UNIT 1

Problem solving Harkness

1. The sum of two odd numbers is 80 and their difference is 6.

Work out these numbers.

(2 marks)

2. Find three different prime numbers that add up to 21.

(2 marks)

3. Sanjiv has £46 and Joshua has £38.

Sanjiv gives Joshua some of his money, so that they both have the same amount.

How much do they each have now?

(2 marks)

4. Zarah pays the same amount for each text message she sends.

In June she pays £4.16 for 52 text messages.

In July she pays £5.12.

How many text messages did she send in July?

(2 marks)

UNIT 1

5. Three whole numbers have a total of 80.

The first number is a factor of 32.

The second number is 5 times the third number.

Work out the three numbers.

(3 marks)

6. Saima is organising a conference.

She wants to give each persona pen and a notebook.

Pens are sold in packs of 15.

Notebooks are sold in packs of 18.

Saima bought exactly the same number of pens and notebooks.

a) What were the smallest numbers of packs Saima could have bought of pens and notebooks?

………. pack of pens

………. packs of notebooks

(4 marks)

b) How many people did she provide pens and notebooks for?

(1 mark)

UNIT 2

Problem solving Harkness

1. In one month, Paul buys c CDs.

Each CD contains 10 songs.

Write an expression for the total number of songs Paul has bought.

(1 mark)

2. The formula used to find the delivery cost of a parcel is

Cost = (80 × mass in kilograms + 50) pence.

a) Jo posts a parcel that weighs 150 g.

How much will it cost to post?

(2 marks)

b) Mike’s parcel weighs 300 g. He says:

‘The cost for posting my parcel will be twice the cost of posting Jo’s parcel.’

Is he correct?

Explain your answer.

(1 mark)

UNIT 2

3. a) Write 2(3x + 5) + 4(2x – 1) + 4 in the form

a(bx + c)

where a, b and c are integers and a > 1.

(3 marks)

b) Is 2(3x + 5) + 4(2x – 1) + 4 divisible by an even number?

Explain your answer.

(2 marks)

UNIT 3

(3 marks)

Problem solving Harkness

1. Ben asked 70 students to choose from tennis, hockey or badminton.

30 of the students were girls.

10 girls chose tennis.

11 out of the 26 students who chose hockey were girls.

22 students altogether chose badminton.

How many boys chose badminton?

(4 marks)

3. The table shows the population of England (in millions) from 2002 to 2014.

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

49.6 49.9 50.1 50.5 50.8 51.1 51.5 51.8 52.2 52.7 53.1 53.6 54.0

a Use this information to draw a graph and predict the population in England in 2020.

(4 marks)

UNIT 3

(3 marks)

b Make two comments to explain why your prediction in part a may not be reliable.

(1)

(2)

(1 mark)

A shopkeeper did a survey to find his customers’ favourite fruits.

He asked 90 customers.

The bar chart shows the results.

Use the information from the bar chart to complete the pie chart.

UNIT 4

Problem solving Harkness

1. Sarah buys 15 boxes of cupcakes.

The cost of each box of cupcakes is £8.

Each box holds 20 cakes.

Sarah sells 40% of the total number of cupcakes for 80 p each.

She then sells all the remaining cakes for 40 p each.

Work out the total profit Sarah makes.

(5 marks)

2. John, Colin and Bob share the petrol costs on their way to work each month.

John pays 30% of the cost.

Colin pays 12

of the cost.

Bob pays £25.

How much do they pay altogether for the petrol?

(4 marks)

UNIT 4

3. Students were given a choice of 3 language study options.

30% of students chose Spanish.

14

of students chose French.

The remaining students chose German.

a What fraction of students chose German?

(3 marks)

90 students chose Spanish.

b How many students chose French?

(3 marks)

4. Naomi, Caitlin and Rebecca share a house and split the electricity bill between them.

Naomil pays 25

of the bill.

Caitlin pays 12

of the bill.

Rebecca pays £15.

How much do they pay altogether for the electricity bill?

(4 marks)

UNIT 5 – F

Problem solving Harkness

1. Hasan, Umar and Bill play a computer game.

Hasan scores 20 more points than Bill.

Bill scores 8 more points than Umar.

Altogether they score 72 points.

How many points does each of them score?

Hasan………………

Bill…………………

Umar………….……

(4 marks)

UNIT 5 – F

2. Sarah, Parveen and Jen shared out some pencils.

Sarah had twice as many pencils as Parveen.

Parveen had 6 more pencils than Jen.

They had a total of 42 pencils.

How many pencils does Parveen have?

(5 marks)

3. The diagram shows a rectangle. The length of the rectangle is greater than the height.

Complete this inequality.

x > …………………

(3 marks)

UNIT 6 – F

Problem solving Harkness

1. ABC is an isosceles triangle.

Write an expression in terms of x for angle ABC.

(2 marks)

2. The diagram shows three straight lines.

Calculate the value of a.

Give your reasons.

(6 marks)

UNIT 7 – F

Problem solving Harkness

1. Bernie and his five friends timed themselves to complete a puzzle.

The times, in seconds, of his five friends are shown.

12.4, 12.5, 12.6. 12.7, 12.9

The mean time for Bernie and his friends was 12.55 seconds.

How long did Bernie take?

(3 marks)

2. Here are some fractions.

14

1

12

16

13

524

a) Work out the mean.

(1 mark)

b) Work out the range.

(1 mark)

3. The mean age of the first 10 patients to arrive at a doctor’s surgery is a.

The mean age of the next 20 is b.

Write an expression, in terms of a and b, for the mean age of all 30 patients.

(2 marks)

UNIT 8 – F

Problem solving Harkness

1. The diagram shows a shaded triangle inside a square.

Work out the area of the shaded triangle.

…………………cm2

(5 marks)

UNIT 8 – F

2. A square has an area of 144 cm2.

Calculate its perimeter.

................................... cm

(2 marks)

3. The area of the rectangle is four times the area of the triangle.

Calculate the height x of the rectangle.

................................... cm

(4 marks)

UNIT 9 – F

Problem solving Harkness

1. Here is a rectangle.

The height of the rectangle is twice the length of the base.

a Write an expression for the height of the rectangle.

(1 mark)

The perimeter of the rectangle is 36 cm.

b Calculate the area of the rectangle.

.................................. cm2

(4 marks)

UNIT 9 – F

2. Jeffrey leaves home at 10:00 on a cycling tour.

a How many minutes did Jeffrey rest for?

........................... minutes

(1 mark)

b What is his average speed for the last part of his journey? Give your answer in km/hour.

.......................... km/hour

(3 marks)

c Between which times did he cycle faster?

(1 mark)

UNIT 9 – F

3. Orange squash is made from concentrated orange juice and water in the ratio 2 : 5.

a Draw a graph that can be used to work out the amount of water needed, given the amount of concentrated orange juice.

Your graph must show up to 8 litres of concentrated orange juice.

(3 marks)

b How much water needs to be mixed with 5 litres of concentrated orange juice?

(1 mark)

UNIT 10 – F

Problem solving Harkness

1. The perimeter of this rectangle is 64 cm.

Calculate the length of each side of the rectangle.

................................... cm

................................... cm

(4 marks)

2. The perimeter of the triangle is the same as the perimeter of the square.

Calculate the area of the square.

(5 marks)

UNIT 10 – F

3. A square with an area of 16 cm2 is enlarged.

The area of its image is 144 cm2.

What is the scale factor of enlargement?

(3 marks)

4. Point A with coordinates (2, 4) has been translated to point B with coordinates (5, –3).

a Write the translation vector from A to B.

(2 marks)

b Calculate the shortest distance between point A and point B. Write your answer to 1 decimal place.

(3 marks)

UNIT 11 – F

Problem solving Harkness

1. The perimeter of the rectangle is the same as the perimeter of the square.

Find an expression for the length of one side of the square, in terms of x.

(4 marks)

2. Chloe saved £4000 for a holiday.

She spent 20% on flights.

She shared the rest of the money between accommodation, food and spending money in the ratio 3 : 2 : 3.

How much did she spend on accommodation and food?

............. Accommodation

................................ Food

(4 marks)

UNIT 11 – F

3. Joe, Rajiv and Sharon shared some money in the ratio 3 : 5 : 6.

Rajiv got £90 more than Joe.

How much did Sharon get?

(3 marks)

4. Mark and Simon shared some money in the ratio 3 : 12.

Simon gave half of his money to Mark.

Write the ratio of Mark’s money to Simon’s in its simplest form.

(3 marks)

Q Grade Answer Mark Comment

1 3 37 + 43 = 80

43 – 37 = 6

M1 M1 for finding two odd numbers that add to make 80 or have a difference of 6.

37, 43 A1 A1 cao

2 4 3, 7, 11 M1 M1 for finding 3 prime numbers.

or

3, 5, 13

A1 A1 cao

3 4 46 + 38 = £84

£84 ÷ 2 = £42

M1 M1 for 46 + 38 = 84 oe

£42 A1 A1 for 84 ÷ 2 = 42 cao

4 4 4.16 ÷ 52 = £0.08

or 416 ÷ 52 = 8

M1 M1 for 4.16 ÷ 52 = £0.08

or 416 ÷ 52 = 8

5.12 ÷ 0.08 = 64

or 512 ÷ 8 = 64

A1 A1 for (5.12 ÷ their 0.08) = 64

5 5 8, 60, 12

or

2, 65, 13

B3 B3 Any order

B2 for three numbers with two of these criteria:

A factor of 32

Two numbers in the ratio 5 : 1

Sum 80

B1 for a factor of 32 or two numbers in the ratio 5 : 1 or three numbers with a sum of 80.

6a 5 Pens: 15, 30, 45, 60, 75, 90, 105

Notebooks: 18, 36, 54, 72, 90, 108

M3 M1 for attempting multiples of either 15 or 18 (at least 3 but condone errors if intention is clear).

M1 for attempting multiples of both 15 and 18 (at least 3 but condone errors if intention is clear).

M1 (dep on M1) for a division of 15 or 18 or counts up ‘multiples’ (implied if one answer is correct or answers reversed).

6 packs of pens; 5 packs of notebooks

A1 A1 pens (packs) 6, notebooks (packs) 5

6b 5 90 A1 A1 pens and notebooks 90 or ft on both of their packs or ft ‘common multiple’.

Q Grade Answer Mark Comment

1 4 10c A1 cao

2a 4 80 × 0.15 + 50 M1 M1 for 0.15 kg seen.

= 62p A1 cao

2b 5 No; 80 × 0.3 + 50 = 74p B1 B1 for ‘No’ and working showing that it is not 2 × 62 or ‘No’ and reasons.

3a 5 6x + 10 + 8x – 4 + 4

= 14x + 10

M2 M1 for expanding one of the brackets correctly.

M1 for 14x + 10.

= 2(7x + 5) A1

3b 6 Yes because 2 is a factor A2 Yes with correct explanation, such as ‘an even number is divisible by 2 and 2(7x + 5) is divisible by 2’.

1

1 See the table M3 M1 for 22 – 10 (=12)

M1 for 26 – 11 (=15)

M1 for 70 – 30 (=40)

Tennis Hockey Badminton Total

Boys 12 15 13 40

Girls 10 11 9 30

Total 22 26 22 70

A1 A1 for 40 – 12 – 15 (= 13).

NB working may appear in table.

Q Answer Mark Comment

2a See the graph P3 P1 for interpreting the data and deciding to draw a graph to represent the data.

P1 for correct process to label axes or communicate the data connections. P1 for drawing an appropriate line of best fit.

59–60 A1 A1 for correctly reading off the value at 2020 in the range 55.5–56.5.

2b C1 C1 for valid comment e.g. Cannot assume it will be a straight-line graph or will have a linear relationship or it is unreliable to predict a date that is so far away from the original.

3 See the pie chart M1 M1 for 360 ÷ 90 or 4 seen or one angle correct in pie chart ± 2° ignore labels, or

2

Apple = 2090 × 360o = 80o

Pear = 3590 × 360o = 140o

Banana = 2590 × 360o = 100o

Peach = 2590 × 360o = 40o

one correct angle calculated.

1

1 Cost price is 15 × 8 = £120

Total number of cakes is 300

0.4 × 300 = 120

Income from these is

120 × 80p = £96.00

Income from the remainder is

180 × 40 = £72.00

P4 P1 for the process to find the total cost of 15 boxes of cupcakes and the total number of cakes bought, for example, 15 × 8 (= 120) and 15 × 20 (= 300)

P1 for the process to calculate the number of cupcakes sold for 80p, for example, 0.4 × 300 (= 120).

P1 for the process to find the income from the cupcakes sold for 80p, for example, 120 × 80 (= £96).

P1 for the process to find the income from the remaining cupcakes at 40p each, for example, (300 – 120) × 40 oe.

Profit = £96 + £72 – £120 = £48 A1 cao

2 3 5 8 410 10 10 5+ = =

15 of the petrol cost is £25

25 × 5

M3 M1 for 3 5 810 10 10

+ = oe (or using

percentages or decimals).

M1 for 15 or 2

10 is £25 oe.

M1 for 25 × 5 oe.

= £125 A1 A1 cao

3 30% = 310

3 6 5110 4 20 20+ = +

= 1120

1 – 1120 9

20=

M2 M1 for adding 310 and 1

4 or 30% and

25%.

M1 for 1 – ‘their fraction or percentage’.

Alternative method

14 = 25%

30% + 25% = 55%

100% – 55% = 45%

45% = 45100

= 920 A1 A1 cao

3b 30% is 90

10% is 30

100% is 300

M2 M1 for finding that 10% is 30.

M1 for finding 100% is 300 or 5% is 15

2

14 of 300 and using a complete method to find 1

4 .

= 75 A1 A1 cao

4 5 91 2 42 5 10 10 10+ = + =

1 – 910 = 1

10

110 is £15

£15 × 10

M3 M1 for 5 9410 10 10+ = oe (or using

percentages or decimals).

M1 for 110 or 10% is £15 oe.

M1 for 15 × 10 oe.

£150 A1

Problem-solving pack

1

Q Answer Mark Comment

1 Hasan – x + 20

Umar – x – 8

Bill – x

x + x + 20 + x – 8 = 72

3x + 12 = 72

3x = 60

x = 20

M3 M1 for any two consistent expressions, for example, x + 20, x – 8, x or x + 28, x, x + 8.

M1 for forming an equation, for example, x + x + 20 + x – 8 = 72.

M1 for solving their equation to calculate x.

Alternative method

Hasan – x + 28

Umar – x

Bill – x + 8

x + 28 + x + x + 8 = 72

3x + 36 = 72

3x = 36

x = 12

Hasan 40

Bill 20

Umar 12

B1 B1 for Hasan 40 or Umar 12 or Bill 20.

2 Jen had x pencils

Parveen had x + 6

Sarah had 2(x + 6) = 2x + 12

Total pencils is

x + x + 6 + 2x + 12 = 42

4x + 18 = 42

4x = 24

x = 6

Parveen has 6 + 6

M4 M1 for strategy to start to problem solve, for example, x, x + 6, 2x + 12.

M1 for process to form an equation in x.

M1 for complete process to find x.

M1 for ‘their x’ + 6.

12 A1 A1 cao

3 8x – 20 > 3x + 10

5x > 30

M2 M1 for forming an inequality in x.

M1 for isolating x and number terms oe.

x > 6 A1