A1) & 5 x 3 = 15 (A1)

Ken £5

Susan £15

(Total 3 marks)

Question 2.

Amy, Beth and Colin share 36 sweets in the ratio 2 : 3 : 4 Work out the number of sweets that each of them receives.

2 + 3 + 4 = 9 or 36 ÷ 9 = 4 (M1)

4 x 2 = 8 (A1), 4 x 3 = 12 & 4 x 4 = 16 (A1)

Amy 8 sweets

Beth 12 sweets

Colin 16 sweets

(Total 3 marks)

Question 3.


Peter won £75 as a prize. He gave 4/5 of the prize money as a present to Roger and Bethan. Roger and Bethan shared the present in the ratio 2:3 Work out how much they each got. (£75 ÷ 5) x 4 = 60 (M1) 2 + 3 = 5 or 60 ÷ 5 = 12 (M1) 12 x 2 = 24 (A1) 12 x 3 = 36 (A1)

Roger 24

Bethan 36

(Total 4 marks)

Total /10


Ratio and fractions 2 Grade 4 Solutions Objective: Relate ratios to fractions and linear functions

Question 1

The ratio of girls to boys in a school is 5 : 7 . What fraction of these students are girls?

5 + 7 = 12 512 A1

. . . . . . . . . . . . . . . . . . . . . .

(Total 1 mark)

Question 2

Ann and Bob shared £240 in the ratio 3 : 5

Ann gave a half of her share to Colin.

Bob gave a tenth of his share to Colin.

What fraction of the £240 did Colin receive?

240 ÷ 8 = 30 M1

3 × 30 = £90

90 ÷ 2 = £45 (Colin) M1

5 × 30 = £150

150 ÷ 1- = £15 ( Colin) M1

Colin 45 + 15 = £60 M1 60240 = 14 A1

. . . . . . . . . . . . . . . . . . . . . .

(Total 5 marks)


Question 3

Peter won £75 as a prize. He gave 45 of the prize money as a present to Roger and Bethan.

Roger and Bethan shared the present in the ratio 2:3

Work out how much they each got.

# 60 ÷ 5 = 12 M1

2 × 12 M1

or

3 × 12

£24 (Roger) A1

£36 (Bethan) A1

........................................................................................................................

(Total 4 marks)

Total /10


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)


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)


Question 3

Scale factor = 43 (M1)

RU = 6 x 43 = 8 (M1)

VU = 2 cm (A1)

VU = ……………………………………………..cm

(total 3 marks)

Total marks / 10


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 £200 £350

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 5100 = 60 (A1)

1400 x 10100 = 140 (A1)

530 + 60 + 140 = 730

£730 (A1)

(Total 5 marks)


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 4100 = 100 (M1)

100 x 2 = £200 (A1)

Scheme B: 2500 x 1.0392 = 2698.80 (M1)

2698.80 – 2500 = £198.80 (A1)

Scheme A gives more interest by £1.20 (C1)

(Total 5 marks)

Total /10


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 kt2 (M1)

30 = k x 402 301600 = k 3160 = k (M1 dep)

D = t2 x 3160 (A1)

(3) (b) Calculate the value of D when t = 64

D = t2 x 3160

D = 642 x 3160

D = 4096 x 3160

D = 76.8 (A1)

(1) (c) Calculate the value of t when D = 12 Give your answer correct to 3 significant figures

D = t2 x 3160

12 = t2 x 3160 (M1)

640 = t2 √640 = t

25.2982… = t

25.3 = t (A1)

(2)

(Total 6 marks)


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) t2

8.5 = �16

16 x 8.5 = k

136 = k (M1)

q = 136 (A1) t2

(3) (b) Calculate the value of q when t = 5 q = 136 52

q = 5.44 (A1)

(1)

(Total 4 marks)

Total /10

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