scholar study guide national 5 mathematics course

SCHOLAR Study Guide

National 5 Mathematics

Course MaterialsTopic 7: Similar shapes

Authored by:Margaret Ferguson

Reviewed by:Jillian Hornby

Previously authored by:Eddie Mullan

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2016 by Heriot-Watt University SCHOLAR.

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SCHOLAR Study Guide Course Materials Topic 7: National 5 Mathematics

1. National 5 Mathematics Course Code: C747 75

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1

Topic 1

Similar shapes

Contents

7.1 Length scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

7.1.1 Similar triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7.2 Area scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.3 Volume scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.4 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.5 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 TOPIC 1. SIMILAR SHAPES

Learning objectives

By the end of this topic, you should be able to:

• identify similar shapes;

• identify enlargement and reduction scale factors;

• use a scale factor to find an unknown length;

• work with similar triangles;

• identify an area scale factor;

• use an area scale factor to find an unknown area;

• identify a volume scale factor;

• use a volume scale factor to calculate an unknown volume.

© HERIOT-WATT UNIVERSITY

TOPIC 1. SIMILAR SHAPES 3

1.1 Length scale factors

Similar shapes

Shapes are said to be congruent if they are identical in every way.

These prints of the painting by Claude Monet are congruent because one is an exactcopy of the other.

Shapes are said to be similar if one is an enlargement or reduction of the other.

These prints of the Mona Lisa by Leonardo da Vinci are similar because the one on theright is a reduction of the one on the left. One is a scaled version of the other.

Shapes are also said to be similar if one is a mirror image of the other.

These dogs are similar because the one on the right is a reflection and enlargement ofthe one the left.



Scale factors

Go onlineThese rectangles are congruent because pairs of corresponding sides are equal andcorresponding angles are equal.

To identify the scale factor choose a pair of corresponding sides and write down theratio.

55 = 1 5

5 = 1 33 = 1 3

3 = 1 → Congruent shapes have a length scale factor of1.


1020 = 1

236 = 1

2 → A length scale factor less than 1 is a reduction scale factor.

These rectangles are similar. The rectangle on the right is a reduction of the one on theleft.


84 = 2 6

3 = 2 105 = 2 → A length scale factor greater than 1 is an enlargement

scale factor.

These triangles are similar. The triangle on the right is an enlargement of the one onthe left.

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Examples

1. Using scale factors

Problem:

These shapes are similar.

Calculate the breadth x.

Solution:

The side that we are looking for is on the larger shape. We need an enlargement scalefactor.

We know the height of each heart so we have one pair of corresponding sides.

There are two possible ways to display the ratios, 728 = 1

4 or 287 = 4.

An enlargement scale factor is greater than 1.

The enlargement scale factor = 287 = 4.

All we have to do is scale the corresponding side of x.

x = 4 × 6 = 24 cm

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2. Using scale factors

Problem:

These shapes are similar.

Calculate the height y.

Solution:

The side that we are looking for is on the smaller shape. We need a reduction scalefactor.



We have one pair of corresponding sides and two possible ratios, 1·52 = 3

4 or 21·5 = 4

3 .

A reduction scale factor is less than 1.

The reduction scale factor = 1·52

y = 1·52 × 12 = 9m

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Key point

If two shapes are similar then:

• pairs of corresponding angles are equal; and

• the ratios of pairs of corresponding sides are equal.

Using scale factors exercise

Go online

Q1: Are the rectangles below similar, Yes or No?

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Q2:

These rhombi are similar. What is the reduction scale factor?

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



Q3:

These regular octagons are similar. What is the enlargement scale factor?

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

Q4: The shapes below are similar.

Do you need an enlargement scale factor or reduction scale factor to find the unknownlength?

a) enlargementb) reduction

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Q5: What is the scale factor needed to find the length of the big sun?

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Q6: What is the length of the big sun?

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Q7: The mountain bikes below are similar.

Do you need an enlargement scale factor or reduction scale factor to find the height ofthe small bike?


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Q8: What is the scale factor needed to find the height of the small bike?

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Q9: What is the height of the small bike?

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Q10:

These prints of Leonardo da Vinci’s "The Last Supper" are similar.

Calculate the height of the small print.

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Q11:

The smiley faces are similar.The length of the mouth on the small smiley face is 0·8 cm.

Calculate the length of the mouth on the big smiley face.

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1.1.1 Similar triangles

Key point

Triangles are special. If two triangles are similar then:

• pairs of corresponding angles are equal; or

• the ratios of pairs of corresponding sides are equal.

When all pairs of corresponding angles in two triangles are equal we say that thetriangles are equiangular.

Similar triangles

Go online

Similar triangles are special.

There are two triangles in the above diagram, ABC and ADE. From our knowledge ofangle properties we can identify that corresponding angles are equal.



• Both triangles have a common angle at A.

• Angles ABC and AED are corresponding or "F" angles and are equal.

• Angles ACB and ADE are corresponding or "F" angles and are equal.

We say that these triangles are equiangular and as such are similar.

Triangle ABC is similar to triangle ADE.

There are three pairs of corresponding sides;

• AD and AC;

• DE and CB;

• AE and AB.

In some triangles corresponding sides are harder to spot. There are two triangles in thisdiagram ABC and CDE.

From our knowledge of angle properties we can identify that alternate angles andvertically opposite angles are also equal.



A

D

C

E

B

• Angles ACB and DCE are vertically opposite or "X" angles and are equal.

• Angles ABC and CDE are alternate or "Z" angles and are equal.

• Angles CAB and DEC are alternate or "Z" angles and are equal.

These triangles are also equiangular and similar.

Triangle ABC is similar to triangle CDE. There are three pairs of corresponding sidesbut they are harder to spot.

• AC is opposite the green star so its corresponding side is CE because it is alsoopposite the green star.

• DE and AB are corresponding sides because they are both opposite the red star.

• CD and CB are corresponding sides because they are both opposite the blue star.

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Example

Problem:

Calculate the length of the side marked x.

Solution:

These triangles are equiangular and therefore similar.

It is often easier to pull the similar triangles apart in a sketch.

Notice 40 cm = 24 cm + 16 cm

The side that we are looking for is on the smaller triangle. We need a reduction scalefactor.

We know one pair of corresponding sides.

For a reduction scale factor the ratio must be less than 1 so the smaller of thecorresponding sides goes on the numerator.

The reduction scale factor = 2440 or 3

5 .


x = 24/40 × 45 = 27 cm

When identifying the scale factor in a non-calculator question it is best to simplify thefraction if you can.

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Example

Problem:

Calculate the length of the side marked x.

Solution:

These triangles are equiangular and therefore similar.

We have one pair of corresponding sides.It is easier to mark alternate angles in this question to help identify the correspondingsides.

The corresponding sides are opposite the green alternate angles so we want the 20 mmand 32 mm sides.(Notice that the 19 mm side is not required.)

The side that we are looking for is on the larger triangle. We need an enlargement scalefactor.

For an enlargement scale factor the ratio of corresponding sides must be greater than 1so the larger of the sides goes on the numerator.

The enlargement scale factor = 3220 or 8

5 .


x = 32/20 × 25 = 40 mm

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Similar triangles exercise

Go online

Q12: Looking at the image above, do you need an enlargement or reduction scale factorto find y?


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Q13: What is the scale factor?

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Q14: What is the length of y?

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Q15: What is the length of z?

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Q16: Looking at the image above, do you need an enlargement or reduction scale factorto find x?




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

Q17: What is the scale factor?

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Q18: What is the length of x?

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Q19:Looking at the image again, do you need an enlargement or reduction scale factor tofind y?


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Q20: What is the length of y?

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A piece of wire is bent to form the shape below. The verticals are parallel.

Q21: What is the height h?

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Q22: What is the length d?

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Q23:

Some pieces of wood have been nailed together. The pieces labelled BC and KL areparallel. AK = 51 cm, KL = 45 cm and BC = 60 cm as shown.

Calculate the length of CK.

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1.2 Area scale factors

Area scale factors

Go online

The length scale factor for enlargement is 30/10 = 3.

The area scale factor for enlargement is 180/20 = 9.



Key point

9 = 32

so Area scale factor = (Length scale factor)2

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Example

Problem:

The bolts of lightening are similar. Calculate the area of the large lightening bolt.

Solution:

Since we want the area of the large shape we need the scale factor for enlargement.

The length scale factor for enlargement

=12 · 57 · 5

The area scale factor for enlargement =(12·57·5

)2

The area of the large lightening bolt =(12·57·5

)2 × 36 = 100 cm2

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Area scale factors exercise

Go onlineQ24: If the length scale factor for enlargement is 5, what is the area scale factor forenlargement?

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Q25: If the length scale factor for reduction is 1·52·5 , what is the area scale factor for

reduction?

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The mirrors above come in 2 sizes and are similar. The area of glass in the small mirroris 540 cm2.

Q26: What is the length scale factor for enlargement?

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Q27: What is the area scale factor?

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Q28: What is the area of glass in the large mirror?

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The regular seven pointed stars are similar.

Q29: What is the length scale factor for reduction?

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Q31: What is the area of the small star?

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The rectangles below are similar .


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Q34: What is the area of the large rectangle?

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Q35: What is the area of the small rectangle?

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1.3 Volume scale factors

Volume scale factors

Go online

The length scale factor for enlargement is 6/2 = 3.

The volume scale factor for enlargement is 1080/40 = 27.



Key point

27 = 33

so Volume scale factor = (Length scale factor)3

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Examples

1.

Problem:

The cylinders below are similar. Calculate the volume of the small cylinder.

Solution:

For the small cylinder we need a reduction scale factor.

The length scale factor for reduction = 2346

The volume scale factor for reduction =(2346

)3

The volume of the small cylinder =(2346

)3 × 19112 = 2389 cm3

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2.

Problem:

The bottles of perfume are similar. Calculate the cost of the large bottle of perfume.



Solution:

For the large bottle of perfume we need the enlargement scale factor.

The length scale factor for enlargement = 10·56

Cost is dependent on Volume. You are not told this in the question but you will beexpected to know it.

Since cost is dependent on volume we need the volume scale factor.

The volume scale factor =(10·56

)3

Cost of the large bottle =(10·56

)3 × 16 = £85 · 75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Volume scale factors exercise

Go onlineQ36: If the length scale factor for enlargement is 4, what is the volume scale factor forenlargement?

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Q37: If the length scale factor for reduction is 1·32·4 , what is the volume scale factor for

reduction?

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The bottles below are similar.


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Q39: What is the volume scale factor for reduction?

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Q40: What is the volume of the smaller bottle?

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The tubes of toothpaste are similar.

Q41: What is the length scale factor for enlargement?

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Q42: What is the volume scale factor for enlargement?Give your answer as a fraction to a power (x/y)

z.

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Q43: What is the volume of the large tube of toothpaste?

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These cans of beans are similar. The large can is 15 cm tall and costs 65 pence. Thesmall can is 12 cm tall.


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Q45: What is the volume scale factor for reduction?

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Q46: Calculate the cost of the small can of beans to the nearest penny.

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1.4 Learning points

Similar shapes

• Congruent shapes are an exact copy of each other.

• Congruent shapes have a length scale factor of 1.

• Similar shapes are an enlargement or reduction of each other.

• If two shapes are similar then:

◦ pairs of corresponding angles are equal; and◦ the ratios of pairs of corresponding sides are equal.

• An enlargement scale factor is greater than 1 e.g. 5, 22/7.

• A reduction scale factor is less than 1 e.g. 1/2, 12/25.

• If two triangles have equal angles they are equiangular.

Similar triangles

• If two triangles are similar then:

◦ pairs of corresponding angles are equal; or◦ the ratios of pairs of corresponding sides are equal.

• Triangles which are equiangular are similar.

• In similar triangles:

◦ corresponding sides are opposite corresponding angles;

◦ there are corresponding, alternate and/or vertically opposite angles.

Area scale factor

• Area scale factor = (length scale factor)2

Volume scale factor

• Volume scale factor = (length scale factor)3



1.5 End of topic test

End of topic 7 test

Go online

Length scale factor

Q47:

These guitars are similar and come in adult and child sizes. The adult size guitar is 39cm wide.

What is the width of the child size guitar?

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Q48:

The lamps below are similar and come in two sizes, small and medium.

What is the height of the medium size lamp?

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Similar triangles

Q49:

The diagram shows a 30◦, 60◦, 90◦ set square.

Calculate the length of x.

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∠EAC and ∠CBD are equal.

Q50: Calculate the length of AC.

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Q51: Calculate the length of BD.

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Area scale factor

Q52:

The two Christmas trees are similar.

Calculate the area coloured green on the large tree.

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Q53:

The sheepskin rugs below are similar.

Calculate the area of the small sheepskin rug.

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Volume scale factor

Q54:

The glasses of milk below are similar.

Calculate the volume of the small glass of milk.

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Q55:

The two children’s balls below are similar.

Calculate the cost of the large ball to the nearest penny.

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28 ANSWERS: TOPIC 7

Answers to questions and activities

7 Similar shapes

Using scale factors exercise (page 6)

Q1:

Hint:

• Although corresponding angles in rectangles are all right angles, the ratios of pairsof corresponding sides are not the same e.g. 80

30 �= 3612 or 30

80 �= 1236

Answer: No

Q2: 26

Q3: 53

Q4: a) enlargement

Q5:

Hint:

• An enlargement scale factor is greater than 1.

Answer: 4

Q6:

Hint:

• scale factor × length of little sun = length of big sun

• So, 4 × 7 = ?

Answer: 28 cm

Q7: b) reduction

Q8:

Hint:

• A reduction scale factor is less than 1.

• So, 25/150 = ?

Answer: 16

Q9:

Hint:

• scale factor × height of big bike = height of small bike

• So, 1/6 × 72 = ?

Answer: 12 cm


ANSWERS: TOPIC 7 29

Q10:

Steps:

• Do we need an enlargement or reduction scale factor? reduction

• What is the scale factor? 30/80

• Use the scale factor to calculate the height of the small print.

Answer: Height = 18 cm

Q11:

Steps:

• Do we need an enlargement or reduction scale factor? enlargement


• Use the scale factor to calculate the length of the mouth on the large face.

Answer: Length = 2·4 cm

Similar triangles exercise (page 14)

Q12: b) reduction

Q13:

Hint:


Answer: 10/18 or simplified to 5/9

Q14:

Hint:

• scale factor × 36 = y

• So, 10/18 × 36 = ?

Answer: 20

Q15:

Hint:

• 36 − y = z

• So, 36 − 20 = ?

Answer: 16

Q16: b) reduction

Q17:

Hint:


30 ANSWERS: TOPIC 7


Answer: 8/12, or simplified to 4/6 or 2/3

Q18:

Hint:

• scale factor × 9 - y = x

• So, 8/12 × 9 = ?

Answer: 6

Q19: b) reduction

Q20: 10

Q21:

Steps:

• Do you need an enlargement or reduction scale factor to find h? reduction

• Identify the pair of corresponding sides. 20, 32


• Use the scale factor to find the height h.

Answer: Height h = 13 · 75 mm

Q22:

Steps:

• Do you need an enlargement or reduction scale factor to find d? enlargement


• Use the scale factor to find the length d.

Answer: Length d = 33 · 6 mm

Q23:

Steps:

• Do we need an enlargement or reduction scale factor to find AC? enlargement


• What is the length of AC? AC = 68 cm

• Use this answer to calculate CK.

Answer: Length CK = 19 cm


ANSWERS: TOPIC 7 31

Area scale factors exercise (page 17)

Q24: 25

Q25: (1.5/2.5)2, (3/5)

2 and (6/10)2 are also correct

Q26: 45/20, or simplified to 9/4

Q27: (45/20)2, or simplified to (9/4)

2

Q28: 2733·75 cm2

Q29: 4·56 , other simplified scale factors are 45/60,

9/12 and 3/4

Q30:(4·56

)2, other simplified scale factors are (45/60)2, (9/12)

2 and (3/4)2

Q31: 29·25 cm2

Q32: 18/30, or simplified to 9/15,6/10,

3/5 or 18/30


2, (6/10)2, (3/5)

2 or (18/30)2

Q34: 1890 mm2

Q35: 680·4 mm2

Volume scale factors exercise (page 21)

Q36: 64

Q37:(1·32·4

)3, other simplified answers accepted would be(1324

)3

Q38: 18/24, or simplified to 9/12,6/8 or 3/4


3, (6/8)3 or (3/4)

3

Q40: 81 cm3



3

Q43: 1331 cm3



3

Q46: 33 pence


32 ANSWERS: TOPIC 7

End of topic 7 test (page 24)

Q47:

Steps:



Answer: Width = 26 cm

Q48:

Steps:



• Use the scale factor to calculate the height of the medium size lamp.

Answer: Height = 84 cm

Q49:

Steps:



• Use the scale factor to calculate the length x.

Answer: Length x = 10 · 6 cm

Q50:

Steps:

• Do we need an enlargement or reduction scale factor to find AC? enlargement

• What is the scale factor? 2·42·1

• Use the scale factor to calculate the length of AC.

Answer: AC = 1 · 6 m

Q51:

Steps:

• Do we need an enlargement or reduction scale factor to find BD? reduction

• What is the scale factor? 2·12·4

• Use the scale factor to calculate the length of BD.

Answer: BD = 2 · 45 m

Q52:

Steps:


ANSWERS: TOPIC 7 33


• What is the length scale factor? 76·59

• What is the area scale factor?(76·59

)2

• Use the area scale factor to calculate the area coloured green on the largeChristmas tree.

Answer: Area = 2023 cm2

Q53:

Steps:



• What is the area scale factor?(1·64

)2

• Use the area scale factor to calculate the area of the small sheepskin rug.

Answer: Area = 0·8 m2

Q54:

Steps:


• What is the length scale factor? 18/28

• What is the area scale factor? (18/28)3

• Use the volume scale factor to calculate the volume of milk in the small glass.

Answer: Volume = 66·4 ml

Q55:

Steps:



• What is the volume scale factor? 5·23

• Use the volume scale factor to find the cost of the large ball. Give your answer tothe nearest penny.

Answer: Cost = £9·84 (to the nearest penny)


scholar study guide national 5 mathematics course

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