Contents...1 Copyright © Mathematics Mastery 2018 Contents Unit 5: Triangles, quadrilaterals and angles in parallel lines ..... 2

1 Copyright © Mathematics Mastery 2018

Contents Unit 5: Triangles, quadrilaterals and angles in parallel lines ....................................................... 2

5.1: Identify and name triangles .......................................................................................................... 2

5.2: Calculate missing angles in triangles ........................................................................................ 4

5.3: Constructing triangles..................................................................................................................... 6

5.4: Define and recognise properties of quadrilaterals .............................................................. 9

5.5: Constructing quadrilaterals ........................................................................................................ 11

5.6: Angles in parallel lines .................................................................................................................. 12

Unit 6: Length and area including units, parallelograms and trapezia ................................... 23

6.1: Convert between mm², cm² and m² ........................................................................................ 23

6.2: Finding the area and perimeter of composite shapes (rectangles and triangles) . 29

6.3: Area of a parallelograms and trapezia.................................................................................... 35

6.4: Mixed questions including problems in context ................................................................. 39

Unit 7: Percentages...................................................................................................................................... 48

7.1: Convert between fractions decimals and percentages .................................................... 48

7.2: Express one quantity as a percentage of another .............................................................. 53

7.3: Percentage change.......................................................................................................................... 56

7.4: Finding the original value ............................................................................................................ 60

Unit 8: Ratio.................................................................................................................................................... 64

8.1: Understand and interpret ratio in the form a:b, where a and b are whole numbers

........................................................................................................................................................................ 64

8.2: Equivalent ratios ............................................................................................................................. 67

8.3: Comparing quantities by ratio ................................................................................................... 69

8.4: Divide a quantity into a given ratio ......................................................................................... 71

8.5: Find one quantity given the other quantity and its ratio ................................................ 75

8.6: Understand the relationship between ratio and proportion ......................................... 77

8.7: Mixed problems in context.......................................................................................................... 80

8.8: Rate of change – Speed ................................................................................................................. 83


Unit 5: Triangles, quadrilaterals and angles in parallel lines

5.1: Identify and name triangles

1. Name three different triangles in the diagram.

2. In the diagram, 𝐴𝐸 = 𝐵𝐸 = 𝐵𝐷 = 𝐷𝐸 and 𝐴𝐵𝐶 is a straight line.

a) What mathematical name is given to ∆𝐴𝐵𝐸? ……………………………………………

b) What mathematical name is given to ∆𝐵𝐷𝐸? ……………………………………………

c) Give the three-letter name of the scalene triangle in the diagram. …………………….

Concept corner

Shapes are named by labelling each vertex with a capital letter.

Triangle ABC can be written as ∆ 𝐴𝐵𝐶.

Triangle ABC is formed by sides 𝐴𝐵, 𝐵𝐶, and 𝐴𝐶.

Triangles and lines are often named in alphabetical order.

∆ 𝐴𝐵𝐶 is the same as ∆ 𝐵𝐶𝐴.

The angle marked on the diagram is angle 𝐴�̂�𝐵 or ∠𝐴𝐶𝐵.

The middle letter is the vertex where the angle is formed.

A

B

C


3. A triangle has a perimeter of 24 cm.

Give an example of what the side lengths could be if the triangle was:

a) An equilateral triangle ……………………………………………………………………………………

b) An isosceles triangle……………………………………………………………………………………….

c) A scalene triangle. ………………………………………………………………………………………….

d) Sketch each of these triangles.

4. Which of these statements are always, sometimes or never true?

a) Triangles have a right angle. ………………………………………

b) All angles in a triangle are equal in size. ………………………………………

c) Triangles have exactly three lines of symmetry. ………………………………………

d) Triangles have no reflection symmetry. ………………………………………

e) In a triangle none of the sides are the same length.……………………………………..

f) Exactly two sides of triangles are equal in length. ………………………………………


5.2: Calculate missing angles in triangles

1. These triangles have not been drawn accurately.

Work out the size of the angles marked with letters.

𝑏 =………………

𝑝 =……………… 𝑞 =………………

𝑑 = ………………

𝑒 = ……………… 𝑓 =………………

What angle facts

are you using?


2. The following diagrams have not been drawn accurately.

Form equations and solve them to work out the size of the missing angles.

𝑎 = ………………

𝑏 = ………………

𝑐 = ………………


5.3: Constructing triangles

1. Draw the following triangles accurately and measure the sides and angles not given

in the diagram.

Concept corner

In order to construct a triangle, you need to have some information:

1. All 3 sides: Side Side Side (SSS)

2. Two side lengths and the angle between them: Side Angle Side (SAS)

3. Two angles and the length of the side between them: Angle Side Angle (ASA)

Note: AAS does not give a unique triangle. For example:

7 cm 7 cm

56°

56° 72°

72°

A A C

B B

C


2. Draw ∆ CED when ED = 6 cm, EC = 4 cm and ∠𝐶𝐸𝐷 = 40°.

Measure the sides and angles not given in the diagram.


3. Draw an isosceles triangle that has two sides of length 6 cm with the angle between

them being 70°.

a) Measure the length of a base and the corresponding height of the triangle to the

nearest millimetre.

b) What is the area of the triangle?

4. Construct an equilateral triangle of side length 5 cm.

a) Measure the height of the triangle.

b) What is the area of the triangle?


5.4: Define and recognise properties of quadrilaterals

1. Join dots to complete these quadrilaterals.

Rectangle Square Kite

Trapezium Rhombus Parallelogram

Concept corner

Match each word to the corresponding shape(s).

Trapezium Rhombus Rectangle Square Parallelogram Kite


2. Tick where the statement is always true.

Square Rectangle Parallelogram

Trapezium (not an isosceles

trapezium) Rhombus

The diagonals meet at right angles.

The diagonals meet at right angles and are equal in length.

The diagonals bisect each other.

The diagonals are lines of symmetry

of the shape.

3. Always, sometimes or never true?

a) The diagonals of a rhombus bisect each other. ……………………………………..

b) The diagonals of a parallelogram bisect at right angles. .……………………………………..

c) A square has equal length diagonals. .……………………………………..

d) The diagonals of a parallelogram are equal in length. ……………………………………..

Which of these quadrilaterals could you accurately construct using just the

information given? Give reasons for your answers.

8 cm

5.2 cm


5.5: Constructing quadrilaterals

1. Using a ruler and protractor draw the following quadrilaterals accurately. Measure

the sides and angles not given in the diagram.

3 cm


5.6: Angles in parallel lines

5.2 cm

Concept corner

Lines which never meet and are always the same distance

apart are …………………………………………..

When two .............................................. ……………………….. are

crossed by a ……………………………………… line, these sets of angles are formed:

Transversal

Parallel lines

Parallel

Corresponding angles

are equal

𝑥 + 𝑦 = 180°

Alternate angles are equal

Allied angles add up to 180°


1. Work out the size of the angle marked with letters.

Give reasons for each answer.

a)

𝑎 = ……………… reason ……………………………………………………………..

b)

𝑔 = ……………… reason ……………………………………………………………..

c)

𝑝 = ……………… reason ……………………………………………………………..

d)

𝑚 = ……………… reason ……………………………………………………………..

m

Diagrams not drawn

accurately


2. Calculate all unknown angles:

3. Write down an angle that:

a) …………….. corresponds to 𝑎

b) …………….. is alternate to 𝑥

c) …………….. is allied with 𝑐

d) …………….. is vertically opposite to 𝑦

Diagram not drawn

accurately


4. True or false?

Give reasons for your answers when true.

a) ∠ 𝑎 and ∠ ℎ are equal angles…………….…………………………………..……………………………

b) ∠ 𝑎 and ∠ 𝑐 are equal angles…………….…………………………………..……………………………

c) ∠ 𝑎 and ∠ 𝑤 are equal angles…………….…………...………………………..…………………………

d) ∠ 𝑎 and ∠ 𝑑 sum to180°……..…………….…………………………………..……………………………

e) ∠ 𝑑 and ∠ ℎ sum to180°……..…………….…………………………………..……………………………

f) ∠ 𝑥 and ∠ ℎ sum to180°……..…………….…………………………………..……………………………

g) ∠ 𝑑 and ∠ 𝑒 are equal angles…………….…………………………………..……………………………

h) ∠ ℎ and ∠ 𝑐 are equal angles…………….…………………………………..……………………………

i) ∠ ℎ and ∠ 𝑤 are equal angles…………….…………...………………………..…………………………


5. Work out the size of the angle marked with letters.

Give reasons for each answer.

a)

𝑎 = …………… reason ………………………….…………………………………………….

𝑏 = …………… reason ………………………….…………………………………………….

b)

𝑟= …………… reason ………………………….…………………………………………….

𝑠 = …………… reason ………………………….…………………………………………….

c)

𝑔 = …………… reason ………………………….…………………………………………….

ℎ = …………… reason ………………………….…………………………………………….

Diagrams not drawn

accurately


6. Work out the sizes of the angles marked with letters.

Label the diagram clearly and show all necessary steps, giving reasons

for your answers.

a)

𝑧 =………………

b)

𝑥 =………………

c)

𝑦 = ………………

Diagrams not drawn

accurately


d)

𝑏 =…………… 𝑐 = ……………

e)

𝑓 = …………… 𝑔 =……………

f)

𝑥 = ………………

Diagrams not drawn

accurately


7.

𝐴𝐵 is parallel to 𝐶𝐷.

a) Work out the size of angle 𝑥. Give a reason for your answer.

b) Work out the size of angle 𝑦. Give a reason for your answer.

8. In the diagram the line 𝐴𝐵 is parallel to line 𝐶𝐷.

a) Work out the size of angle 𝑎. Give a reason for your answer.

b) Work out the size of angle 𝑏. Give a reason for your answer.

c) Work out the size of angle 𝑐. Give a reason for your answer.

Diagrams not drawn

accurately


9. Find the sizes of angles 𝑝, 𝑞 and 𝑟. Give reasons for your answers.

𝑝 =…………….. reason ………………………….………………………….…………………………………

𝑞 =…………….. reason ………………………….…………………………………………………………….

𝑟 =…………….. reason ………………………….…………………………………………………………….

Diagram not drawn

accurately


10. Work out the sizes of the angles marked with letters.

a)

𝑥 =……………..

b)

𝑎 =…………….. 𝑏.……………..

Diagrams not drawn

accurately


Reflections

This space is for you to write your reflections on the whole unit on draining accurate triangles

and quadrilaterals and finding unknown angles in parallel lines.

You may wish to write about:

Things you’ve learnt

Things you found difficult

Other areas of maths you used in this topic

Topics you need to revisit/revise in the future


Unit 6: Length and area including units, parallelograms and trapezia

6.1: Convert between mm², cm² and m²

1. John has worked out the perimeter of the shape below. What has he done wrong?

Correct his work.

John’s work

2. Calculate the perimeter of this rectangle.

Concept corner

Length

1 cm = 10 mm

1 m = 100 centimetres (cm)

1 m = 1000 millimetres (mm)

Fill in the gaps

100 m = ……………………… cm 3500 mm = ………………………m

100 cm = ……………………… mm 6.5 m = ……………………… mm

2.5 cm

15 mm

636 cm 1.9 m

Centi means hundredth, 1

100.

So, a centimetre is one hundredth of a metre.

The perimeter is

15 mm + 2.5 cm + 15 mm + 2.5 cm =

30 + 5 = 35 mm

Diagrams not drawn

accurately


3. Draw five different rectangles with the area of 12 cm².

Work out the perimeter of each of your rectangles.


4. Complete the tables for the squares below.

Square A

Side length Area

…….……… mm …….…… mm2

…….……… cm …….…… cm2

…….……… m …….…… m2

Square B

Side length Area

……..……… mm ………..… mm2

…….……… cm ………..… cm2

…….……… m ………..… m2

Square C

Side length Area

…….……… mm …….…… mm2

…….……… cm …….…… cm2

…….……… m …….…… m2

Concept corner

1 m = 100 cm

1 m2 = 100 cm × 100 cm = 10 000 cm²

How many mm² are equal to equal 1 cm²?

……………………………………………………………………………………………………………………………

Area = 10 000 cm²

100 cm

100 cm

A

B

C

20 cm

0.6 m

35 mm


5. Circle the correct answer.

Express in cm²

a) 4 m² 400 cm² 4000 cm² 40 000 cm²

b) 0.5 m² 50 cm² 5000 cm² 50 000 cm²

c) 300 mm² 0.3 cm² 3 cm² 30 cm²

Express in mm²

d) 16 cm² 160 mm² 1600 mm² 16 000 mm²

e) 9.5 cm² 95 mm² 950 mm² 9500 mm²

Express in m²

f) 760 cm² 0.076 m² 0.0076 m² 0.76 m²

6. True or false?

a) To convert mm² to cm², divide by 100 ………………………………………………………………

b) To convert cm² to m², divide by 100 …………………………………………………………………

c) To convert mm² to m², multiply by 1 000 000…………………………………………………….


7. Complete the table below:

mm² cm² m²

0.4

640

12500

𝑦

𝑥

8. A rectangular rug measures 6 m by 4 m.

What is the total cost of cleaning this rug at £1.20 per square metre?

9. A roll of wallpaper is 10 m long and 50 cm wide.

Calculate its area in square metres.


10. A school hall measuring 10 m by 15 m is to be covered with square floor tiles with a

side length of 50 cm.

How many tiles are required to cover the school hall?

11. Put the correct symbol, either =, < or >, in each circle:

a) 75 cm² 7.5 m²

b) 35 m² 350 000 cm²

c) 125 00 cm² 12.5 m²

d) 0.81 m² 81 cm²


6.2: Finding the area and perimeter of composite shapes (rectangles and

triangles)

1.

a) Calculate the perimeter and area of each shape.

Total area = ……………………….m²

Total area = ……………………….cm²

Total perimeter = …………………….m

Total perimeter = ……………………cm

Area A =

Area B =

Total area = ……………………….m²

Total area = ……………………….cm²

Total perimeter = ……………………….m

Total perimeter = ……………………….cm

700 cm

500 cm

Diagrams not drawn

accurately


b) Calculate the shaded area of each of these shapes.

Total shaded area = ……………………….m²

Total shaded area = ……………………….cm²

Total shaded area = ……………………….m²

Total shaded area = ……………………….cm²

Diagrams not drawn

accurately


2. Calculate the shaded area of the shapes below.

Express your answer in m² and cm².

a)

……………………….cm²

……………………….m²

b)

[Hint: Area of the large triangle ― Area of the small triangle]

……………………….cm²

……………………….m²

Diagrams not drawn

accurately


3. A rectangular flag has two colours. The measurements are as shown in the diagram.

The line AB is drawn from the midpoints of either side.

a) Calculate the area of the whole flag.

b) Calculate the shaded area of the flag.

c) Calculate the white area of the flag.

d) The material to make the flag cost £10.80 per m².

How much will the material for the flag cost?

A

B Diagram not drawn

accurately


4. The diagram shows how the material required for one side of the tent is cut out.

a) Work out the area of the material shown if:

𝑥 = 2 m, 𝑦 = 2 m and 𝑧 = 3.2 m.

b) Calculate the area if 𝑥 = 2 m, 𝑦 = 160 cm and 𝑧 = 340 cm.


5. Simon has made some mistakes in each of the homework questions below. Correct

each question.

Correct Simon’s homework

Calculate the area of each of the shapes below:

1)

2)

3)

4)

6 × 4 × 6 × 4 = 576 cm²

4.8 × 5.2 = 24.96 cm²

4.2 × 7

6= 4.9 cm²

2.5 × 800 = 20 000 cm²

3.5 × 150 = 525 cm²

20 000 + 525 = 20 525 cm²


6.3: Area of a parallelograms and trapezia

1. Find the area of each shape below.

a)

b)

Concept corner

DX is perpendicular to BC.

AD is not perpendicular to BC.

Area of a parallelogram = base × perpendicular height

Area of a trapezium = 1

2 (𝑎 + 𝑏) × ℎ

Note: the height, h, must be perpendicular to

the parallel sides a and b.

B X C

Diagrams not drawn

accurately


2. For each of the following shapes, draw the perpendicular height with reference to

the given base. You must indicate the right-angle where necessary.

3. Work out the missing lengths in each shape below:

a)

b)

Area = 9 m²

Area = 21 cm²

Diagrams not drawn

accurately


4. The table below shows the dimensions and area of different parallelograms.

Complete the table.

Base Height Area

a) 12 cm 7 cm

b) 6 m 42 m²

c) 6.8 m 129.2 3 m²

5. Work out the missing lengths in each shape below.

6. The table below shows the dimensions and area of different trapezia.

Complete the table.

Parallel side 1 Parallel side 2 Height Area

a) 9 cm 13 cm 7 cm

b) 23 m 12 m 210 m²

c) 5.8 m 3.3 m 39.6 m²

Area = 72 cm²

Diagrams not drawn

accurately


7. Which of the two shapes below has the larger area?

8. Correct the student’s homework below:

Homework

a.

b.

9 × 5 = 45 cm²

6 × 82 × 2 = 48 cm²

Diagrams not drawn

accurately


6.4: Mixed questions including problems in context.

1. The diagram shown is the end wall of a wooden hut,

a) Work out the area of this end of the hut.

The other end of the hut is identical. The sides are made up of two rectangles of

length 4 m.

b) Work out the area of each side of the hut.

Calculate the total area of the walls of the hut.

Diagram not

drawn accurately


2. Calculate the area of the following shapes.

Diagrams not drawn

accurately


3. Work out the total area of the shaded parallelograms.

4. The rules for an art competition state that the area of the canvas must be exactly

100 cm².

a) Laura uses a square canvas. What will the length of one side of this canvas be?

b) Sketch and label the dimensions of a possible parallelogram-shaped canvas.

c) Sketch and label the dimensions of a possible trapezium-shaped canvas.

Diagram not

drawn accurately


5. Find the area of the parallelogram if:

a) ℎ = 2 cm

b) ℎ = 4 cm

c) ℎ = 5 cm

d) Can ℎ be longer than 6 cm? Why?

Diagram not

drawn accurately


6. The area of the trapezium is equal to the area of the parallelogram.

Work out the missing value 𝑥.

7. The figure shows a rectangle 𝐴𝐶𝐷𝐹 that has an area of 50 cm².

If 𝐴𝐵 = 𝐵𝐶 = 𝐶𝐷 = 𝐷𝐸 = 𝐸𝐹 = 𝐴𝐹, find

a) The area of ABEF,

b) The length AF,

c) The area of the parallelogram BCEF.

cm

Diagrams not drawn

accurately


8. Work out the shaded area.

9. Barry wants to lay turf in an area in his garden (the shaded region). A 30 m2 roll of

turf costs £25, how much will it cost him to grass the shaded area?

Diagram not

drawn accurately


10. A picture frame is made by joining 4 trapezium-shaped pieces of wood together.

a) Find the area of each trapezium and the total area of the frame.

b) Describe a different way to work out the area of the frame.

17 cm

Diagram not

drawn accurately


11. In the figure, 𝐴𝐵𝐹𝐺 and 𝐶𝐷𝐸𝐹 are two parallelograms such that the sum of their

areas is 1554 m². If 𝐴𝐵 = 𝐶𝐷 = 𝐸𝐹 = 𝐹𝐺 = 1

2𝐵𝐶, work out the area of the

shaded region.

What do you notice?

Diagram not

drawn accurately


Reflections

This space is for you to write your reflections on length and area including units,

parallelograms and trapezia. You may wish to write about:






Unit 7: Percentages

7.1: Convert between fractions decimals and percentages

1. Circle the odd one out.

a) 30% 3

100

0.3 3

10

b) 0.6 60% 3

5

6

100

c) 6% 6

100

0.6 0.06

Concept corner

Per means ‘out of’ and cent means ‘100’. Therefore, percent means ‘out of 100’.

The diagram shows a hundred square which is a large

square divided into 100 equal smaller squares.

43 of the squares are shaded.

43

100 of the hundred square is shaded.

43

100 = 0.43 = 43 %.

So 43% of the hundred square is shaded.

For each diagram, state what percentage of one hundred square is shaded.

Diagram 1 Diagram 2 Diagram 3

The shaded region of these 100-square could be represented

as a vulgar fraction, a decimal fraction or a percentage.

Percentages over 100 can represented as a mixed or

improper fraction, a decimal fraction or a percentage.

…….. % …….. %

…

…….. %

…

150% = 1.5 = 11

2

20% = 0.2 = 1

5


2. Mary scored 84 out of 120 in a test.

a) Express this as a fraction.

b) Write this as a decimal

c) Write her score as a percentage?

3. Change each of these marks to a percentage.

a)

Science: 22

25

Art: 24

30

History: 54

60

Maths: 34

40

b) Put these marks in descending order.

4. Write in order of size, lowest first:

a) 2

3, 0.6,

3

4, 55%...............................……………………………………………………………………………

b) 42%, 11

25, 0.43,

9

20……………………………………………………………………………………………

c) 21

80, 27%,

57

200, 0.280………………………………………………………………………………………


5. Which diagram has the greater percentage shaded? Give reasons for your answer.

A

B

6. Put the correct symbol, either =, < or >, in the circle:

a) 2.9 14

5

b) 300% 3

c) 17

4

420%

d) 2

5+

1

3−

1

8

0.15 × 4


7. Complete the diagram below, using the grey box as a starting point.

8. Complete the diagram below.

100% = £36

10% = 20% =

80% =

5% =

15% =

300% = 600% = 30% =

3% = 60% =

100% = £152

10% = 70% = 80% =

300% = 600% = 30% =

15% = 60% = 900% =

450% = 45% =


9. Put the correct symbol, either =, < or >, in the circle:

a) 20% of £80 80% of £24

b) 30% of £60 75% of £22

c) 45% of £25 5% of £200

d) 21% of £212 95% of £52

Concept corner

Work out 30% of £70

Why do all the calculations have the same answer?

0.3 × 70 = £21

3

10 × 70 = £21

7

10 × 70 = £21

70

100 × 30 = £21

30

100 × 70 = £21


7.2: Express one quantity as a percentage of another

1. Circle the correct answer:

Work out in your head and then check your answers with a calculator.

a) What is 20 as a percentage of 50? 20% 40% 80%

b) What is 48 as a percentage of 200? 24% 48% 96%

c) What is 72 pence as a percentage of £2? 36% 72% 144%

d) What is 150g as a percentage of 1 kg? 1.5% 15% 150%

e) What is 60° as a percentage of 360°? 16.7% 30% 60%

f) What is 335 cm as a percentage of 5 m? 3.35% 33.5% 67%

g) What is 4 months as a percentage of 1 year? 33𝟏

𝟑% 40% 300%

Concept corner

Express one quantity as a percentage of another.

Eric scored 24 out of 30 in a Science test and 29 out of 40 in a maths test.

In which subject did he achieve a higher percentage score?

Science

maths

Eric achieved a higher percentage score of ……… % in …………………………

𝟐𝟒

𝟑𝟎× 𝟏𝟎𝟎 =

𝟐𝟗

𝟒𝟎× 𝟏𝟎𝟎 =

……… %

……… %


2. A bar of chocolate has 32 squares. Laura eats 12 squares.

What percentage of the bar does she eat?

3. A new car costs £12 500. The car dealer gives a discount of £18 75.

Work out the percentage discount.

4. I can buy a scooter for one cash payment of £227, or pay a deposit of 20% and then

six equal monthly payments of £32.

How much extra will I pay in the second option?

5. A lady buys a car for £2500 and sells it for £1800.

Work out her percentage loss.

6. An elastic band which is 72 cm long is stretched to 90 cm.

Work out the percentage increase in its length.

7. A stereo system has been reduced from £320 to £272. What is the percentage

reduction?


8. The area of the parallelogram is 30% of the area of the trapezium.

Work out the missing height of the parallelogram.

9. Nate earns £1750 each month.

In one month he spent 20% of his salary on rent, £580 on food and £850 on other

expenses.

a) How much did he overspend by?

b) Express the amount he overspent as a percentage of his monthly salary,

giving your answer correct to 1 decimal place.

Diagrams not drawn

accurately


10. There are 800 pages in a book.

Jack reads 150 pages of the book on Monday and 40% of the remaining pages on a

Tuesday.

Express the number of pages remaining on Wednesday as a percentage of the total

number of pages in the book, giving your answer correct 1 decimal place.

7.3: Percentage change

Concept corner - Percentage change

Original amount

20% increase

20% decrease So, 120% of 60 kg = 1.2 × 60 = 72 kg. 80% of 60 kg = 0.8 × 60 = 48 kg.

100%

100% 20%

100% = 1

120%

80%

120% = 120

100 = 1.2

100%

80% = 80

100 = 0.8


1. Complete the following sentences:

a) If something increases by 100%, it ……………………………………………..

b) If something increases by 500%, it increases by ………………………… times itself,

and is then …………………….. times its original size.

c) I have …………………… left after my £10 decreased by 100%.

2. If a number is increased by 35%, what percentage is the new number of the original

number?

3. If a number is decreased by 35%, what percentage is the new number of the original

number?

4. Match each statement to the correct multiplier:

Increase by 70% 0.3

Decrease by 30% 1.3

Increase by 30% 0.7

Decrease by 70% 1.7


5. Each of these people get a pay rise. Work out how much they earn now.

£200 per week

Pay rise of 7.5%

£1240 per month Pay rise of 4.5%

£26 000 per annum

Pay rise of 6%

6. Match the calculations which are of equal value:

Increase £60 by 20% Decrease £362.50 by 40%

Increase £110 by 25% Decrease £115 by 20%




7. A shop has a sale, for each item in the sale work out the sale price.

8. Tim says:

Show calculations to explain whether Tim is correct.

Dress £68 SALE

20% off

Bed £350 SALE

40% off

Table £480 SALE

30% off

I add 30% to a value.

I then take away 30% of this new

value.

I should then get my original value.


7.4: Finding the original value

1. A shop sells T-shirts with a 20% discount.

Jan buys a T-shirt and pays £10.

How much does the T-shirt normally cost?

2. A coat is on sale at £55.25, which is 85% of its original price.

What was its original price?

Concept corner

Jay receives a 25% pay rise.

His new wage is £175 per week.

What was Jay’s wage before his pay rise?

Compare the different calculation strategies. What do you notice?

on

………………………………………………………………………………………………………………………………

125 %

£175

100 %

?

Each equal part: 𝟏𝟕𝟓 ÷ 𝟓 = 𝟑𝟓

Original wage: 𝟑𝟓 × 𝟒 = £𝟏𝟒𝟎

Original

wage

£175 per

week

× 1.25

÷ 1.25

𝟏𝟕𝟓 ÷ 𝟏. 𝟐𝟓 = £𝟏𝟒𝟎


3. Larry gets a 5% wage rise.

His new wage is £252 per week.

What was Larry’s wage before his wage rise?

4. If 10% is deducted from a restaurant bill, £40.95 remains to be paid.

How much is the original bill?

5. I bought a bicycle in a sale and saved £49. The label said that it was a ‘20%

reduction’. What was the original price of the bicycle?

6. A football team plays one game each month.

12 500 people attended the game in June.

This was an increase of 25% on the previous month.

How many people attended the football match in May?

7. Neil sells his bike to Alex.

Alex sells it to John for £194.40.

Both Neil and Alex makes a 10% loss.

How much did Neil pay for the bike?


8. Dan sells his Smartphone to Katy and makes a 15% profit.

Katy then sells the Smartphone to Ben for £195.50.

Katy makes a 15% loss.

How much did Dan pay for the Smart phone?

Explain why it’s not £195.50.

9. Circle the correct working out for each of the following:

a) Jenny earns £88 a day. She has been told that she will receive a 15% pay rise.

How much will she earn now?

88 × 0.15 88 ÷ 0.85 88 × 1.15 88 × 0.85

b) Clive earns £270 each week. He donates 12% of his wages to charity. How much

money does Clive donate to charity each month?

270 × 1.12 270 ÷ 0.88 270 × 0.12 270 × 0.88

c) A coat costs £90 in a shop. The shop has a sale and reduces the price of the coat

by 10%. How much is the coat in the sale?

90 × 0.1 90 × 0.9 90 ÷ 1.1 90 ÷ 0.1

d) After a 20% increase, Ian earns £72 a day. What was his original wage?

72 × 0.2 72 ÷ 0.8 72 ÷ 1.2 72 × 1.2


Reflections

This space is for you to write your reflections on percentages. You may wish to write

about:






Unit 8: Ratio

8.1: Understand and interpret ratio in the form a:b, where a and b are whole

numbers

1. Fill in the gaps.

a) For every 5 black beads there are white beads.

b) The ratio of black beads to white beads is 5 : …….


a) For every ………… black beads there is 1 white bead.

b) The ratio of black beads to white beads is ……. : 1.

Concept corner

The ratio of black to white is 3 : 2 in the string of 10 beads.

This means that for every three black beads, there are two white beads.

Ratios compare part to part relationships.


3.

The ratio of white beads to black beads is :

4. Colour the beads to match the given ratio.

a) The ratio of black to white is 3 : 2.

b) The ratio of white to black is 1 : 1.

c) The ratio of black to white is 4 : 1.

d) The ratio of black to white is 1 : 3.

5. Shade the rectangles in the given ratios:

a) 1 : 3

b) 2 : 1

c) 1 : 1

d) 1 : 5

e) 8 : 4

f) 5 : 7

g) 1 : 11

h) 6 : 2

i) 3 : 3


6. The ratio of black beads to white beads is 3 : 1.

a) How many more white beads do you need to add to make the ratio of black

beads to white beads 3 : 2?

………… white bead(s)

b) From the original bead string how many more black and white beads do you

need to add to make the ratio of black beads to white beads 2 : 3?

………… black bead(s) .………… white bead(s)

c) From the original bead string how many more white beads do you need to add

to make the ratio of black to white beads 1 : 3?

………… white bead(s)

7. True or false?

a) The ratio of cats to dogs is 4 : 6. …………………………………….

b) The ratio of rabbits to dogs is 6 : 2. …………………………………….

c) Half of the animals are dogs. …………………………………….

d) The ratio of cats to dogs to rabbits is 6 : 2 : 4 …………………………………….

e) There are a third as many rabbits as there are dogs…………………………………

Dog Cat Rabbit Dog Dog Dog Dog Dog Cat Cat Cat Rabbit


8.2: Equivalent ratios

1. Match the equivalent ratios:

Concept corner

Ratios are used only to compare quantities.

They do not give information about actual values.

Some possible number of beads in the necklace are shown in the table.

The ratios 2 : 1, 4 : 2, 6 : 3, … are different forms of the same ratio.

They are equivalent ratios.

For example:

A necklace is made of blue and

yellow beads in the ratio 2 : 1.

This gives no information about the actual number of beads in the necklace.

Blue beads

Yellow beads

Total beads

2 1 3

4 2 6

10 5 15

30 15 45

1 : 3 9 : 6

3 : 3 3 : 9

3 : 2 9 : 12

3 : 4 6 : 6


2. Fill in the blanks.

a)

b)

c)

1 : 4

2 : ………..

……….. : 100

……….. : 36

3 : ………..

100 : ………..

10 : ………..

5 : 3

15 : ………..

……….. : 90

……….. : 30

20 : ………..

100 : ………..

10 : ………..

……….. : 2

12 : 8

15 : ………..

……….. : 90

……….. : 30

21 : ………..

30 : ………..


3. Odd one out?

Circle the ratio which is not equivalent to the others.

a) 15 : 10, 5 : 2, 30 : 20, 3 : 2, 9 : 6

b) 9 : 15, 3 : 5, 6 : 10, 3 : 4, 18 : 30

c) 7 : 2, 14 : 4, 7a : 2a, 28 : 7, 21 : 6

8.3: Comparing quantities by ratio

4. Write each of these ratios in its simplest form.

a) 90 p : 33 p

b) 165 g: 15 g

c) 35 cm : 280 cm

d) 414 mm: 162 mm

e) 12 cm : 9 cm : 3 cm

Concept corner

A ratio with whole numbers with no common factor is in its simplest form.

All quantities in a ratio must have the same units before the ratio can be simplified.

For example,

£3.50 : 50 p ≡ 350 p : 50 p ≡ 350 : 50 ≡ 7 : 1


5. Write each of these ratios in its simplest form.

a) £2.40 : 20 p

b) 20 p : £4.50

c) 20 seconds : 5 minutes

d) 6 m : 280 cm

e) 1

2 minute : 45 seconds

f) 500 mm : 75 cm : 2.5 m

6. A necklace has 20 pink beads and 35 purple beads.

What is the simplest form of the ratio of pink beads to purple beads on the necklace?

7. Charlie spends £4.90 on Monday and 70 p on Tuesday.

Work out the ratio of money spent on Monday to the money spent on Tuesday.

8. What is the ratio of lengths of the sides of this triangle?

Leave your answer in its simplest form.

2.1 cm

1.2 cm

2.7 cm


9. Max walks 2 km to school in 40 minutes and Dave cycles 5 km to school in 15

minutes. What is the ratio of:

a) Max’s distance to Dave’s distance

b) Max’s time to Dave’s time.

10. If 𝑥 ∶ 𝑦 = 2 ∶ 3, find the ratio 6𝑥 ∶ 2𝑦 in simplest form.

11. If 𝑎 ∶ 𝑏 = 3 : 8 and 𝑎 ∶ 𝑐 = 2 ∶ 3 ,

Work out 𝑎 ∶ 𝑏 ∶ 𝑐.

8.4: Divide a quantity into a given ratio

Concept corner

John and Gemma are sharing some sweets in the ratio 2 : 3.

Find John’s share if they share a total of 60 sweets.

1 share = 60 ÷ 5 = 12 sweets

So John’s share = 2 × 12 = 24 sweets

? ?

60 sweets

John Gemma


1. Tom spent his savings of £80 on shoes and clothes in the ratio 1 : 3.

How much did he spend on clothes?

2. Coffee is made from two types of beans, from Java and Colombia, in the ratio 2 : 3.

Complete the bar model below to work out how much of each type of bean will be

needed to make 1 kg of coffee.

11. Share the following amount in the given ratios:

£80

Clothes Shoes

………. kg

………………

.

………….

£84

1 : 6

2 : 5

3: 4

£84

£84

£84


12. Share the following amounts in the given ratio:

13. 28 children are at a party.

They are in the ratio of 3 girls to 4 boys.

How many more boys are there than girls?

14. Nate and Henry share a bag of 48 sweets in the ratio 7 : 5.

How many sweets does each person get?

What’s the difference between the larger share and the smaller share?

[Draw a bar model to help.]

£60

2 : 3

£85

£230

?


15. Pastry is made from flour and fat in the ratio 2 : 1.

How much flour will make 270 g of pastry?

16. The ratio of men to women to children visiting the Tower of London one day was

4 : 5 : 6.

If 975 people visited the Tower of London, find out how many more children there

were then men.

17. The angles 𝑥, 𝑦 and 𝑧 in a triangle are in the ratio 5 : 1 : 3.

Work out the size of angles 𝑥, 𝑦 and 𝑧.

18. The angle 𝑤, 𝑥, 𝑦 and 𝑧 is in a quadrilateral are in the ratio 1 : 2 : 4 : 3.

Work out the sizes of angles 𝑤, 𝑥, 𝑦 and 𝑧.

Not drawn to scale

Not drawn to scale


8.5: Find one quantity given the other quantity and its ratio

1. Sugar and butter are mixed in the ratio of 2 : 3.

How much sugar is used with 900g of butter?

2. The ratio of girls to boys in a school is 4 : 5.

There are 100 girls.

How many boys are there?

Concept corner

John and Gemma are sharing some sweets in the ratio 2 : 3.

Find John’s share if Gemma’s share is 45 sweets.

1 share = 45 ÷ 3 = 15 sweets

So John’s share = 2 × 15 = 30 sweets

? ? 15 15 15

? 45

John Gemma

? 900g

sugar butter

? 100

girls boys


3. Lottery winnings were divided in the ratio 3 : 5.

Wayne got the smaller amount of £1500.

How much in total were the lottery winnings?

4. Compost is made from loam, peat and sand, in the ratio 7 : 3 : 2 respectively.

A gardener used 1.2 litres of sand to make some compost.

a) How much loam did she use?

b) How much peat?

5. Pancake batter is made from milk, fat, flour and sugar in the ratio 4 : 1 : 4 : 2.

A chef used 80 grams of sugar.

a) Draw a bar model to represent this problem.

b) How much fat did he use?

c) How much milk did he use?

1.2 litres

loam peat sand

? ?

? £1500

Wayne Pat


8.6: Understand the relationship between ratio and proportion


a) 1 in every …… beads is white.

b) The fraction of white beads is …

c) The fraction of black beads is …………

2.

a) What fraction of the beads are black?

b) What fraction of the beads are white?

c) What is the ratio of black to white beads?

d) In another string of beads 1

6 are black. What is the ratio of white to black beads?

D D D

Concept corner

The ratio of black to white beads is one part black to four parts white or 1 : 4.

This proportion of black to the whole string is 1 out of 5, 1

5 or 20%

Proportion compares part to whole relationships.


3. A bag contains red and green sweets in the ratio of 1 : 2.

What fraction of the sweets are red?

4. The ratio of white to black beads is 3 : 7.

a) What percentage of the beads are white? …………………………

b) What percentage of the beads are black? …………………………

c) In another string of beads 40% are white.

What is the ratio of white to black beads? …………………………

5. The ratio of black to white beads is 4 : 1.

a) What fraction of the beads are black? …………………………

b) What percentage of the beads are white?

..………………………

6. In a class the ratio of the number of boys to girls is 2 : 3.

Work out the percentage of boys in the class.

..………………………


7. Amy is 11 years old and Lance is 14 years old.

They share some chocolate in the ratio of their ages.

What percentage of the chocolate does Amy get?

..………………………

8. In a game, Nick scored 6, Sam scored 8, and Alice scored 10.

a) What is the ratio of their scores?

b) What proportion did Nick score represent?

9. In a survey, students were asked if they owned a mobile phone

Results: 5

8 of the students said ‘Yes’.

3

8 of the students said ‘No’.

66 more students said ‘Yes’ than said ‘No’.

Altogether, how many students were in the survey?


8.7: Mixed problems in context

1. Complete the table below for the cupcake recipe:

a)

18 cupcakes 9 cupcakes 27 cupcakes 45 cupcakes

Butter 100g

Sugar 200g

Flour 75g

Eggs 2

b) Emma has plenty of butter, sugar and eggs, but only one kilogram of flour.

How many cupcakes can she make?

2. Here are the ingredients needed to make 12 biscuits.

a) Jan makes some biscuits; she uses 450 g of flour.

How many biscuits does Jan make?

b) May has 500 g of sugar, 1000 g of butter, 1000 g of flour and 500 ml of milk.

Work out the greatest number of biscuits May can make.

3.

50 g of sugar

200 g of butter

200 g of flour

10 ml of milk


4. A meal in a restaurant costs the same for each person.

For 4 people the total cost is £88

What is the total cost for 5 people?

5. In this design, the ratio of grey to black is 3 : 1

a) What percentage of the design is grey?

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

b) In this design, 40% is grey and the rest is black.

What is the ratio of grey to black?

Write your ratio in its simplest form.

............. : .............

……………. : …………….

6. 2 parts of blue paint mixed with 3 parts of red paint makes purple.

If you have 50 ml of blue paint and 100 ml of red.

What is the maximum amount of purple you can make?

£88

?

blue red


7. True or false?

a) Ten counters are shared between the sets A and B in the ratio 6 : 7. ………………………

b) The ratios of the counters in the intersection to set A is 1 : 2………………….………………

c) The number of counters in the intersection as a fraction of the total number of

counters is 3

10.

…………………………..

d) The ratio of counters in set B to those not in set B is 7 : 6 ………………………………………

e) The number of counters in set A as a fraction of the total number of counters is 3

10.

…………………………..


8.8: Rate of change – Speed

Do not use a calculator.

1. Find the missing values in the table below.

Total distance travelled

Total time taken Average speed

100 km 4 hours

120 km 3 hours

40 km 10 km/h

4 hours 50 km/h

2 hours 75 km/h

30 km 60 km/h

2. Jade runs at 7 km/h for 1

2 hour.

How far does she run?

Concept corner

Speed is a measure of how fast something is travelling. Speed involves two other

measures, distance and time.

It can be worked out using this formula.

𝑆𝑝𝑒𝑒𝑑 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑇𝑖𝑚𝑒

In most situations the idea of average speed is used.

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 = 𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑

𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛


3. Dan cycles 18 miles in 11

2 hours.

What is the average speed in miles per hour?

4. A train travels 75 miles at an average speed of 25 mph.

How long does the journey take?

5. Find the missing values in the table below.

Total distance travelled

Total time taken Average speed

20 m 2.5 seconds

210 m 30 m/min

0.5 seconds 20 cm/s

6. On the first part of the journey a car travels 160 km in 3 hours.

On the second part of the journey the car travels 140 km in 2 hours.

a) What is the total distance travelled on the journey?

b) What is the total time taken on the journey?

c) What is the average speed of the car over the whole journey?


Reflections

This space is for you to write your reflections on ratio. You may wish to write about:






Notes


Notes


Notes

Contents...1 Copyright © Mathematics Mastery 2018 Contents Unit 5: Triangles, quadrilaterals and angles in parallel lines ..... 2

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