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Pacific Educational Press

MathWorks 10
Workbook
MathWorks 10
Workbook

Pacific Educational Press

Vancouver, Canada
Copyright Pacific Educational Press 2010

ISBN 978-1-89576-694-3

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying, recording,
or otherwise without the prior written permission of the publisher.

Writer
Katharine Borgen, Vancouver School Board and University of British Columbia
Consultants
Katharine Borgen, PhD, Vancouver School Board and University of British Columbia
Jordie Yow, Mathematics Reviewer
Design, Illustration, and Layout
Sharlene Eugenio
Five Seventeen
Editing
Catherine Edwards
David Gargaro
Leah Giesbrecht
Nadine Pedersen
Katrina Petrik

Photographs

Cover photo courtesy Colin Pickell;

Men fixing stairs: Leah Giesbrecht;
Woman measuring fabric: Sharlene Eugenio;
Surveyor: Jacqueline Chandler/ Public domain/ Wikimedia Commons/ FEMA-42251;
Man mowing lawn: Selmer Van Alten/ www.flickr.com;
Veterinarian: Veronica Pierce/ Public domain/ Wikimedia Commons/ VIRIN 050722-F-3177P-047;
Chapter 1: ©Jean-marie Guyon | Dreamstime.com;
Chapter 2: Courtesy U.S. Army Element Assembled Chemical Weapons Alternatives;
Chapter 3: Courtesy Ashlea Earl-Robinson;
Chapter 4: Morgan M. Wayling;
Chapter 5: Photo by Gary Schotel, Quesnel, BC;
Chapter 6: Leah Giesbrecht;
Chapter 7: Leah Giesbrecht.

SW-COC-002358

Printed and bound in Canada.

 Contents
How to Use this Book 7

1 Unit Pricing and

Currency Exchange 9

1.1 Proportional Reasoning 9

1.2 Unit Price 26
1.3 Setting a Price 36
1.4 On Sale! 49
1.5 Currency Exchange Rates 59
Chapter Test 70

2 Earning an Income 74

2.1 Wages and Salaries 74

2.2 Alternative Ways to Earn Money 87
2.3 Additional Earnings 97
2.4 Deductions and Net Pay 106
Chapter Test 114

3 Length, Area, and

Volume 118

3.1 Systems of Measurement 118

3.2 Converting Measurements 131
3.3 Surface Area 142
3.4 Volume 158
Chapter Test 169
Contents continued

4 Mass,Temperature, and
Volume 175

4.1 Temperature Conversions 175

4.2 Mass in the Imperial System 183
4.3 Mass in the Système International 195
4.4 Making Conversions 203
Chapter Test 210

5 Angles and Parallel Lines 214

5.1 Measuring, Drawing, and Estimating Angles 214

5.2 Angle Bisectors and Perpendicular Lines 225
5.3 Non-Parallel Lines and Transversals 231
5.4 Parallel Lines and Transversals 239
Chapter Test 248

6 Similarity of Figures 253

6.1 Similar Polygons 253

6.2 Determining if Two Polygons Are Similar 265
6.3 Drawing Similar Polygons 271
6.4 Similar Triangles 277
Chapter Test 284

7 Trigonometry of
Right Triangles 288

7.1 The Pythagorean Theorem 288

7.2 The Sine Ratio 297
7.3 The Cosine Ratio 308
7.4 The Tangent Ratio 318
7.5 Finding Angles and Solving Right Triangles 324
Chapter Test 336

Glossary 340

Answer Key 342

 7

How to Use This Book

This workbook is a companion to MathWorks 10 student resource, the authorized

resource for the WNCP course, Apprenticeship and Workplace Mathematics. The
MathWorks 10 Workbook is a valuable learning tool when used in conjunction with the
student resource, or on its own. It emphasizes mathematical skill-building through
worked examples and practice problems.
Here, you will learn and use the practical mathematics required in the workplace.
Whether you plan to enroll in college, learn a trade, or enter the workforce after
graduating from secondary school, the mathematical skills in this workbook will
support you at work and in your daily life.
The MathWorks 10 Workbook contains seven chapters. Chapters are divided into sections,
each focussing on a key mathematical concept. Each chapter includes the following
features.

Review
Each chapter opens with a review of mathematical processes and terms you will need to
understand to complete the chapter’s lessons. Practice questions are included.

Example
Each example includes a problem and its solution. The problem is solved step by step.
Written descriptions of each mathematical operation used to solve the problem are
included.

New Skills
The chapter’s core mathematical concepts are introduced here. This section often
includes real-world examples of where and how the concepts are used.

Build Your Skills

This is a set of mathematical problems for you to solve. It appears after each new
concept is introduced and will let you practise the concepts you have just learned about.
Build Your Skills questions often focus on workplace applications of mathematical
concepts.

Practise Your New Skills

The section’s key concepts are presented as review problems at the end of each chapter
section.
8 MathWorks 10 Workbook

Chapter Test
At the end of each chapter, a chapter test is provided for review and assessment of
learning.

Definitions
New mathematical terms are defined in the sidebar columns. They are also included in
the glossary at the back of the book.

Answer Key
An answer key to this workbook’s questions is located at the back of the book.

Glossary
Definitions for mathematical terms are provided here. To increase understanding, some
glossary definitions include illustrations.
 9

Chapter 1Unit Pricing and Currency Exchange

Canadian wheat is processed into flour and sold nationally, where most of it becomes baked goods. Canada also exports about
20 million tonnes of its wheat and grain each year.

Proportional Reasoning
1.1
REVIEW: WORKING WITH FRACTIONS

In this section, you will use fractions to solve for unknown values.
 10 MathWorks 10 Workbook

Example 1

Simplify 18 .
27

SOL U T ION

factor: one of two or To simplify a fraction, state the fraction in its lowest terms. To find the lowest terms,
more numbers that, when divide the numerator and the denominator by their largest common factor.
multiplied together, form a
18 ← numerator
product. For example, 1, 2,
27 ← denominator
3, and 6 are factors of the
product 6 because: Identify the factors of 18 and 27.
1×6=6 18 = {1, 2, 3, 6, 9, 18}
2×3=6
27 = {1, 3, 9, 27}
3×2=6
The largest factor common to both 18 and 27 is 9. Divide both the numerator and
6×1=6
the denominator by 9.

18 = 18 ÷ 9
27 27 ÷ 9
18 ÷ 9 = 2
27 ÷ 9 3

18 = 2
27 3

A LT ERN AT I V E SOL U T ION

18
The fractions 27 ,  sing the largest common factor when simplifying fractions takes the least number
U
6 2
9 , and 3 are of steps, but you can use any common factor.
equivalent fractions 18 = 18 ÷ 3 Simplify, using the factor 3.
because they 27 27 ÷ 3
represent the same
18 ÷ 3 = 6
amount. 27 ÷ 3 9

18 = 6
27 9
6 = 6÷3 Simplify further, again using the factor 3.
9 9÷3
6÷3 = 2
9÷3 3

6 = 2
9 3
 Chapter 1 Unit Pricing and Currency Exchange 11

BUILD YOUR SKILLS

1. Simplify these fractions to their lowest terms.

a) 4 = b) 3 = c) 25 =
16 12 75

d) 15 = e) 8 = f) 45 =
21 18 100

g) 20 = h) 3 = i) 7 =
50 21 56

Example 2

Solve for x in this equation containing fractions.

x = 5
16 24

SOL U T ION

To solve an equation, the same operation must be applied to both sides of the multiple: the product of
equation. a number and any other
number. For example,
Begin by multiplying both sides of the equation by the same number so that you
2, 4, 6, and 8 are some
can clear it of fractions (by eliminating the denominators). This will allow you to
multiples of 2 because:
isolate x.
2×1=2
The simplest multiplier is the lowest common denominator of the two fractions.
2×2=4
Identify the multiples of 16 and 24. 2×3=6

16 = {16, 32, 48, 64, 80, . . .} 2×4=8

24 = {24, 48, 72, 96, . . .}

12 MathWorks 10 Workbook

The lowest common multiple between 16 and 24 is 48, so begin solving the
equation by multiplying both fractions by 48.

x = 5
16 24

48 × x = 5 × 48 Multiply by 48.
16 24
48 x = 240 Simplify.
16 24
3x = 10

3 x = 10 Divide by 3 to isolate x.
3 3

x = 10 or 3 1
3 3

A LT ERN AT I V E SOL U T ION

You did not need to use 48 as your multiplier. Any common multiple will work, and
people often choose to multiply by the product of the denominators.

In this question, the product would be 16 multiplied by 24. You will get the same
answer in the end, but you will work with larger numbers.
x = 5
If x is not a whole 16 24
number, it is best to x = 5 × (16 × 24 )
( 24 × 16 ) × 16 24
Multiply both sides by
leave the answer as
the product of the denomiinators.
a fraction or in mixed
384 × x = 5 × 384 Simplify.
numeral form rather 16 24
than as a decimal
384 x = 1920
because the decimal 16 24
answer would often
24 x = 80
have to be rounded.
24 x = 80 Divide both sides by 24 to isoolate x.
24 24

x = 80
24

x = 80 ÷ 8 Simplify, using the factor 8.

24 ÷ 8

x = 10 or 3 1 or 3.3
3 3
 Chapter 1 Unit Pricing and Currency Exchange 13

BUILD YOUR SKILLS

2. Solve for x.

a) x = 40 b) 12 = 18
10 50 16 x

c) 56 = x d) 18 = 36
64 8 27 x

e) x = 3 f) 3 = 15
2056 4 12 x

g) 3 = x h) 25 = 40
5 460 x 200
 14 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH RATIO AND PROPORTION

ratio: a comparison When a carpenter bonds two pieces of wood with epoxy resin, she must first mix the
between two numbers epoxy with a hardener. She mixes these materials in a ratio of 10 to 1, where there are
measured in the same 10 parts of epoxy to 1 part of hardener. This ratio can be written as 10:1 or as a
units fraction, 10 .
1
I f the carpenter wanted to use 150 parts of epoxy, she would need 15 parts of hardener.
This would give her a ratio of 150 to 15 between the amount of epoxy and the amount
of hardener. You can write this as 150:15 or as 150 .
15

The ratio 150 can be simplified to 10 .

15 1
150 = 150 ÷ 15
15 15 ÷ 15

150 ÷ 15 = 10
15 ÷ 15 1

150 = 10
15 1

It is common to express ratios as fractions when doing calculations.

proportion: a fractional When you state that two ratios are equal, as they are in the following equation, you have
statement of equality written a proportion.
between two ratios
150 = 10
15 1

For more information, see page 12 of MathWorks 10.

 Chapter 1 Unit Pricing and Currency Exchange 15

Example 3

Charles works as a cook in a restaurant. His chicken soup recipe contains:

•• 11 cups of seasoned broth

•• 5 cups of diced vegetables

•• 3 cups of rice

•• 3 cups of chopped chicken

He wants to make the recipe at home for his parents. To reduce the recipe yield, he
needs to know what the ratios are between the quantities of ingredients.

a) What is the ratio of vegetables to chicken?

b) What is the ratio of broth to vegetables? A ratio can have

a numerator
c) What is the ratio of chicken to rice?
larger than the
d) What is the ratio of the chicken to the total ingredients in the recipe? denominator.
Because a ratio

SOL U T ION compares two

numbers, do not
a) Since there are 5 cups of vegetables and 3 cups of chicken, the ratio of vegetables rewrite it as a mixed
to chicken is 5:3 or 5 . fraction.
3
b) Since there are 11 cups of broth and 5 cups of vegetables, the ratio of broth to
vegetables is 11:5 or 11 .
5
1
c) T
 here are 3 cups of chicken and 3 cups of rice, so the ratio of chicken to rice is Although 1 is
3:3 or 3 . This can be simplified to 1:1 or 1 . usually written as
3 1
the number 1, keep it
d) There are 3 cups of chicken in the recipe, and a total of 22 cups of ingredients.
as a fraction when it
The ratio of chicken to the total amount of ingredients is 3:22 or 3 .
22 is a ratio comparing
two numbers.
16 MathWorks 10 Workbook

BUILD YOUR SKILLS

3. For a silk screening project, Jan mixes a shade of orange ink. She uses a ratio of red
ink to yellow ink of 2:3 and yellow ink to white ink of 3:1.

a) How many mL of yellow ink would she need if she used 500 mL of white ink?

b) How many mL of red ink would she need if she used 750 mL of yellow ink?

4. On a bicycle with more than one gear, the ratio between the number of teeth on the
front gear and the number of teeth on the back gear determines how easy it is to
pedal. If the front gear has 30 teeth and the back gear has 10 teeth, what is the ratio
of front teeth to back teeth?

5. Some conveyor belts have two pulleys. If one pulley has a diameter of 45 cm and
another has a diameter of 20 cm, what is the ratio of the smaller diameter to the
larger diameter?
 Chapter 1 Unit Pricing and Currency Exchange 17

6. Bank tellers use ratios when converting currencies. If $1.00 CAD equals approximately
1.13 Australian dollars, what is the ratio of Canadian dollars to Australian dollars?

7. What is the ratio of 250 mL of grape juice concentrate to 1 L of water? (Hint:

Convert both measurements to the same units. There are 1000 mL in 1 L.)

8. A mechanic mixes oil with gas to lubricate the cylinders in a motorcycle engine. He
uses 1 part oil and 32 parts gas. What is the ratio of oil to gas?

Example 4

Tom and Susan make $180.00 from holding a garage sale. Because Tom contributed
fewer items to the sale, the money is to be divided between Tom and Susan in the ratio
of 1:2. How much money will each person receive?

SOL U T ION

Since the ratio is 1:2, this means that for every $1.00 Tom receives, Susan
receives $2.00. Stated another way, this means that for every $3.00 earned, Tom gets
$1.00 and Susan gets $2.00.

Therefore, Tom receives 1 of the money and Susan receives 2 of it.

3 3
1 × 180 = 60
3
2 × 180 = 120
3
Tom gets $60.00 and Susan gets $120.00.
18 MathWorks 10 Workbook

BUILD YOUR SKILLS

9. The ratio of flour to shortening in a recipe for piecrust is 2:1. If a baker makes
30 cups of piecrust, how many cups of flour and shortening does he use?

10. A compound of two chemicals is mixed in the ratio of 3:10. If there are 45 litres of
the compound, how much of each chemical is in the mixture?

11. Cheryl is an automotive repair technician. She mixes paint and thinner to apply to
a bus. The instructions say to mix paint with thinner in the ratio of 5:3. If Cheryl
needs 24 L of paint/thinner mixture, how much of each will she use?
 Chapter 1 Unit Pricing and Currency Exchange 19

NEW SKILLS: WORKING WITH RATE

A rate is a ratio comparing two numbers measured in different units. rate: a comparison
between two numbers
Some examples of rates are:
measured with different
$1.69
•• $1.69/100 g or
100 g for the cost of ham at the deli units

•• 80 km/h or 80 km for how fast a car travels

1h
•• $38.00/4 h or $38.00 for how much you earn at work
4h

For more information, see page 17 of MathWorks 10.

Example 5

The amount of fuel consumed by a vehicle when it is driven 100 km is referred to as the
rate of fuel consumption. Write a rate statement that indicates that a car uses 6.3 litres
of gas for every 100 km driven.

SOL U T ION

6.3 L:100 km, 6.3 L , or 6.3 L/100 km

100 km

BUILD YOUR SKILLS

12. Write a rate statement that indicates that you earned $65.00 interest on your
investment in the last 3 months.

13. Write a rate statement that indicates how much you earn in an 8-hour day if you are
paid $9.25 for each hour you work.

14. Write a rate statement that indicates that 1 cm on a map represents 2500 km in
real distance.
20 MathWorks 10 Workbook

Example 6

If you earn $150.00 in 12 hours, how much will you earn if you work 40 hours?

SOL U T ION

$150.00 = $ x
12 h 40 h
150 = x In your calculation, omit the units.
12 40

(12 × 40) × 150 = x × ( 40 × 12) Multiply booth sides of the equation

12 40 by the product of the denominators.

480 × 150 = x × 480 Simplify.

12
40 × 150 = x × 12

6000 = 12x

6000 = 12 x Divide both sides by 12 to isolate x.

12 12
6000 = x
12
500 = x

You will earn $500.00 in 40 hours.

BUILD YOUR SKILLS

15. If a type of salami at the deli costs $1.59 per 100 g, how much will you pay
for 350 g?
 Chapter 1 Unit Pricing and Currency Exchange 21

16. As a janitor, Janine makes a cleaning solution by mixing 30 g of concentrated

powdered cleanser into 2 L of water. How much powder will she need for 5 L of water?

17. An office has decided to track how much paper it uses to reduce waste. At the end of
each month, the secretary records the total number of sheets used and their weight.
If paper weighs 4.9 kg for every 500 sheets, how much will 700 sheets weigh?

PRACTISE YOUR NEW SKILLS

1. F
 ind the unknown value in each of the following proportions. Give answers to the While calculating,
nearest tenth of a unit (to one decimal place). omit the units.

a) 24 = x b) 168 km = 548 km
18 12 2h xh
22 MathWorks 10 Workbook

40 = 60 6 pizza slices x pizza slices

c) d) =
28 x 2 people 21 people

e) 87 blankets = 24 blankets f) 12 = 25
x bundles 8 bundles 25 x

g) 7 = x h) 12 = 16
15 1 45 x

2. A hairdresser mixes brunette hair colouring for a client using 20 mL hair colour,
40 mL colour developer, 15 mL conditioner, and 3 mL thickener. Find the following
ratios and simplify them to their lowest terms. Express your answers as fractions.

a) The ratio of hair colour to thickener.

 Chapter 1 Unit Pricing and Currency Exchange 23

b) The ratio of thickener to conditioner.

c) The ratio of colour developer to hair colour.

d) If this treatment costs the customer $68.00 and the cost of labour and materials
used is $14.20, what is the ratio of customer price to actual cost?

3. If the ratio of yellow pigment to blue pigment in a shade of green paint is 2:3, how
many drops of yellow pigment will be needed if 12 drops of blue are used?
24 MathWorks 10 Workbook

4. If 5 cm on a map represents 2.5 km of actual ground, how many centimetres would

15 km of actual ground be on the map?

5. If a can of paint will cover 48 m2 of wall space, how many cans will you need to
paint 220 m2?

6. The ratio of teeth on a pair of driving gears is 13:6, with the larger gear having more
teeth. If the larger gear has 52 teeth, how many does the smaller gear have?
 Chapter 1 Unit Pricing and Currency Exchange 25

7. If Stephie can type 75 words per minute, how long will it take her to type an
800-word term paper? Is the solution a rate or a ratio? Explain your answer.

8. If the ratio of flour to sugar in a recipe is 3:2, how much flour would you need if you
used 1.5 cups of sugar?

9. If a machine can produce 85 parts in 40 minutes, how many parts can it produce
in 8 hours?
 26 MathWorks 10 Workbook

1.2 Unit Price

NEW SKILLS: WORKING WITH UNIT PRICE

unit price: the cost of When items are sold in quantities of more than 1, the unit price indicates how much 1
one unit; a rate expressed of the items would cost. For example, if you buy a package of 3 pencils, the unit price is
as a fraction in which the the price of 1 pencil.
denominator is 1
For more information, see page 23 of MathWorks 10.

Example 1

If a carton of one dozen eggs costs $3.29, how much are you paying for 1 egg?

SOL U T ION

Solve using a proportion. Let x represent the cost of 1 egg.

$3.29 = x
12 eggs 1 egg

3.29 = x Omit the units in the calculation.

12 1

12 × 3.29 = x × 12 Multiply both sides of the

12 1 equation by 12.
3.29 = 12x Simplify.

3.29 = 12 x
Divide booth sides by 12 to isolate x.
12 12
3.29 = x
12
0.27 ≈ x    

One egg costs approximately $0.27.

 Chapter 1 Unit Pricing and Currency Exchange 27

BUILD YOUR SKILLS

1. If 12 apples cost $10.20, how much does one apple cost?

2. Lindsay is the kitchen manager of an Ethiopian café. She purchases a case of

1000 paper coffee cups for $94.83. How much does one cup cost?

3. Frank is a locksmith who owns his own business. He buys a case of 144 bulk brass
padlocks for $244.97. He sells each lock for $5.50.

a) How much does each lock cost Frank?

b) How much profit does Frank make when he sells one lock?
28 MathWorks 10 Workbook

Example 2

People often think Often unit price is used to compare costs.

that larger-sized
A 48-oz can of tomatoes costs $2.99. An 18-oz can costs $1.19. Which is a better buy?
packages cost less
per unit, but this is
not always so. SOL U T ION

Calculate the cost of 1 oz of tomatoes for each size of can.

Calculate the cost of 1 oz from the 48-oz can.

$2.99 ÷ 48 ≈ $0.062 Divide the total cost by the total number of ounces.

1 oz of tomatoes costs approximately $0.06.

Calculate the cost of 1 oz from the 18-oz can.

$1.19 ÷ 18 ≈ $0.066 Divide the total cost by the total number of ounces.

1 oz of tomatoes costs approximately $0.07.

It costs more for one ounce of tomatoes from the 18-oz can, so the 48-oz can is the
better buy.
 Chapter 1 Unit Pricing and Currency Exchange 29

BUILD YOUR SKILLS

4. La Boutique du Livre is a francophone bookstore in St. Boniface, MB. It sells

notebooks in packages of 12 for $15.48. Another bookstore sells the same notebooks
in packages of 15 for $19.65. Which is the better price?

5. Which is the better buy: 6 muffins for $7.59, or one dozen muffins for $14.99?

6. Johnny can buy two 8-foot pieces of 2″ by 4″ lumber at $2.60 each, or three 6-foot
pieces for $1.92 each. Which is the better buy per foot?
30 MathWorks 10 Workbook

Example 3

Shirin is the manager of a fabric store and is training a new employee. Shirin wants to
make an easy reference chart that lists the prices of different lengths of a fabric. One
metre of the fabric costs $8.42. Fill in the rest of the chart.

FABRIC COST BY LENGTH

Length of fabric Cost of fabric
0.5 m
1m $8.42
1.75 m

SOL U T ION

You can solve this using proportions. First find the cost of 0.5 m.

$8.42 = $x
1m 0.5 m
8.42 = x
1 0.5 Omit the units during calculations.

0.5 × 8.42 = x × 0.5 Multiply both

h sides of the equation by the
0.5 product of the deenominator, 0.5.
0.5 × 8.42 = x Simplify.

4.21 = x

The cost of 0.5 m of fabric is $4.21.

Then find the cost of 1.75 m.

8.42 = x
1 1.75

1.75 × 8.42 = x × 1.75

1.75

1.75 × 8.42 = x

14.74 = x

FABRIC COST BY LENGTH

Length of fabric Cost of fabric
0.5 m $4.21
1m $8.42
1.75 m $14.74
 Chapter 1 Unit Pricing and Currency Exchange 31

A LT ERN AT I V E SOL U T ION

You can use unit rates rather than proportions.

1 m costs $8.42. Use this unit cost to calculate the cost of the other lengths

$8.42 × 0.5 = $4.21

$8.42 × 1.75 = $14.74

BUILD YOUR SKILLS

7. Sasha is a landscape gardener. He sees that a 200-foot roll of string trimmer line
costs $18.75. A 150-foot roll of line costs $15.21.

a) Which roll of line is the least expensive per foot?

b) What is the difference in price, per foot?

8. If 2 1 kg of tomatoes cost $8.25, how much will you pay for 7 kg?
2

9. If Wayne bought 5 litres of gas for his lawnmower for $5.45, how much would he
have to pay to fill his car with 48 litres of gas?
32 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. During the summer, Dean works as a cashier in a store near Saskatchewan’s

Greenwater Lake Provincial Park. The store sells a case of 12 bottles of water for
$8.50 and individual bottles of the same brand of water for $1.55.

a) Approximately how much does each bottle of water in the case of 12 cost?

b) How much would a customer save by buying a case of water, rather than
12 individual bottles?

2. Maureen purchased enough carpet to cover a rectangular room measuring 7 metres

by 12 metres. The carpet costs $8.15 per square metre.

a) How much carpet did Maureen buy?

b) How much did the carpet cost?

 Chapter 1 Unit Pricing and Currency Exchange 33

3. Tyler is a self-employed sheet metal worker. He purchases 25 sheets of aluminum

that measure 4 feet by 8 feet. The cost is $4000.00 before tax and shipping.

a) How much does 1 sheet cost?

b) What is the price per square foot?

4. A painting business buys 3-inch wide paintbrushes from a supplier in cases of 6.

One case costs $31.29.

a) How much do two brushes cost?

b) If a customer buys two or more cases, the supplier reduces the price of the
case by 10 percent. How much would 3 cases of paintbrushes cost? How much
would each brush cost?
34 MathWorks 10 Workbook

5. Which is the better buy: 8 ounces of Brie cheese for $4.95 or 12 ounces for $7.49?

6. Debbie is a cook in a restaurant that is open 6 days a week. She is responsible for
recording and monitoring the amount of money she spends on food. In the summer,
she uses an average of 9 loaves of bread per day.

a) On average, how many loaves of bread does Debbie use each week?

b) If bread costs $1.25 per loaf to buy from a wholesale distributor, how much
money should Debbie budget to purchase it, for the month of June? Assume that
there are just 4 weeks in June.
 Chapter 1 Unit Pricing and Currency Exchange 35

7. The cost of a 355-mL can of juice is $1.25 in a vending machine. A 1.89-L carton of
the same juice costs $3.89 at the grocery store. How much would you save per mL if
you bought juice from the grocery store instead of the vending machine? (Hint: 1 L
equals 1000 mL.)

8. Patricio is ordering cartons of detergent for resale in his store. He can order a carton
of 12 for $34.68 plus $5.45 for delivery, or a carton of 18 for $51.30 plus $6.25 for
delivery. Which is a better buy, and by how much per unit?
36 MathWorks 10 Workbook

1.3 Setting a Price

REVIEW: WORKING WITH PERCENTS AND DECIMALS

I n this section, you will calculate percentages and convert between percents
and decimals.

Example 1

Convert 65% to a decimal.

percentage: a ratio with SOL U T ION

a denominator of 100;
To convert a percentage to a decimal, first convert it to a fraction with a
percent (%) means "out
denominator of 100, then divide the numerator by the denominator.
of 100"
65% means 65 out of 100, or, stated as a fraction, 65 .
100
Dividing by 100 Divide 65 by 100 to express 65% as a decimal.
gives the same
65 ÷ 100 = 0.65
result as moving the
decimal two places 65% is equal to 0.65.
to the left. Many
people use this as a
shortcut.
 Chapter 1 Unit Pricing and Currency Exchange 37

BUILD YOUR SKILLS

1. Convert the following percentages to decimals.

a) 78% b) 93%

c) 125% d) 324%

e) 0.5% f) 0.38%

g) 1.2% h) 100%
38 MathWorks 10 Workbook

Example 2

Calculate 20% of 45.

SOL U T ION

To find a percentage of a number, you can use proportional reasoning.

Let x represent 20% of 45. Use proportional reasoning to solve for x.

20 = x
100 45

( ) ( )
4500 20 = x 4500
100 45
Multiply both sides of the equation by the
producct of the denominators.
45 × 20 = 100 x Simplify.
900 = 100 x
900 = 100 x Divide both siides by 100 to isolate x.
100 100
9= x

20% of 45 is 9.

A LT ERN AT I V E SOL U T ION

First convert 20% to a decimal.

20 ÷ 100 = 0.20

Then calculate 20% of 45 by multiplying 0.20 by 45.

x = 0.20 × 45

x=9
 Chapter 1 Unit Pricing and Currency Exchange 39

BUILD YOUR SKILLS

2. Calculate the following percentages. If the percentage is

larger than 100, then
a) 15% of 300 b) 45% of 1500
your answer will
be larger than the
number you started
with.

c) 140% of 70 d) 175% of 24

e) 7.8% of 50 f) 0.3% of 175

g) 200% of 56 h) 135% of 25
40 MathWorks 10 Workbook

Example 3

What percent is 5 of 20?

SOL U T ION

To calculate what percent one number is of another means that you need to
determine what number out of 100 is equivalent to your ratio.

You can use proportional reasoning to solve the question.

x = 5
100 20
100 × x = 5 × 100 Multiply both sides of the equation
100 20 by 100.
x = 5×5 Simpliify.
x = 25

5 is 25% of 20.

A LT ERN AT I V E SOL U T ION

5 of 20 is the same as 5 . You can find the percent by dividing 5 by 20.

20

5 ÷ 20 = 0.25

This is 25 hundredths, or 25%.

 Chapter 1 Unit Pricing and Currency Exchange 41

BUILD YOUR SKILLS

3. Calculate what percentage the first number is of the second (to the nearest tenth of
a percent).

a) 65 of 325 b) 135 of 405

c) 68 of 42 d) 13 of 65

e) 1 of 12 f) 625 of 50

NEW SKILLS: WORKING WITH MARKUPS AND TAXES

When a person owns a business, she usually buys merchandise at a wholesale price. She markup: the difference
then sells it at a higher price, called a retail price, to make a profit. The retail price is between the amount a
often determined by adding a certain percentage of the wholesale price to the wholesale dealer sells a product
price. The amount added is referred to as the markup. for (retail price) and the
amount he or she paid for
Sales taxes are usually added to the retail price. There are three kinds of taxes that you
it (wholesale price)
may need to consider when purchasing items: Goods and Services Tax (GST), Provincial
Sales Tax (PST), and Harmonized Sales Tax (HST).

For more information, see page 28 of MathWorks 10.

42 MathWorks 10 Workbook

Example 4

Melanie owns a clothing store. Her standard markup is 85%. She bought a coat from the
wholesaler at $125.00.

a) What would the markup be?

b) How much would Melanie charge her customer for the coat?

SOL U T ION

a) Change 85% to a decimal.

85 ÷ 100 = 0.85

Multiply the wholesale price by the markup.

$125.00 × 0.85 = $106.25

The markup is $106.25.

b) Melanie would charge her customer the wholesale price plus the markup.

$125.00 + $106.25 = $231.25

With a markup of 85%, Marnie would charge her customer $231.25.

BUILD YOUR SKILLS

4. The markup on a bicycle in a sporting goods store is 125%. The bicycle’s wholesale
price is $450.00. What is the markup in dollars?
 Chapter 1 Unit Pricing and Currency Exchange 43

5. Marco buys a certain brand of shampoo from a supplier at $7.25 per bottle. He sells
it to his customers at a markup of 25%. What would the markup be?

6. A hanbok is a formal, traditional outfit of Korean clothing. At a markup of 60%,

what would be the retail price for a child’s hanbok with a wholesale price of $117.45?

7. What should Max charge for a package of paper plates in his store if he bought them
for $9.00 and wants to make a 75% profit?
44 MathWorks 10 Workbook

Example 5

Quentin wants to buy a pair of steel-toed boots listed at $179.95. How much will the
boots cost if 5% GST and 6% PST are charged?

SOL U T ION

Calculate the GST by changing 5% to a decimal and multiplying by the retail price.

5 ÷ 100 = 0.05

0.05 × $179.95 = $9.00

Calculate the PST by changing 6% to a decimal and multiplying by the retail price.

6 ÷ 100 = 0.06

0.06 × $179.95 = $10.80

The final price is calculated by adding the tax amounts to the retail price.

$179.95 + $9.00 + $10.80 = $199.75

Quentin will have to pay $199.75.

A LT ERN AT I V E SOL U T ION 1

5% + 6% = 11% Add the two taxes together.

11 ÷ 100 = 0.11 Change to a decimal.

0.11 × $179.95 = $19.79 Multiply to find 11% of the retail price.

While it is $179.95 + $19.79 = $199.74 Add the total tax to the retail price.
convenient for you to
This is different than the first solution because rounding occurs at a different spot.
use the methods in
Alternative Solutions
1 or 2, the store A LT ERN AT I V E SOL U T ION 2

would have to use  ou can think about the retail price as 100% of the cost. If you add the two sales
Y
the first method taxes (5% plus 6% equals 11%) to this, the total cost is 111%.
because they must
keep track of how Convert 111% to a decimal and multiply this by the initial cost.
much GST and PST 111 ÷ 100 = 1.11
they collect. $179.95 × 1.11 = $199.74
 Chapter 1 Unit Pricing and Currency Exchange 45

BUILD YOUR SKILLS

8. What would you have to pay for a jacket that is listed at $99.95 if you live in
Nunavut, where the only tax is 5% GST?

9. Find the total cost of a washing machine that is being sold for $944.98 in
Saskatchewan, where PST is 6% and GST is 5%.

10. Calculate the HST (12%) on a return flight from Cranbrook, BC to Vancouver, BC
that costs $372.00.

11. Saskatchewan charges 5% GST and 6% PST. How much GST and PST would Krista
pay on a can of paint priced at $45.89?
46 MathWorks 10 Workbook

12. If the markup is 125% on a certain brand of jeans that have a wholesale price of
$30.00, what will the consumer pay, if GST and PST are each 5%?

13. The markup on a restaurant meal is 250%. A meal costs $7.25 to produce. How
much will the customer be charged, after markup and 5% GST are applied?

PRACTISE YOUR NEW SKILLS

1. Calculate the following percentages.

a) 5% of 72 b) 275% of 8

c) 152% of 200 d) 6 3 % of 700

4
 Chapter 1 Unit Pricing and Currency Exchange 47

2. An electrician buys his material at the local hardware store, then charges
his customer 20% more. The material for a given project is $253.75 at the
hardware store.

a) What is the markup?

b) How much does he charge the customer for the material?

3. Maria is a florist in a small boutique. If Maria paid her supplier $8.50/doz for roses
and sold them for $19.95, what was the percent markup?

4. Garth buys snow tires from a dealer in Thunder Bay, ON. The tires cost $79.00
each. To make a profit, he must mark them up 40%. How much must a customer
pay for 4 tires if there is a 12% HST on the final sale?
48 MathWorks 10 Workbook

5. Because the cost of ingredients has gone up, Maurice has decided to increase the
cost of meals in his restaurant by 12%. How much will he now charge for a grilled
salmon fillet that used to cost $17.95?

6. The wholesale cost of a certain brand of T-shirts is $132.00/dozen. To cover costs

and make a profit, Lorena marks them up by 75%. If GST is 5% and PST is 7%,
what will a customer have to pay for 1 T-shirt?

7. An MP3 player that Harry wants to buy sells for $89.95 in BC where there is
12% HST. When he is on holiday in Alberta (where there is only the 5% GST), Harry
sees the same MP3 player for $94.89. Which is a better buy and by how much?
 Chapter 1 Unit Pricing and Currency Exchange 49

On Sale!
1.4
NEW SKILLS: SALE PRICES

When you go shopping, you see signs that advertise promotions, such as “For Sale,” “Up promotion: an activity
to 50% Off,” and “Discounted Prices,” that mean you will pay less than the price that increases awareness
on the tag. of a product or attracts
customers
For more information, see page 34 of MathWorks 10.

Example 1

Samantha wants to buy a new TV. The model she likes costs $675.95, but the clerk
tells her that it is going on sale next week at 20% off. If Samantha waits one week, how
much will she save on the price of the TV?

SOL U T ION

The discount is 20%, so find 20% of $675.95 by multiplication.

Convert 20% to a decimal.

20 ÷ 100 = 0.20

Multiply 0.20 by the original price.

0.20 × $675.95 = $135.19

The original price will be discounted by this amount, so Samantha would

save $135.19.
50 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. How much will Jordon save on the price of a computer listed at $989.98 if it is
discounted by 30%?

2. The Midtown Bakery sells Chinese specialties such as pineapple buns and lotus
seed buns. Day-old goods at the bakery are sold at a discount of 60%. If the original
price of a loaf of sweet bread was $2.98, how much would you save by buying a
day-old loaf?

3. If the sale price is 25% off, what will you save if you buy a sofa regularly priced
at $999.97?
 Chapter 1 Unit Pricing and Currency Exchange 51

Example 2

Michelle is a member of the Xat’sull First Nation and is fluent in the Shuswap language.
She works as a language instructor and gift shop cashier at the Xat’sull Heritage Village,
near Williams Lake, BC.

The gift shop is selling off summer inventory. What will be the cost of a carving that
was priced at $149.95 if the sale sign says “Reduced by 60%”?

SOL U T ION

Calculate 60% of $149.95 to determine the savings.

First, convert 60% to a decimal.

60 ÷ 100 = 0.60

Multiply the original price by 0.60.

0.60 × $149.95 = $89.97

The original price will be reduced by this amount, so subtract $89.97 from $149.95.

$149.95 − $89.97 = $59.98

The cost of the carving will be $59.98.

A LT ERN AT I V E SOL U T ION

If the price of the carving is reduced by 60%, that means that the customer will pay
only 40% of the cost (100% minus 60% is 40%).

Convert 40% to a decimal.

40 ÷ 100 = 0.40

Multiply the original price by 0.40.

0.40 × $149.95 = $59.98

The carving will cost $59.98.

52 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. Sarbjit charges $24.95 for a haircut in her beauty salon, but gives students a 30%
reduction on Thursday evenings. How much would you have to pay to have your
hair cut on a Thursday evening?

5. Margariet manages a second-hand clothing store in Flin Flon, Manitoba. The store
advertises that if you buy 3 items, you will get 15% off the most expensive item,
20% off the second most expensive, and 30% off the cheapest item. You choose
three items costing $10.00, $25.00, and $12.00. How much will you pay for the
three items?

6. Normally Chiu charges $75.00 to paint a room. However, if he paints 3 or more

rooms for you, he gives you a 15% discount. How much will he charge if he paints
4 rooms for you?
 Chapter 1 Unit Pricing and Currency Exchange 53

Example 3

Lisa specializes in selling products from the Philippines, including rattan, bamboo, and
palm baskets. Medium-sized bamboo baskets are regularly priced at $19.98. They are on
sale, advertised as “Buy one, get the second at half price.” What is the discount rate, as
a percent?

SOL U T ION

Calculate the regular cost of 2 baskets.

$19.98 × 2 = $39.96

In buying 2 baskets, you save half the price of 1 basket.

1 ($19.99) = $9.99
2

Calculate the percent savings by dividing the discount by the regular price, and
multiplying by 100.

$9.99 × 100% = 25%

$39.96

The discount rate is 25%.

54 MathWorks 10 Workbook

BUILD YOUR SKILLS

7. What is the percentage markdown if a $175.00 item sells for $150.00?

8. In a store opening promotion, Fred advertises T-shirts: “Buy 4 get 1 free.” If the cost
of 1 T-shirt is $15.97, what is the discount rate, as a percent?

9. Cameron is buying new computers for his office. Each computer costs $789.00.
He is told that if he buys 5 computers, he can get a 6th one free. What will be his
percent saving compared to buying all 6 at the regular price?
 Chapter 1 Unit Pricing and Currency Exchange 55

10. Shelly works as an optician in Whitehorse, YT. Her store is selling last year’s glasses
frames at a savings of 30%. What will you pay for frames that were originally priced GST and PST are
at $149.00 if 5% GST is charged? paid on the selling
price, not the original
price.

11. Nicole wants to buy a coat originally priced at $249.95. It is on sale at 25% off. How
much will she pay if 5% GST and 5% PST are charged?

12. Yasmin owns a kitchen and bath fitting store. She is selling a kitchen sink at a
reduction of 40% because of a scratch in the finish. The original price was $249.95.

a) Determine the total savings to the customer, including 5% GST and a

PST of 8%.

b) Calculate the percentage of savings.

56 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. In Abbotsford, BC, Mack works as a used car salesman. He offers a 15% reduction
to repeat customers. If the price marked on a car is $9879.00, how much will it be
reduced for a repeat customer?

2. A fishing rod originally priced at $49.98 is reduced by 30%.

a) How much is the discount?

b) What will the cost be before tax?

3. Senior citizens are offered a 20% discount on their lunch on Tuesdays in Hay River’s
local diner. How much will Rita and Dick, both senior citizens, save if they order
the teriyaki chicken salad at $14.98 and the pork cutlets at $17.98? (Ignore taxes.)
 Chapter 1 Unit Pricing and Currency Exchange 57

4. A can of paint costs $59.95. There is a 20% price reduction for contractors. How
much will the contractor save if he buys 5 cans?

5. A furniture store offers a “Closing Out Sale: Everything 80% Off.”

a) How much will you pay for a bedroom suite that originally cost $2989.97 if GST
is 5% and PST is 7%?

b) What are your total savings on this purchase?

6. Robert sells bicycles, skateboards, and snowboards at his sporting goods store.
A bicycle that was originally priced at $785.00 sold for $553.00. What percent
markdown did Robert offer?
58 MathWorks 10 Workbook

7. The wholesale cost of a tawa, a griddle used to cook Indian flatbread, is $53.00. A
merchant marks it up 65%. At the end of the season, he sells the remaining stock
at 60% off.

a) What was his original asking price?

b) At the original price, how much would a customer pay with 5% GST
and 5% PST?

c) What was the end-of-season sale price?

d) How much would a customer pay when it was on sale, including 5% GST
and 5% PST?

e) What would the total savings be if it were bought on sale?

f) What would be the percentage savings?

 Chapter 1 Unit Pricing and Currency Exchange 59

Currency Exchange Rates

1.5
NEW SKILLS: EXCHANGE RATES

Different countries use different monetary units and/or different currencies. It is exchange rate: the price
important when travelling to consider exchange rates, or the value of one monetary of one country's currency
unit compared to another. in terms of another
country's currency
For more information, see page 41 of MathWorks 10.

Example 1

 ucas is a glazier who operates a window installation business. He regularly travels

L Using approximation
to the United States to buy supplies. Before travelling, he converts $500.00 CAD into when working with
American dollars for personal expenses. If one Canadian dollar is worth 0.94192 of exchange rates is
an American dollar, how many American dollars will Lucas receive in exchange for useful if you want
$500.00 CAD? to quickly convert
between currencies.

SOL U T ION You can think of

the exchange rate
Use unit pricing. in Example 1 as

$1.00 CAD = $0.94192 USD meaning that for

every Canadian
$500.00 CAD = $0.94192 USD × 500 dollar, you will

$500.00 CAD = $470.96 USD lose approximately

6 cents when
Lucas will receive $470.96 USD. converting to
American dollars.
60 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Ray purchased $500.00 CAD worth of parts from Hungary for use in his garage. If
the exchange rate is one Canadian dollar to 180.0779 Hungarian forints (Ft), how
many forints will you receive for $500.00 CAD?

2. If one Canadian dollars is worth 0.5911 British pounds sterling (£), calculate how
many pounds sterling you would get for $200.00 CAD.

3. Madeline is attending a trade show in Denmark. She runs short of spending money
and must convert $100.00 CAD into Danish kroner (kr). The exchange rate is
5.3541 Danish krone for one Canadian dollar. How many kroner will she receive?
 Chapter 1 Unit Pricing and Currency Exchange 61

Example 2

One Thai baht is worth 0.023541 of a Canadian dollar. How many bahts would a tourist
in Thailand receive for $200.00 CAD?

SOL U T ION

1.00 baht = $0.023541 CAD

200 ÷ 0.023541 = 8495.82

For $200.00 CAD, a tourist in Thailand would receive 8495.82 bahts.

BUILD YOUR SKILLS

4. If the exchange rate for converting a Canadian dollar to the euro is 0.7180 on a
particular day, how many euros would you get for $300.00 CAD?

5. The exchange rate for converting a Canadian dollar to the Swiss franc (SFr) is
1.0542. How many Swiss francs will you get for $400.00 CAD?

6. Canada imports steel, iron, and organic and inorganic chemicals from Trinidad
and Tobago. The exchange rate for converting the Canadian to the Trinidad and
Tobago dollar is 6.1805. How many Trinidad and Tobago dollars will you get for
$200.00 CAD?
62 MathWorks 10 Workbook

7. Using the following exchange rates, calculate how much foreign currency you would
receive for $200.00 CAD.

a) $1.00 CAD is worth 1.72904 Brazilian reals

b) $1.00 CAD is worth 8.71137 Moroccan dirhams

c) $1.00 CAD is worth 7.72277 Ukrainian hryvnia

d) $1.00 CAD is worth 3.19889 Polish zloty

8. Calculate the value in Canadian dollars of an item that costs $449.75 Singapore
dollars. Assume the exchange rate for one Canadian dollar is 0.75529.
 Chapter 1 Unit Pricing and Currency Exchange 63

9. Henry returns home to Whale Cove, Nunavut, after a trip to Europe. On his travels
he purchased a jacket for 125.98 euros. Calculate the value of Henry’s jacket in
Canadian dollars. Assume that a euro is worth 1.3987 of a Canadian dollar.

10. The exchange rate between the South African rand and the Canadian dollar is Exchanging money is
0.138469 (1 rand equals $0.138469 CAD). What is the cost in Canadian dollars of not quite as simple
an item priced at 639.00 rand? as the transactions
here show. Although
the calculations will
be the same, you
have to consider
bank buying rate
and bank selling rate
when exchanging
currency. When the
bank buys foreign
currency from you,
they pay you less
than they charge
when they sell it
to you.
64 MathWorks 10 Workbook

Example 3

Anne works for an automotive parts distributor and visits Switzerland to source new
products. On a given day, the bank selling rate of the Swiss franc compared to the
Canadian dollar is 1.0501 and the buying rate is 1.0213.

a) How many Swiss francs would Anne receive for $400.00 CAD?

b) If Anne sold them back to the bank, how much would she receive?

c) What would her net loss be?

SOL U T ION

a) The bank will sell Swiss francs to Anne, so use the selling rate. 1 Swiss franc is
worth $1.0501 CAD.

1 SFr = x
$1.0501 $400.00
1 = x
1.0501 400.00 Omit the units.

(400.00 × 1.0501 ) × 1 = x × (1.0501 × 400.00 ) Multiply both sides of the equatiion

1.0501 400.00 by the product of the denominators.
400.00 = 1.0501x Simplify..
400.00 = x
Divide both sides by 1.0501 to isolate x.
1.0501
380.92 = x

Anne would receive 380.92 Swiss francs.

A LT ERN AT I V E SOL U T ION

a) 1 Swiss franc is worth $1.0501 CAD.

$1.00 is therefore worth 1 Swiss francs. Multiply this by 400 to calculate

1.0501
what $400.00 CAD is worth.

400.00 × 1 = 380.92 Swiss francs

1.0501
 Chapter 1 Unit Pricing and Currency Exchange 65

b) The bank buys the Swiss francs back at a rate of 1.0213 Swiss francs per
$1.00 CAD.

1 SFr = 380.92 SFr

$1.0213 x
1 = 380.92
1.0213 x Omit the units.

(1.0213 × x ) × 1 = 380.92 × ( x × 1.0213) Multiply both sides of the equatiion

1.0213 x by the product of the denominators.
x = 380.92 × 1.0213 Simplify..
x = 389.03

Anne would receive $389.03 CAD.

A LT ERN AT I V E SOL U T ION

b) 1 Swiss franc is worth $1.0213 CAD. Unit rates often work

best for exchanging
Multiply 380.92 Swiss francs by 1.0213 to find the value in Canadian dollars.
currency.
380.92 × 1.0213 = $389.03 CAD

c) Anne’s net loss would be $400.00 minus what she received back. When customers
exchange money
$400.00 − $389.03 = $10.97
at a bank or other
Anne would lose $10.97 in the transaction. institution, the bank
will usually only deal
with paper money,
not coins.
66 MathWorks 10 Workbook

BUILD YOUR SKILLS

11. If the exchange rate of a country compared to the Canadian dollar is 0.00519, will
you get more or less of their currency units when you exchange money?

12. Dianne works as a bank teller in Canmore, AB. A customer wishes to buy
250 British pounds at a rate of 1.5379 $CAD. How many Canadian dollars would
the British pounds cost?

13. If the selling rate of a euro (€) is 1.4768 and the buying rate is 1.4287, how much
would you lose if you exchanged $1000.00 CAD for euros and then converted them
back to $CAD on the same day?
 Chapter 1 Unit Pricing and Currency Exchange 67

PRACTISE YOUR NEW SKILLS

1. Using the following information, calculate how much of the foreign currency you
would get for $500.00 CAD. Round to the nearest unit.

a) $1.00 is worth 95.4911 Japanese yen b) $1.00 is worth 1.41046 Turkish lira

c) $1.00 is worth 0.680228 euro d) $1.00 is worth 6.43033 Chinese yuan

2. Damien is training to become a customer service representative at a credit union

in Saskatoon, SK. Given the following exchange rates compared to the Canadian
dollar, calculate how much foreign currency Damien would give to a customer who
wished to convert $500.00 CAD.

a) Mexican peso, 0.0818085 b) Estonian kroon, 0.0939564

c) British pound, 1.3376 d) South Korean won, 0.000922277

e) Indian rupee, 0.0229526 f) Russian ruble, 0.0352667

68 MathWorks 10 Workbook

3. Using the rates from question #1, calculate the amount you would get in Canadian
dollars if you sold the following.

a) 8750 Japanese yen b) 900 Turkish lira

c) 250 euros d) 3000 Chinese yuan

4. Use the exchange rates from question #2. Calculate how many Canadian dollars you
would get for each of the following.

a) 6750 Mexican pesos b) 145 British pounds

c) 15 000 Indian rupees d) 750 Russian rubles

5. If the exchange rate is 0.1736 between the Norwegian krone and the Canadian
dollar, what would the price be in Canadian dollars of an item that cost 275 kroner?
 Chapter 1 Unit Pricing and Currency Exchange 69

6. A hand-woven shawl costs 35 Botswana pula. How much does it cost in Canadian
dollars if the exchange rate is 0.1515?

7. If the selling rate of the Omani rial is 2.96845 and the buying rate is 2.86145, how
much would you lose if you bought and then sold $800.00 on the same day?
70 MathWorks 10 Workbook

CHAPTER TEST

1. When he works as a landscape gardener, Mark sometimes uses a powdered

fertilizer. It must be mixed at a rate of 1 part of powder to 14 parts of water. How
much water will Mark use for 3 litres of powder?

2. A car uses 5.5 litres of gas when it travels 100 km.

a) Express this as a rate of fuel consumption.

b) How much fuel would be needed for a 400-km trip?

3. Loretta works as a surveyor near Burwash Landing, YT. Her map uses a scale of
2.5 cm:100 km. On her map, two sites she must visit are 7.4 cm apart. What is the
actual distance between the two sites?
 Chapter 1 Unit Pricing and Currency Exchange 71

4. If a package of 12 pens cost $38.98, what is the cost of 1 pen?

5. If the wholesale price of 10 packages of smoked salmon is $99.50, what will the cost
be for one package after a markup of 45%?

6. The bakery in Lund, BC is selling day-old buns at a 40% reduction. If the regular
price is $4.79/doz, what is the reduced price?

7. A sofa in a furniture store was originally $1899.00. The price was “reduced by 35%
for quick sale.” When it did not sell, the manager offered another reduction of 20%.

a) What was the final price of the sofa with 5% GST and 7% PST?

b) Is this the same as a 55% reduction? Show why or why not.

72 MathWorks 10 Workbook

8. A furniture store in The Pas, Manitoba, advertises: “All weekend, no GST and no
PST.” If GST and PST are usually 5% each, what is the actual saving as a percent on
an item that costs $24.97?

9. After prime planting season was over, a horticulturist sold lilac bushes for $15.00. If
the original price was $39.00, what is the percentage markdown?

10. Non-profit agencies get a 12% reduction from Polly’s Printers. How much will they
save on a printing job that regularly costs $865.00?
 Chapter 1 Unit Pricing and Currency Exchange 73

11. James works for an industrial lighting company. He travels to Hong Kong to attend
a trade show. James sees a fluorescent track lighting unit priced at 1295.31 Hong
Kong dollars. What is the cost in Canadian dollars if $1.00 CAD is worth 7.3181
Hong Kong dollars?

12. Marian travels to Spain to visit her mother and father.

a) $1.00 CAD is worth €0.680228. If Marian converts $450.00 CAD into euros,
how many euros does she receive?

b) During her visit, Marian buys a leather purse for €125.00. What is the cost in
Canadian dollars?
74 MathWorks 10 Workbook

Chapter 2Earning an Income

Many tradespeople such as electricians, welders, carpenters, and plumbers are employed by the construction industry. These men are
installing rebar, the steel rods used to reinforce the walls and floors of concrete buildings.

2.1 Wages and Salaries

REVIEW: WORKING WITH MIXED FRACTIONS

A proper fraction is a fraction where the numerator is smaller than the denominator, for
example, 2 and 8 .
3 12
An improper fraction is one in which the numerator is greater than or equal to
the denominator, for example, 3 and 12 . Improper fractions can be changed to
2 8
mixed numerals.

A mixed numeral is a number represented as a whole number and a fraction, for

example, 2 3 and 12 5 . In most cases, the fraction is simplified to its lowest terms.
4 8
 Chapter 2 Earning an Income 75

Example 1

Change the improper fraction to a mixed numeral, expressed in its simplest form.

188
12

SOL U T ION

To change an improper fraction to a mixed numeral, divide the numerator by the divisor: in a division
denominator and write the remainder as a fraction of the divisor. operation, the number by
which another number is
188 ÷ 12 = 15, remainder 8
divided; in a ÷ b = c,
When 188 is divided by 12, the quotient is 15 and the remainder is 8. The mixed b is the divisor
numeral is 15 8 . However, the fraction 8 can be further simplified. quotient: the result of a
12 12
division; in a ÷ b = c,
8 = 8÷4
12 12 ÷ 4 c is the quotient

8÷4 = 2
12 ÷ 4 3
8 = 2
12 3

The simplified mixed numeral is written as 15 2 .

3

A LT ERN AT I V E SOL U T ION

First simplify the improper fraction, then divide the numerator by the
denominator.

188 = 188 ÷ 4 Simplify.

12 12 ÷ 4
188 ÷ 4 = 47
12 ÷ 4 3
188 = 47
12 3
47 ÷ 3 = 15, remainder 2 Divide.

The quotient is 15 and the remainder is 2, so the mixed numeral is 15 2 .

3
76 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Change the improper fractions to mixed numerals.

a) 29 b) 493 c) 1005
7 9 29

d) 45 e) 398 f) 1000
6 16 15

Example 2

Change the mixed numeral 2 3 to an improper fraction.

4

SOL U T ION

The mixed numeral can be broken up as follows:

23 = 2+ 3
4 4

Change 2 to a fraction whose denominator is 4.

2 = 2×4
1 1× 4
2×4 = 8
1× 4 4
2 = 8
1 4

Now substitute this improper fraction into the expression above.

23 = 2+ 3
4 4
2+ 3 = 8+3
4 4 4
8+3 = 8+3
4 4 4
23 = 11
4 4
 Chapter 2 Earning an Income 77

BUILD YOUR SKILLS

2. Change the mixed numerals to improper fractions.

a) 5 6 b) 4 7 c) 15 8
11 9 17

d) 7 5 e) 12 4 f) 10 7
8 5 12

NEW SKILLS: WORKING WITH INCOME

A salary, a wage, or an income is the amount of money you receive for work you do. In
some jobs, pay is calculated by the hour, while other jobs offer an annual income (paid
weekly, biweekly, or monthly).

Gross pay is the amount you make before deductions. Deductions will be discussed in gross pay: the total
section 2.4. amount of money earned
before deductions; also
For more details, see page 54 of MathWorks 10.
called gross earnings or

Example 3 gross income

deduction: money taken

Marcus works as an electrician and earns $24.68/h. If it takes him 15 hours for one job,
off your paycheque to
how much will he earn?
pay taxes, union fees,
and for other benefits and
SOL U T ION programs

Multiply his hourly wage by the number of hours he works.

$24.68 × 15 hours = $370.20

1 hour

He will earn $370.20 on the job.

78 MathWorks 10 Workbook

BUILD YOUR SKILLS

3. Martha works as a window dresser in her hometown of Victoria, BC. She charges
$16.72/h and it takes her 5 hours to finish the window at a local department store.
How much will her gross pay be for the job?

4. Ben works as a carpenter for $20.87/h. How much will he earn in a 40-hour
work week?

5. Harpreet works in the trucking business. He charges $35.75/h to haul materials for
a local contractor. Last week he worked the following hours:

•• 6 hours on Monday

•• 8 hours on Tuesday

•• 8 hours on Wednesday

•• 12 hours on Thursday

What was his gross income for the week?

 Chapter 2 Earning an Income 79

Example 4

Last week, Chi worked 34 hours cutting lawns. His gross income was $329.12. What
was his hourly wage?

SOL U T ION

Divide his gross income by the number of hours he works to calculate his
wage per hour.

$329.12 = $9.68
34 hours 1 hour

Chi earns $9.68/h.

BUILD YOUR SKILLS

6. Last year, Liliana earned $45 183.36 working in a Grande Prairie hair salon.

a) What was her average monthly income?

b) What was her average weekly income?

80 MathWorks 10 Workbook

7. If Janny works a 40-hour work week as a receiving clerk in the Powell River
Hospital and earns $552.88 per week, what is her hourly wage?

8. Emile is a flag person and earned $321.25 last week. If he worked 32.5 hours, what
was his hourly salary?

Example 5

Antonio is a cashier in a store that sells Caribbean products, such as ginger syrup, ackee
(a type of fruit), and dasheen (also known as taro, a root vegetable). His time card for one
week is shown below.

Time Card: Antonio

Day Morning Afternoon Total Hours
IN OUT IN OUT
Monday 9:00 11:45 12:45 3:00
Tuesday 8:45 10:45 2:00 5:30
Wednesday 9:00 12:00 1:00 4:00
Thursday 9:30 12:00 1:15 3:45
Friday 9:00 11:30 12:15 3:15

a) How many hours did he work?

b) If he earns $15.85 per hour, how much did he earn that week?
 Chapter 2 Earning an Income 81

SOL U T ION

a) On Monday, Antonio worked 2 3 hours in the morning and 2 1 hours in the

4 4
afternoon for a total of 5 hours.

On Tuesday, he worked 2 hours in the morning and 3 1 hours in the afternoon

2
for a total of 5 1 hours.
2
On Wednesday, he worked 3 hours in the morning and 3 hours in the afternoon
for a total of 6 hours.

On Thursday, he worked 2 1 hours in the morning and 2 1 hours in the

2 2
afternoon for a total of 5 hours.

On Friday, he worked 2 1 hours in the morning and 3 hours in the afternoon

2
for a total of 5 1 hours.
2
Add these amounts together.

5 + 5 1 + 6 + 5 + 5 1 = 27
2 2

Antonio worked 27 hours.

b) Multiply the number of hours he worked by his hourly wage.

27 × $15.85 = $430.70

Antonio earned $430.70 that week.

82 MathWorks 10 Workbook

BUILD YOUR SKILLS

9. Monty works after school at a gas station in Swift Current, SK. He earns $9.45/h.
How much would he earn if the time card below represents his work week?

Time Card: Monty

Day Total Hours
IN OUT
Monday 3:30 6:45
Tuesday
Wednesday 5:00 9:30
Thursday 5:00 9:30
Friday 3:30 7:00

Hae-rin often works 10. Hae-rin works as a part-time warehouse technician. She gets paid $12.76/h and
a split shift, where keeps her own time card. How much did she earn during the week?
her work day is split
into two time blocks. Time Card: Hae-rin
Day Morning Afternoon Total Hours
IN OUT IN OUT
Monday 7:45 9:00 5:00 7:45
Tuesday 4:00 8:00
Wednesday 9:00 11:00
Thursday 9:00 11:00 3:00 5:00
Friday 3:00 6:00
Saturday 9:00 12:00
 Chapter 2 Earning an Income 83

NEW SKILLS: WORKING WITH OVERTIME PAY

Many full-time jobs have a 40-hour work week, but others may have different regular
hours. If you work more than the regular number of hours, it is classified as overtime
and you will earn overtime pay for those extra hours. Overtime is often paid at “time
and a half”—that is, 1.5 times your regular wage—but can be any other agreed-
upon amount.

Example 6

Marcel works for a construction company and earns $15.82/h for a 37 1 -hour work
2
week. He is paid time and a half for any time that he works in excess of 37 1 hours. If
2
he works 42 1 hours during one week, how much will he earn?
4
SOL U T ION

Calculate Marcel’s overtime wage. It is time and a half, which means 1.5 times his
regular wage.

1.5 × $15.82 = $23.73

His overtime salary is $23.73/h.

Calculate how many hours of overtime he worked by subtracting 37 1 from 42 1 .

2 4
(Hint: Change these values to decimals to make subtracting easier.)

42.25 − 37.5 = 4.75

He worked 4.75 h overtime.

Find his total income by calculating his regular income and then his overtime
income. Add the two amounts together.

Regular income:

37.5 × $15.82 = $593.25

Overtime income:

4.75 × $23.73 = $112.72

Total income:

$593.25 + $112.72 = $705.97

Marcel earned $705.97 during the week.

84 MathWorks 10 Workbook

BUILD YOUR SKILLS

11. Pete works in road construction as a grader operator. His regular work week is
40 hours. During the busy season, he often has to work overtime. For overtime
hours worked Monday to Friday, he earns time and a half. If he has to work on
Saturday, he earns double time and a half. How much will Pete make if he works
45.25 hours during the week and 5.75 hours on Saturday? His regular salary
is $15.77/h.

12. Ingrid works as a medical receptionist at a rate of $11.82/h. She regularly works
35 hours per week, but her clinic wants to increase her work week to 42 h. She
agrees to do this if they will pay her overtime, at time and a half, for the extra hours.
If they agree to pay this amount, what will her weekly pay be?

13. Nathalie works as a playground supervisor for 8 weeks during the summer at a rate
of $15.27/h for a 40-hour week. If she averages 3 hours of overtime each week, paid
at time and a half, how much will she earn during the summer?
 Chapter 2 Earning an Income 85

PRACTISE YOUR NEW SKILLS

1. What is your daily income if you earn $10.75/h as a camp counsellor and you work
10 hours per day?

2. Lauren worked as an assistant at the National Métis Youth Conference. Her job was
to provide support to people giving workshops. Lauren worked for 7 1 hours at a
2
rate of $12.36/h. How much did she make?

3. Juanita has been offered a job that pays $497.35 for a 35-hour work week. A second
company offers her a job at $16.75/h, but will only guarantee 30 hours per week.
Which job would you advise Juanita to take?
86 MathWorks 10 Workbook

4. Rita’s annual income at her part-time job walking dogs is $6758.00. Assuming she
works the same amount of time each week, what is her weekly salary?

5. Abdul’s time card is below. If his hourly wage is $9.05, how much did he earn
during the week?

Time Card: Abdul

Day Morning Afternoon Total Hours
IN OUT IN OUT
Monday 9:05 12:15 1:20 5:00
Tuesday 9:02 12:13 1:12 4:25
Wednesday 8:58 12:14 1:05 4:19
Thursday 9:02 12:12 12:58 4:14
Friday 8:45 12:35 1:05 4:15

6. Tandor has begun a job as an animal trainer in the movie industry. His starting
wage is $10.53/h for 30 hours per week. If he works more than 30 hours, he is
paid 1.25 times his regular salary. How much will he earn if he works 35 hours
in one week?
 Chapter 2 Earning an Income 87

Alternative Ways to Earn Money 2.2

NEW SKILLS: WORKING WITH INCOME OPTIONS

Not all working people earn wages or a salary. There are other ways to earn income,
including piecework, commission, salary plus commission, and contract work.

For more information, see page 64 of MathWorks 10.

Example 1

Greg works as a tree planter during the summer and he earns his income through
piecework. He is paid $2.50 for each seedling he plants. If he plants 45 seedlings in a
day, how much will he earn?

SOL U T ION

Multiply $2.50 by 45.

$2.50 × 45 = $112.50

He will earn $112.50 that day.

BUILD YOUR SKILLS

1. Thomasina knits sweaters and sells them in her craft shop. She has been hired to
knit sweaters for a team of 4 curlers and their spare. If she charges $75.50 for a large
sweater and $69.75 for a medium sweater (because they require different amounts of
wool), how much will she earn if she knits 3 large and 2 medium sweaters?
88 MathWorks 10 Workbook

2. Patricia works in the garment industry. She is paid $1.50 for each hem and $2.25 for
each waistband. How much does she earn by hemming 12 dresses and attaching 15
waistbands?

3. Jack cleans windows for extra income. He charges $3.00 for a main floor window
and $5.00 for a second-storey window. How much will he earn if he cleans the
windows on a house that has 7 main floor windows and 6 second-storey windows?

Example 2

Mary works as a flower arranger. She is paid $143.75 for making 25 identical flower
arrangements for a wedding. How much was she paid per arrangement?

SOL U T ION

Find the unit rate. The easiest way to calculate this is to divide $143.75 by 25.

$143.75 ÷ 25 = $5.75

She is paid $5.75 per arrangement.

 Chapter 2 Earning an Income 89

BUILD YOUR SKILLS

4. Jorge is an auto detailer. If each job costs the same amount and his office grossed
$3048.00 on 12 jobs, what was the cost per job?

5. Karissa picked 18 quarts of strawberries and earned $67.50. How much did she
earn per quart?

6. Joey is a freelance writer. He often writes articles for a local newspaper that pays
$0.35 per word. How long was his article (expressed as the number of words) if he
was paid $192.50?
90 MathWorks 10 Workbook

Example 3

Ming works in a store that sells classic Chinese furniture such as roundback armchairs
and corner-leg stone stools. Ming works on commission at a rate of 6.5% of his gross
sales. If he sold $9865.00 worth of furniture last week, how much commission
did he earn?

SOL U T ION

Multiply his sales by his commission rate (expressed as a decimal).

$9865.00 × 0.065 = $641.23

His commission was $641.23.

BUILD YOUR SKILLS

7. Peter works in a sporting goods store and earns 12% commission on his sales. How
much does he make on a bicycle that sells for $785.95?

8. When selling a home, a real estate agent makes 5% commission on the first
$250 000.00 of the home’s selling price and 2% on any amount over that. How
much will Sue make if she sells a house worth $375 900.00?
 Chapter 2 Earning an Income 91

9. David earns a salary of $375.00 per week plus 5% commission on his sales. If he
sold $6521.00 of goods, how much did he make?

Example 4

Olaf earned $416.03 commission on his sales of $9245.00. What was his rate of
commission?

SOL U T ION

You need to find what percent $416.03 is of $9245.00. Use proportional reasoning.

$416.03 = x
$9245.00 100
100 × $416.03 = x × 100
$9245.00 100
100 × $416.03 = x
$9245.00
4.5 ≈ x

His rate of commission was about 4.5%.

A LT ERN AT I V E SOL U T ION

Divide Olaf’s commission by his sales to find the rate of his commission as a
decimal. Then multiply this by 100 to convert to a percentage.

$416.03 ÷ $9245.00 ≈ 0.045

0.045 × 100 = 4.5

His rate of commission was 4.5%.

92 MathWorks 10 Workbook

BUILD YOUR SKILLS

10. If Freddi earns $6.86 on a $95.95 sale, what was his rate of commission?

11. What is the rate of commission if you make $592.00 on sales of $12 589.00?

12. Don operates a small craft store in which he sells other people’s crafts. He takes a
45% commission from the sales of all crafts. If he earned $958.00 commission last
week, how much did he sell?
 Chapter 2 Earning an Income 93

Example 5

Gurpreet is figuring out what he should charge for repairing the steps on a client’s
house. The cost of materials will be $785.96, and he will have to hire 2 workers for
8 hours each at a rate of $12.85/h. He wants to earn at least $450.00 for himself. What
should he charge the client in the contract?

SOL U T ION

Calculate all of Gurpreet’s costs.

Cost of materials: $785.96

Cost of labour:

2 people × 8 hours × $12.85 = $205.60

1 hour

Gurpreet’s income: $450.00

Add all of these costs to find the total cost of completing the repairs.

$785.95 + $205.60 + $450.00 = $1441.56

Gurpreet should charge $1441.56. Gurpreet would

probably round this
number. He might
charge $1450.00.
94 MathWorks 10 Workbook

BUILD YOUR SKILLS

13. Marcel is bidding on a contract to lay concrete on a patio. He decides to calculate his
actual costs and then add 20% profit for himself. He needs 3 cubic yards of concrete
at $100.00 per cubic yard delivered. He needs to pay 2 employees $12.45/h each for
4 hours to do the job. What should he charge the client?

14. Tien has three employees working for her. Each employee is paid the minimum
wage in BC, $8.00/h, for an 8-hour day. They are also paid a commission of 12%
on all sales they make. If the three employees made sales of $785.96, $452.87, and
$616.42, how much must Tien pay in total for the day?

15. Kate works at the front desk for a sheet metal company that recently completed 5
contracts. The contracts were worth $5600.00, $2800.00, $7450.00, $1900.00, and
$8900.00. Materials, salaries, and other expenses amounted to $23 750.00. What
was the percentage of the profits?
 Chapter 2 Earning an Income 95

PRACTISE YOUR NEW SKILLS

1. A farmer pays his son $8.25 for each bucket of wild blueberries he picks. He then
sells them at a roadside stand for $15.00 per bucket.

a) How much does his son earn if he picks 28 buckets of blueberries?

b) How much does the farmer earn?

2. A car company pays its salespeople 2% commission on the amount of their sales
after cost. Joey sells a car for $23 000.00 that cost the company $15 000.00. How
much did Joey make in commission?

3. Larissa earns a base salary of $500.00 per week plus 4% commission on any sales.
How much will she have to sell to earn $750.00 in a week?
96 MathWorks 10 Workbook

4. Paulette has been hired to make bridesmaids’ dresses for a wedding. She knows that
it will take her approximately 9.5 hours to sew each dress and that the material for
each dress costs $120.00. Paulette decides to charge $240.00 per dress. At this price,
what is her hourly rate?

5. Jeff’s bid on a contract was $15 980.00. His costs would be $12 250.00, plus he
would have to pay a labourer for 36 hours at $12.45/h.

a) Jeff’s potential client indicated that another contractor offered a bid that was
10% lower. Would Jeff be justified in lowering his contract price to the same
amount? Why or why not?

b) Jeff calculated that he will put 50 hours into the job. If he lowers the bid, what
will his hourly income be?
 Chapter 2 Earning an Income 97

Additional Earnings
2.3
NEW SKILLS: WORKING WITH INCOME SUPPLEMENTS

A bonus is an extra amount earned for a job well done or for exceeding expectations,
and is paid in addition to regular pay and/or overtime pay. It may be a lump sum or
a percentage of earnings. Danger pay, isolation pay, a shift premium, and tips are also
paid in addition to regular pay.

For more information, see page 72 of MathWorks 10.

Example 1

Marika works during the summer as a supervisor of a children’s program at the local
community centre. She has a good reputation with the children and her employer
wants her to come back next year. Marika earned $3600.00 during the summer and
her employer offers her a 15% signing bonus if she will sign up for next year. If Marika
signs up, how much will she get as a signing bonus?

SOL U T ION

Find 15% of Marika’s wages.

0.15 × $3600.00 = $540.00

She will make $540.00 as a signing bonus.

98 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Chester wants to clear out his used car sales lot in Surrey, BC. To do so, he offers
his employees an incentive for each car they sell. They will receive bonuses in the
following amounts:

•• $50.00 for each car they sell for more than $20 000.00

•• $40.00 for each car they sell between $15 000.00 and $19 999.00

•• $30.00 for each car they sell between $10 000.00 and $14 999.00

•• $25.00 for each car they sell below $10 000.00

Gerry sells cars worth $14 895.00, $19 998.00, $15 675.00, $7250.00, and
$15 229.00. What is his bonus pay?

2. Raymond receives isolation pay for working in Wabasca, AB. If his regular pay
is $2245.00/month, and he is offered a bonus of 12% or $275.00/month, which
should he take?

3. Darren works as a logging machine operator. His salary is $24.80/h. Due to the
dangerous nature of his job, he makes 38% more per hour than Sean, who is a
forklift operator. How much do Darren and Sean each make in an 8-hour day?
 Chapter 2 Earning an Income 99

Example 2

Conchita works as a bus driver for a transit company. Since busy times of the day require
more drivers than midday, the company requires that some employees work split shifts.
The bus company offers a 12% shift premium for the second shift to anyone that has a
4-hour break between the end of one shift and the beginning of the next.

If Conchita earns $17.82/h, how much will she earn during the week?

Time Card: Conchita

Day Shift 1 Shift 2
Monday 6:15–11:00 3:30–6:45
Tuesday 6:15–10:15 3:30–7:15
Wednesday 12:00–4:00 6:00–9:00
Thursday 3:30–7:30
Friday 8:00–11:00 2:00–4:45

SOL U T ION

Determine the days on which Conchita will receive a shift premium.

Monday: 4 1 hours – Premium

2
Tuesday: 5 1 hours – Premium
4
Wednesday: 2 hours – No premium
Thursday: no break – No premium
Friday: 3 hours – No premium

Calculate the shift premium rate by find 112% of her regular wage (as a decimal, 1.12).

1.12 × $17.82 = $19.96

Calculate the hours worked in each shift and the income per shift.

Time Card: Conchita

Day Shift 1 Shift 2
Monday 4.75 × $17.82 = $84.65 3.25 × $19.96 = $64.87
Tuesday 4 × $17.82 = $71.28 3.75 × $19.96 = $74.85
Wednesday 4 × $17.82 = $71.28 3 × $17.82 = $53.46
Thursday 4 × $17.82 = $71.28
Friday 3 × $17.82 = $53.46 2.75 × $17.82 = $49.01

Calculate total income by adding.

$84.65 + 3 ($71.28) + 2 ($53.46) + $64.87 + $74.85 + $49.01 = $594.14

Conchita will earn $594.14 during the week.

100 MathWorks 10 Workbook

A LT ERN AT I V E SOL U T ION

Determine the number of hours for which she earned regular pay.

Monday: 4.75 hours

Tuesday: 4 hours
Wednesday: 7 hours
Thursday: 4 hours
Friday: 5.75 hours

Total: 25.5 hours

Add up the number of hours for which she received a shift premium.

Monday: 3.25 hours

Tuesday: 3.75 hours

Total: 7 hours

Calculate her shift premium rate by finding 112% of her regular wage.

112 ÷ 100 = 1.12

1.12 × $17.82 = $19.96

Calculate her regular income and her premium income, and then add the two.

Regular income:

25.5 hours × $17.82 = $454.41

hour
Premium income:

7 hours × $19.96 = $139.72

hour

$454.41 + $139.72 = $594.13

Conchita will earn $594.13 during the week.

This answer differs from the first solution by $0.01, due to rounding in the
first solution.
 Chapter 2 Earning an Income 101

BUILD YOUR SKILLS

4. Regular hours at the computer repair shop where Denise works are 9:00 am to
5:00 pm, Monday to Friday. Her boss has offered a shift premium of $1.75/hour to
anyone who will work after 5:00 pm or on Saturday. Last week, Denise worked the
following hours:
•• Monday: 9:00 am–5:00 pm

•• Tuesday: 2:00 pm–8:00 pm

•• Wednesday: 2:00 pm–7:00 pm

•• Thursday: 12:00 pm–8:00 pm

•• Saturday: 9:00 am–3:00 pm

If her regular pay was $15.25/h, how much did she earn last week?

5. Drivers for the Fast Delivery Parcel Company are offered a shift premium if they
drive the night shift (after 8:00 pm) to deliver parcels by 9:00 the next morning.
Baljeet’s schedule last week was as follows:
•• Monday: 12:00 pm–7:00 pm

•• Tuesday: 9:00 am–5:00 pm

•• Wednesday: 6:00 pm–12:00 am

•• Thursday: 12:00 pm–8:00 pm

•• Friday: 3:00 pm–9:00 pm

His regular pay is $12.75/h, and the shift premium is $7.00/h. How much did he
make last week?
102 MathWorks 10 Workbook

6. Chen is offered isolation pay of $1250.00 for a job in northern Manitoba.

Alternatively, he can have a bonus payment of 28% of his salary. If his salary is
$532.00/wk, which is the better option if it takes 10 weeks to complete the job?

Example 3

Suzette works as a waitress in a local café in Selkirk, MB. Yesterday she made $165.32 in
tips. If this was 15% of the bills she collected, how much were the bills?

SOL U T ION

Let x be the total of the bills. Use proportional reasoning.

15 = 165.32
100 x

( ) (
100 x 15 = 165.32 100 x
100 x ) Multiply both sides by the product of
the denominnators.
15 x = 165.32 × 100 Simplify.
15x = 16 532
15 x = 16 532 Divide both sides by 15 to isollate x.
15 15
x = $1102.13

Her orders totalled $1102.13.

A LT ERN AT I V E SOL U T ION

You know that $165.32 equals 15% of the total orders. Use this to calculate 1% of
the orders.

$165.32 ÷ 15 ≈ $11.0213

Multiply this number by 100 to get 100% of the total orders.

11.0213 × 100 = $1102.13

Suzette’s orders totalled $1102.13.

 Chapter 2 Earning an Income 103

BUILD YOUR SKILLS

7. Kirsten earns a base salary of $8.20/h plus tips. On a typical day, she bills her In most restaurants,
customers $950.00, and her tips average 15%. What is Kirsten’s average daily 15% is the average
income with tips for an 8-hour day? expected tip amount.
A quick way to
estimate a 15% tip is
to round off your bill,
find 10% by moving
the decimal one
place to the left, and
then add half of that
number to itself.
8. Mandeep is at a restaurant in Prince Rupert, BC. He has decided that 15% is too
For example, if the
much to leave for a waiter who did not provide good service. He left $3.00 on a meal
bill is $68.89, round
that cost $24.75. What percentage tip did he leave?
up to $70.00.

10% is $7.00.

Half of $7.00 is
$3.50.

$7.00 + $3.50 =
$10.50

The total tip would

be $10.50. This is
slightly more than
15% because you
9. Rosita earned $408.65 working 35 hours at $8.21/h plus tips. How much did she
rounded up, so you
make in tips?
may want to pay
about $79.00.
104 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. Parminder is working at an isolated weather station in the Yukon. She earns an

annual salary of $45 650.00 plus $780.00/month for isolation pay. If she works at
the station for 8 months of the year, what will her annual income be?

2. Hilda works as a live-in nanny. She earns $11.25/h plus room and board. If Hilda
works over 40 h in one week, her boss gives her a bonus of $8.50/h for each extra
hour. If Hilda works 57.5 h in one week, how much does she earn?
 Chapter 2 Earning an Income 105

3. Restaurant sales totalled $40 568.00 one month, and the average tip was 15%.

a) How much would each of the three waiters make in tips if they shared equally?

b) If they give 25% of their tips to the kitchen staff, how much will each
waiter make?

4. Franco earns $17.23/h, time and a half overtime, and a shift bonus of $2.65 for split
shifts. If he worked a total of 43.5 hours, 18 of which were split shifts, how much
did he earn if a regular work week was 38.5 hours?

5. Horace works as a door-to-door salesman in rural Alberta and must use his own
car. He is paid $0.45/km for each kilometre he drives, plus 8% of sales. If he drove
2354 km and sold $47 854.00 of merchandise, how much would his paycheque be?
106 MathWorks 10 Workbook

2.4 Deductions and Net Pay

NEW SKILLS: WORKING WITH NET INCOME

taxable income: Deductions are amounts of money taken off your gross pay for income tax (federal and
income after before-tax provincial or territorial), union dues, disability insurance, employment insurance (EI),
deductions have been pension plans (including the Canada Pension Plan or CPP), and health or other benefits.
applied, on which federal Income tax is paid on your taxable income.
and provincial taxes are
Each paycheque should list your gross pay, all deductions, and your net income. At
paid
the end of the year, your employer will supply you with a T4 slip that you will use to
net income: income prepare your income tax return.
after all taxes and other
deductions have been For more details, see page 79 of MathWorks 10.
applied; also called take-
Example 1
home pay
John’s group life insurance is 1.5% of his salary of $450.00 every two weeks. How much
does he pay for group life insurance?

SOL U T ION

Some deductions are Change 1.5% to a decimal and multiply by his salary.
taxable, and some
are not. For example,
1.5 ÷ 100 = 0.015
union dues and 0.015 × $450.00 = $6.75
company pension
He pays $6.75 for group life insurance per paycheque.
plans are before-tax
deductions, and so
they are not subject
to federal and
provincial taxes.
 Chapter 2 Earning an Income 107

BUILD YOUR SKILLS

1. If the federal tax rate is 15%, how much is deducted from your $750.00 paycheque? Income tax rates
vary with province
or territory,
salary, and family
circumstances.

2. If your short-term disability insurance rate is 0.5%, what do you pay if your
paycheque is $300.00?

3. If your Canada Pension Plan (CPP) contribution rate is 4.95% and your salary is
$1578.00 every two weeks, what will be the CPP deduction?
108 MathWorks 10 Workbook

Example 2

Jaar had a gross income of $785.00. His net income was $625.42. What percentage of
his gross pay were his deductions?

SOL U T ION

Calculate the amount of the deductions by subtracting his net income from his
gross income.

$785.00 − $625.42 = $159.58

Calculate what percentage $159.58 is of $785.00.

159.58 ÷ 785.00 ≈ 0.2033

0.2033 × 100 = 20.33%

His deductions are about 20.3% of his gross pay.

BUILD YOUR SKILLS

4. Samara’s monthly taxable income was $3276.54. If she paid $757.24 in taxes, what
percentage of her taxable income did she pay?
 Chapter 2 Earning an Income 109

5. Patricia’s before-tax deductions amounted to $75.47 on a gross salary of $700.00.

a) If she paid $93.68 in federal tax, what is her tax rate?

b) If she paid $36.85 in territorial tax, what is her tax rate?

6. Hans paid $37.51 Employment Insurance (EI) on his taxable monthly income of
$2168.21. What is the EI rate?
110 MathWorks 10 Workbook

Example 3

Alphonso has a gross income of $852.00 per week. His before-tax deductions include
union dues of 2.5% of his gross income and a company pension plan contribution of
3%. His federal tax rate is 16.2% and his provincial tax rate is 5.4%. He pays 4.95% to
the CPP and 1.8% for EI. Calculate his net income.

SOL U T ION

Calculate Alphonso’s taxable income by subtracting his union dues and his
company pension from his gross income.

First, calculate the amount of his union dues and company pension.

Union dues:

2.5 ÷ 100 = 0.025

0.025 × $852.00 = $21.30

Company pension:

3 ÷ 100 = 0.03
0.03 × $852.00 = $25.56

Next, subtract these amounts from his gross income to find his taxable income.

$852.00 − ($21.30 + $25.56) = $805.14

Calculate the amount of taxes, CPP, and EI using his taxable income.

Federal tax:

16.2 ÷ 100 = 0.162

0.162 × $805.14 = $130.43

Provincial tax:

5.4 ÷ 100 = 0.054

0.054 × $805.14 = $43.48

CPP:

4.95 ÷ 100 = 0.0495

0.0495 × $805.14 = $39.85
 Chapter 2 Earning an Income 111

EI:

1.8 ÷ 100 = 0.018

0.018 × $805.14 = $14.99

Alphonso’s net income will be his taxable income minus these deductions.

$805.14 − $130.42 − $43.48 − $14.49 = $576.89

Alphonso’s net income will be $576.89.

BUILD YOUR SKILLS

7. Randy works at two jobs. In one job, he earns $325.00/week, and has deductions
of $56.67 federal tax, $13.12 provincial tax, $16.09 CPP, and $4.14 EI. At his other
job, he earns $567.00/week and pays $79.42 federal tax, $16.82 provincial tax, and
$18.12 CPP. What is his net income?

8. As a part-time college instructor, Kathy teaches an introductory course on Mexican

history. She has a biweekly gross income of $3654.75. Her before-tax deductions
include a short-term disability premium of 0.5%, union dues of 3.1%, and a pension
amount of 4%. If she pays federal tax at a rate of 18.5%, provincial tax at a rate of
6.2%, CPP at 4.95%, and EI at 2.2%, what is her net income?
112 MathWorks 10 Workbook

9. Consider the following pay statement.

Employee Name: Hank

Company: Pay Begin Date: 06/01/2010 Net Pay: ???
Pay End Date:06/15/2010 Cheque Date: 06/15/2010

General Taxes Data

Employee ID: Job Title: Description Federal
Address: Pay Rate: $575.00 Claim Code 1
Annual: $29 900.00

Hours and Earnings Taxes

Current Description Current
Description Rate Gross Federal $48.01
Earnings Provincial $21.64
Regular $575.00/wk $575.00 CPP $25.13
EI $9.95
Total $104.73

a) What is Hank’s gross weekly income?

b) What is his net income?

c) What percent of his taxable income did he pay in federal taxes?

 Chapter 2 Earning an Income 113

PRACTISE YOUR NEW SKILLS

1. Juliana has an annual salary of $45 785.00.

a) How much does she pay in union dues if the rate is 2.4%?

b) How much does she pay in CPP if the rate is 4.95% and her taxable income is
$44 686.16?

2. Mario had $685.74 deducted in federal tax. If his taxable income was $2981.52,
what was his tax rate?

3. What will be your net pay if you have deductions of $105.30 federal tax, $23.76
provincial tax, $48.61 CPP, and $14.12 EI from your paycheque of $982.00?
114 MathWorks 10 Workbook

CHAPTER TEST

1. If Brenda earns $12.15/h and gets a 3.2% raise, how much will she earn per hour?

2. How much will you earn in a year as an apprentice metalworker if you are paid
$750.00 every two weeks?

3. What is your annual salary if your monthly salary is $3568.00?

4. As a medical technician, Stephanie has been offered a job that pays $53 000.00 per
year and another job that pays $25.50 per hour. Assuming a 40-hour work week
and all other conditions being the same, at which job will she earn more?
 Chapter 2 Earning an Income 115

5. Tommy made $20.55 commission on a $685.00 sale. What was his rate of
commission?

6. Von works as a car salesman. He earns 8% commission on the after-cost profit when
he sells a car. If he sells a car for $12 795.00 that cost the dealer $9280.00, how
much does he make?

7. Jenny earns $12.42/h, but earns double time and a half when she works on a
statutory holiday. If she works a 6-hour shift on a holiday, how much will she
earn that day?
116 MathWorks 10 Workbook

8. Harold works 40 hours regular time at $18.25 and 5.25 hours overtime at time and
a half. How much does he earn?

9. Nanette crochets scarves and sells them for $15.95 each. If material cost her
$7.52/scarf, how much does she make if she sells 9 scarves?

10. Cho bids $5750.00 for a contract. If he hires 4 men for 2 days (8 h/day) at a rate of
$12.50/h and his materials cost him $1675.84, how much does he earn?

11. How much CPP will be taken off your $782.45 taxable income at 4.95%?
 Chapter 2 Earning an Income 117

12. Padma has been offered isolation pay of $125.00/week to work as a park ranger in
northern Alberta.

a) How much will she make in a 40-hour work week if she is normally
paid $21.52/h?

b) Adding in the isolation pay, what is her hourly rate?

118 MathWorks 10 Workbook

Chapter 3
Length, Area, and Volume

Kristi Hansen is a Red Seal plumber. Calculating the capacity of water lines, determining the length of pipe needed for drainage
systems, and accurately predicting the volume of hot water a building’s system will use are some of her tasks.

3.1 Systems of Measurement

REVIEW: WORKING WITH PERIMETER

perimeter: the sum of the In this section, you will calculate the perimeter of different shapes.
lengths of all the sides of
A square is a quadrilateral with 4 equal sides, so the perimeter can be found by the
a polygon
following formula:

P = 4 × (side length)

The perimeter of a rectangle with length ℓ and width w can be found by the
following formula:

P = 2ℓ + 2w

P = 2(ℓ + w)
 Chapter 3 Length, Area, and Volume 119

Example 1

What is the perimeter of this figure? When the units of

measurement are
4.2 cm
all the same, you
can ignore them
4.4 cm
during calculations.
2.8 cm
Remember to add
the units in at the
end.

2.2 cm
2.1 cm 2.6 cm
2.0 cm

SOL U T ION To make sure that

you don’t miss
 his figure is a heptagon, which means it has 7 sides. Its perimeter, P, is the sum of
T
any sides when
the lengths of all 7 sides.
calculating the
P = 2.8 + 4.2 + 4.4 + 2.6 + 2.0 + 2.1 + 2.2 perimeter of a figure,
start at one vertex
P = 20.3 cm
and work your way
The perimeter is 20.3 cm. around the figure.
120 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Calculate the perimeters of the following diagrams.

A small square a)  18.3 cm

symbol in the corner
of a diagram means
that it is a right 8.5 cm
angle.

b)
12.3 cm

9.6 cm

6.2 cm

5.1 cm
10.4 cm

c) 0.9 m
1.2 m
0.9 m
2.3 m
 Chapter 3 Length, Area, and Volume 121

2. Darma is edging a tablecloth with lace. The tablecloth is 210 cm by 180 cm. How
much lace does she need?

3. Garry installs a wire fence around a rectangular pasture. The pasture measures
15 m by 25 m, and he uses three rows of barbed wire. How much wire did he use?

4. Chantal is building a fence around her swimming pool. The pool is 25 ft long and
12 ft wide, and she wants a 6-ft wide rectangular walkway around the entire pool.
How much fencing will she need?
122 MathWorks 10 Workbook

Example 2

The sides of the flower garden shown below are 4 m long. Each end is a semi-circle with
a diameter of 2 m. What is the perimeter of the flower garden?
4m

2m

SOL U T ION

Break this problem down into two parts, a circle and a rectangle.

circumference: the If you add the two end sections together, they form a circle. You can use the formula
measure of the perimeter for the circumference to find the perimeter:
of a circle
C = πd or 2πr

C is the circumference, r is the radius, d is the diameter, and π is a constant. In this

example, the diameter is 2 m.

Find the circumference of the ends of the flower garden by using this formula.

C = πd

C = π(2)

C ≈ 6.28

Add the lengths of the two straight parts to the circumference of the circle to
calculate the perimeter.

P ≈ 6.28 + 4 + 4

P ≈ 14.28

The perimeter of the flower garden is about 14.28 m.

 Chapter 3 Length, Area, and Volume 123

BUILD YOUR SKILLS

5. What is the circumference of a circular fountain if its radius is 5.3 m?

6. Johnny wants to put Christmas lights along the edge and peak of his roof. How
many metres of lights will he need?
5m
5m

28 m

7. Hershy uses coloured wire to make a model of the Olympic symbol (5 interlocking
circles). If each circle has a radius of 35 cm, how much wire does he need for
the rings?
124 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH SYSTEMS OF MEASUREMENT

Système International Although there are other systems of measurement, the two most common are the
(SI): the modern version Système International (SI) and the imperial system. In Canada, the official system of
of the metric system; uses measurement is the SI. Because of Canada’s close proximity to the United States, you
the metre as the basic should be familiar with both systems. Both are used in certain contexts.
unit of length
Below are listed some common imperial units of length and their relationships.
imperial system: the
system most commonly 12 inches (in or ″) = 1 foot (ft or ′)
used in the United States; 36 inches = 1 yard (yd)
the standard unit of
measurement for length is 3 feet = 1 yard
the foot 5280 feet = 1 mile (mi)

1760 yards = 1 mile

If you look at a ruler For more details, see page 94 of MathWorks 10.
marked in imperial
units, you will notice Example 3
that it is usually Wilhelmina, a seamstress, is sewing bridesmaids’ dresses. She orders the fabric from the
divided into halves, United States, where fabric is measured in yards. Each dress requires 3 3 yards of silk,
4
quarters, eighths, 1 1 yards of lace fabric, and 7 1 yards of trim. How much of each type of material does
2 4
and sixteenths, Wilhelmina need to make 5 dresses?
whereas the SI
system uses tenths.
SOL U T ION

Multiply each amount by 5.

silk = 3 3 × 5
4
silk = 15 ×5 Convert to an improper fraction and multiply.
4
silk = 75
4
silk = 18 3 yd Convert to a mixed fraction.
4

lace fabric = 1 1 × 5
2
3
lace fabric = × 5 Convert to an improper fraction and multiply.
2
 Chapter 3 Length, Area, and Volume 125

lace fabric = 15
2
1
lace fabric = 7 2 yd Convert to a mixed fraction.

trim = 7 1 × 5
4
trim = 29 ×5 Convert to an improper fraction and multiply.
4
trim = 145
4
1
trim = 36 4 yd Convert to a mixed fraction.

Since fabric can be bought in partial yards, Wilhelmina will need to purchase
18 3 yd of silk, 7 1 yd of lace fabric, and 36 1 yd of trim.
4 2 4

A LT ERN AT I V E SOL U T ION

Convert each mixed fraction to a decimal, and then multiply by 5.

silk = 3 3
4
silk = 3.75
silk = 3.75 × 5
silk = 18.75
lace fabric = 1 1
2
lace fabric = 1.5
lace fabric = 1.5 × 5
lace fabric = 7.5
trim = 7 1
4
trim = 7.25
trim = 7.25 × 5
trim = 36.25

Since fabric can be bought in partial yards, Wilhelmina will need to purchase
18.75 yd of silk, 7.5 yd of lace fabric, and 36.25 yd of trim.
126 MathWorks 10 Workbook

BUILD YOUR SKILLS

A 2 by 4 is not 8. B
 ernard is buying some lumber to finish a project. He needs 3 pieces of 2 by 4 that
exactly 2″ by 4″. are each 4 1 feet long, and 10 pieces of 2 by 2 that are each 5 1 feet long. How
2 4
The name comes much of each does he need in total?
from the dimensions
of the lumber before
it is dried; when
the lumber dries, it
shrinks and then is
replaned to make
it a standard size.
9. Benjie is replacing some plumbing pipes. He needs 3 pieces of copper pipe: one
A 2 by 4 is actually
piece is 2 feet long, one is 5 feet 7 inches long, and one is 4 feet long. How much
1 1 ″ by 3 1 ″.
2 2 copper pipe does he need if he loses 1 inch when he cuts the pipe and he can only
Lumber and other
buy it in even numbers of feet?
building supplies are
usually sold using
imperial units.

10. If each board in a fence is 6 inches wide, how many of them will José need to fence
a playground that is 60 feet wide by 125 feet long?

 board is 6″ wide
 Chapter 3 Length, Area, and Volume 127

Example 4

Fatima is trying to calculate how much baseboard she will need for the room
shown below.
15ʹ 8″

24″

9ʹ 2″

2ʹ 6″

6ʹ 6″ 9″ 9″
27″

5ʹ 2″

What is the minimum amount of baseboard she will need?

SOL U T ION

Find the perimeter of the room. Since there is a door, no baseboard will be needed
there. Measurements are given in feet and/or inches.

 o find the perimeter of the room, start at any one point, such as the edge of the
T Where did the
door, and work your way around the room. second 9ʹ 2″ come
from?
P = 9″ + 27″ + 5ʹ 2″+ 27″+ 6ʹ 6″+ 9ʹ 2″+ 15ʹ 8″+ 9ʹ 2″+ 24″+ 24″+ 9″

Add feet to feet and inches to inches to get 44′ 140″.

Convert 140 inches to feet by dividing by 12 (because 12 inches equals 1 foot).

140 = 11 remainder 8, or 11ʹ 8″
12
Add this to the measure in feet.

44ʹ + 11ʹ 8″ = 55ʹ 8″

Therefore, she needs 55ʹ 8″ of baseboard.

128 MathWorks 10 Workbook

A LT ERN AT I V E SOL U T ION

Combine the lengths of the smaller wall segments to simplify the calculation, and
subtract the width of the door.

P = 2(15ʹ 8″) + 2(9ʹ 2″) + 2(27″) + 2(24″) − 2ʹ 6″

P = 30ʹ 16″ + 18ʹ 4″ + 54″ + 48″ − 2ʹ 6″

P = 46ʹ 116″

Convert 116 inches to feet.

116 = 9, remainder 8, or 9ʹ 8″
12
Add this to the measure in feet.

46ʹ + 9ʹ 8″ = 55ʹ 8″

Therefore, she needs 55′ 8″ of baseboard.

BUILD YOUR SKILLS

11. A pet shop stores 5 pet cages that are 2′ 8″ wide, 3 cages that are 4′ 6″ wide, and
2 cages that are 1′ 8″ wide. Can these cages fit side by side along a wall that is
30′ long?

12. A circular garden is 6ʹ 4ʺ in diameter. To plant a geranium approximately every foot

along the circumference, how many geraniums are needed?
 Chapter 3 Length, Area, and Volume 129

13. The height of a basement ceiling is 7′ 2″. A 6″-deep heating pipe runs across the
middle. To enclose it, there must be a 1-inch space between the pipe and the
drywall. Will Craig, who is 6′ 6″ tall, be able to walk under the finished pipe?

PRACTISE YOUR NEW SKILLS

1. Convert the following measurements.

a) 42 inches to feet

b) 16 inches to feet and inches

c) 96 inches to yards

d) 5 miles to yards
130 MathWorks 10 Workbook

2. You are building a fence around your vegetable garden in your backyard. If
the garden is 12′ 8″ long and 4′ 6″ wide, what is the total length of fencing you
will need?

3. Marjorie is building a dog run that is 25′ 8″ long and 8′ 8″ wide. How much fencing
will she need if the opening is 3′ 6″ wide and will not need fencing?

4. A package of paper is 2″ high and 8.5″ wide. If a warehouse shelf is 1′ 5″ high and
2 yards long, how many packages of paper can be put on the shelf?

5. Jennine estimates that each step she takes is 18″ long and that she takes 1550 steps
per block. How many blocks must she walk if she wants to walk 5 miles?
 Chapter 3 Length, Area, and Volume 131

Converting Measurements 3.2

REVIEW: WORKING WITH VARIABLES WITHIN FORMULAS

In this section, you will practise substituting known values into formulas.

In this chapter, you will need to use the following formulas.

A = ℓw Area of a rectangle, where ℓ is the length and w is the width.

A = πr2 Area of a circle, where r is the radius and π is the constant, pi.

A = 1 bh A
 rea of a triangle, where b is the length of the base and h is the height.
2
A = πrs Area of the surface of a cone, where r is the radius and s is the
slant height.

C = 2πr Circumference of a circle, where r is the radius.

Example 1

Sumo is a traditional Japanese martial art. The area of a circular sumo ring, or dohyoi, is
16.26 m2. What is the radius of the ring?

SOL U T ION

Use the formula for finding the area of a circle. You are given the area of the circle,
so substitute it into the formula and solve for the unknown value, r.

A = πr 2
16.26 = πr 2
2
16.26 = π r Divide both sides by π to isolate r2.
π π
16.26 = r
π
2.28 ≈ r

The radius of a sumo ring is 2.28 m.

132 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Ina is laying turf in a yard measuring 38 ft by 20 ft. What is the yard’s area in
square feet?

2. A store advertises a circular rug as being 4.9 m2. Travis wants a rug to fit a
rectangular space that is 2.6 m by 2.6 m. Will this rug fit?

3. You are designing a rectangular label for canned food. The can is 5 cm high, with
a diameter of 9 cm. To plan your design, calculate the label’s length. (The length is
equal to the circle’s cirumference.)

NEW SKILLS: WORKING WITH DIFFERENT SYSTEMS OF MEASUREMENT

The official system of measurement in Canada is the SI, but the United States uses
imperial units. If you are buying products from the United States or are doing business
with a US company, you will need to convert between the two systems of measurement.

Below are some common relationships between SI and imperial units of length.

1 inch ≈ 2.54 centimetres

1 foot ≈ 0.3 metres

1 yard ≈ 0.9 metres

1 mile ≈ 1.6 kilometres

For more details, see page 106 of MathWorks 10.

 Chapter 3 Length, Area, and Volume 133

Example 2

Mary is delivering a load of goods from Vancouver, BC, to Seattle, WA, then in Seattle,
she is picking up another load to deliver to Albuquerque, NM. The distance from
Vancouver to Seattle is 220 km and the distance from Seattle to Albuquerque is 1456
mi. The odometer in Mary’s truck records distance in kilometres.

a) What is the total distance she will travel, in kilometres?

b) If her odometer read 154 987 km when she left Seattle, what did it read when she
left Vancouver?

c) What will her odometer read when she reaches Albuquerque?

SOL U T ION

a) Find the distance in kilometres from Seattle to Albuquerque.

1 mi ≈ 1.6 km

1456 mi ≈ 1456 × 1.6

1456 mi ≈ 2330 km

The distance from Seattle to Albuquerque is 2330 km.

Add this to the distance from Vancouver to Seattle to find the total distance.

220 km + 2330 km = 2550 km

Her trip will be about 2550 km.

b) Subtract the distance she travelled from the odometer reading.

154 987 – 220 = 154 767

Her odometer read 154 767 when she left Vancouver

c) Add the distance from Seattle to Albuquerque to the odometer reading.

154 987 + 2330 = 157 317

Her odometer should read about 157 317 when she reaches Albuquerque.
134 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. Suzanne purchased tiles for her patio that are 8″ by 4″. She measured her patio
in metres and wants to convert the tile dimensions to SI units. What are the
dimensions of the tiles in centimetres?

5. Benjamin owns an older American truck. The odometer shows distance travelled in
miles. On a recent trip to deliver produce for his employer, he drove 1564 mi. His
employer pays him $0.89/km for the use of his own truck. How much will he be
reimbursed for the use of his truck for the trip?
 Chapter 3 Length, Area, and Volume 135

6. Marnie owns a carpet store and sells hallway runners for $9.52/linear foot.

a) How much is this per linear yard?

b) How much is this per linear metre?

c) Ralph needs 3.9 m of the runner for his hallway. How much will it cost?
136 MathWorks 10 Workbook

Example 3

Rebecca is planning to install sod in her backyard, which is 18.2 m by 9.8 m. If sod
costs $0.28/ft2, how much will it cost to sod the backyard?

SOL U T ION

Change the measurements of the backyard to feet, and then find the area.

0.3 m = 1 ft

18.2 m = 18.2 ft
0.3
18.2 ≈ 60.7 ft
0.3
Her yard is approximately 61 ft long.

Similarly, change 9.8 m (980 cm) to feet.

1 m = 1 ft
0.3
9.8 m = 9.8 ft
0.3
9.8 ft ≈ 32.7 ft
0.3
Her yard is approximately 33 ft wide.

Calculate the area of her backyard. The area of a rectangle is calculated by

multiplying the length by the width.

A = ℓw

A = 61 × 33

A = 2013 ft2

She will need approximately 2013 square feet of sod at $0.28/ft2.

2013 × $0.28 = $563.64

It will cost about $563.64 to sod her backyard.

 Chapter 3 Length, Area, and Volume 137

BUILD YOUR SKILLS

7. You could have solved Example 3 by determining the cost of the sod per square
metre. Answer the question using this method. Is your answer the same? Why
or why not?

8. Kuldeep has been hired to lay terracotta tiles on a floor that measures 4.2 m by
3.8 m. The tiles are 9″ by 9″ and come in boxes of 12.

a) How many boxes must he buy? (He cannot buy a partial box.)

b) If the tiles cost $18.95 per box, how much will the tiles cost in total?
138 MathWorks 10 Workbook

9. Toula calculates the cost of cementing the bottom and sides of a circular pond.
When all costs are considered, the job will cost $175.85 per square metre of finished
area. If the pond has a radius of 3 feet and a depth of 2 feet, how much will she
charge for the job?
 Chapter 3 Length, Area, and Volume 139

PRACTISE YOUR NEW SKILLS

1. Juan is a picture framer. He is framing a picture that is 24 inches by 32 inches with

a frame that is 2.5 inches wide. What is the outer perimeter of the framed picture:

a) in inches?

b) in feet and inches?

c) in yards, feet, and inches?

2. Charlie drove from Calgary to Saskatoon, which is a distance of 620 km. How far is
this in miles?
140 MathWorks 10 Workbook

3. A school custodian must mark off a field that is 150 ft by 85 ft. His tape measure
is marked in metres. What are the dimensions of the field in metres (to the nearest
tenth of a metre)?

4. Jeff knows that his semi-trailer truck is 3.2 m high. A tunnel is marked as
“Max height: 10′ 6″.” Will Jeff’s truck fit through the tunnel?

5. Carla needs 3.5 m of cloth. If the cloth she wants to purchase costs $9.78/yd, how
much will the cloth cost?
 Chapter 3 Length, Area, and Volume 141

6. Ari, a gardener, estimates the cost of seeding a 150 m by 210 m area with grass
seed. He needs 3 pounds of seed per 100 000 square feet. How many pounds of
seed will Ari need?

7. A room measures 12′ 8″ by 10′ 9″. Carpeting costs $45.98/m2. A customer will
have to purchase 10% more carpeting than floor area due to waste and he cannot
purchase partial square metres. What is the minimum cost of the carpeting?
142 MathWorks 10 Workbook

3.3 Surface Area

NEW SKILLS: WORKING WITH SURFACE AREA

surface area: the total Surface area is the area that would be covered by a three-dimensional (3-D) object if
area of all the faces, you could lay it out flat. A net is a diagram of a 3-D object seen as a flat surface.
or surfaces, of a three-
If you know how to find the area of 2-D shapes, you can find the surface area of
dimensional object;
3-D objects by breaking them down to their component surfaces and adding the
measured in square units
areas together.
net: a two-dimensional
pattern used to construct For more details, see page 115 of MathWorks 10.
three-dimensional shapes
Example 1
4.4 yd
Akiko has been hired to paint the
exterior of a storage bin. If the bin is
a rectangular prism that measures 2.8 yd

2.3 yards by 4.4 yards by 2.8 yards,

what is the surface area of the bin?
2.3 yd
SOL U T ION

Sketch a net of the bin.

2.3 yd A5

4.4 yd A4 A3 A2 A1

2.3 yd 2.8 yd 2.3 yd

2.3 yd A6

2.8 yd
 Chapter 3 Length, Area, and Volume 143

Calculate the area of each of the rectangles and add together to find the surface
area (SA).

A1 = ℓw

A1 = 4.4 × 2.3

A1 = 10.12 yd2

The area of A3 is the same as A1.

A2 = ℓw

A2 = 4.4 × 2.8

A2 = 12.32 yd2

The area of A4 is the same as A2.

A5 = ℓw

A5 = 2.3 × 2.8

A5 = 6.44 yd2

The area of A6 is the same as A5.

Total surface area:

SA = A1 + A2 + A3 + A4 + A5 + A6

SA = 10.12 + 12.32 + 10.12 + 12.32 + 6.44 + 6.44

SA = 57.76 yd2

The bin has a surface area of 57.76 yd2.

144 MathWorks 10 Workbook

A LT ERN AT I V E SOL U T ION

Many people prefer to measure the different surfaces of the object.

There are:

2 rectangles that are 4.4 yd by 2.8 yd (front and back)

2 rectangles that are 4.4 yd by 2.3 yd (top and bottom)

2 rectangles that are 2.3 yd by 2.8 yd (ends)

Find the area of each rectangle and then add.

A1 + 3 = 2 × 4.4 yd × 2.3 yd

A1 + 3 = 20.24 yd2

A2 + 4 = 2 × 4.4 yd × 2.8 yd

A2 + 4 = 20.24 yd2

A5 + 6 = 2 × 2.3 yd × 2.8 yd

A5 + 6 = 12.88 yd2

SA = A1 + 3 + A2 + 4 + A5 + 6

SA = 24.64 yd2 + 20.24 yd2 + 12.88 yd2

SA = 57.76 yd2

The bin has a surface area of 57.76 yd2.

 Chapter 3 Length, Area, and Volume 145

BUILD YOUR SKILLS

1. Jim has been hired to make a jewellery box. If the box is 12 inches long, 6 inches
deep, and 9 inches tall, how much veneer will it take to cover the exterior,
assuming no waste?

2. Anita is building a greenhouse onto the side of her garage. She wants it to be 6 feet
long, 4 feet wide, and 3 feet high, with the 6-foot long side against the side of her
garage. What area of glass will she need to complete the greenhouse? (Hint: No glass
will be used along the side of the garage or for the floor.)

3. Vicki is tiling her 35″ by 35″ shower stall. The tiles reach the 8-foot ceiling on
3 sides. How many square inches of tiles should she purchase to tile the walls
and floor?
146 MathWorks 10 Workbook

Example 2

A canning factory wants to use as little metal as possible to make its cans. It considers
two can sizes that each hold about the same amount. One is 4.5 inches tall with a radius
of 4.2 inches and another is 9.6 inches tall with a radius of 2.8 inches.

Which can should they use? Why?

2.5″

4.2″
9.6″

4.5″

SOL U T ION

To find out which can uses the least amount of metal, calculate the surface area
of each can.

The top and bottom of the cans are circles. The area of a circle can be found by the
following formula:

A = πr2

If the side of the can is rolled out flat, it will form a rectangle, where the width is the
height of the can and the length is the circumference of the circle.
 Chapter 3 Length, Area, and Volume 147

Can #1 Can #2

Top and bottom: Top and bottom:

A = 2πr2 A = 2πr2

A = 2π(4.2)2 A = 2π(2.8)2

A ≈ 110.8 in2 A ≈ 49.2 in2

Side: Side:

A = 2πrh A = 2πrh

A = 2π(4.2) (4.5) A = 2π(2.8) (9.6)

A ≈ 118.7 in2 A ≈ 168.8 in2

SA = Atop + bottom + Aside SA = Atop + bottom + Aside

SA ≈ 110.8 + 118.7 SA ≈ 49.2 + 168.8

SA ≈ 229.5 in2 SA ≈ 218 in2

Considering surface area, they should use the second can because it uses
less material.

BUILD YOUR SKILLS

4. A cylindrical shipping tube is 48 inches tall and 6 inches in diameter.

a) What is its surface area in square inches?

148 MathWorks 10 Workbook

b) What is its surface area in square feet?

5. Sanjiv designs a cylindrical box to hold 4 tennis balls stacked one on top of the
other. If a tennis ball is approximately 3 1 inches in diameter, what is the surface
4
area of the box? (Ignore the thickness of the material.)

6. Jennifer must make a conical funnel out of sheet metal. If the funnel is 9 inches tall,
has a slant height of 10.7 inches, and has a radius of 5.8 inches at the top, what is
the surface area of the sheet metal in square feet?
 Chapter 3 Length, Area, and Volume 149

Example 3

Harry has to paint the walls and ceiling of a room that is 12 ft long, 10 ft wide, and
8 1 ft high. There is a 6 ft by 4 ft window, a 2 1 ft by 7 ft doorway, and a mirrored
2 2
closet door that is 6 ft by 7 ft. What surface area must he paint?

SOL U T ION

Find the total area of the four walls and the ceiling and subtract the areas of the
window and doors.

Area of ceiling:

A1 = 12 ft × 10 ft
A1 = 120 ft 2

Area of long walls:

A2 = 12 ft × 8 1 ft × 2
2
A2 = 204 ft 2

Area of shorter walls:

A3 = 10 ft × 8 1 ft × 2
2
A3 = 170 ft 2

Area of walls and ceiling, including openings:

A4 = A1 + A2 + A3
A4 = 120 ft 2 + 204 ft 2 + 170 ft 2
A4 = 494 ft 2

Calculate the areas of the window, door, and closet that will not be painted.

Area of window:

A5 = 6 ft × 4 ft
A5 = 24 ft 2

Area of door:

A6 = 2 1 ft × 7 ft
2
A6 = 17 1 ft 2
2
150 MathWorks 10 Workbook

Area of closet door:

A7 = 6 ft × 7 ft
A7 = 42 ft 2

Total area that does not need painting:

A8 = A5 + A6 + A7

A8 = 24 ft 2 + 17 1 ft 2 + 42 ft 2
2
A8 = 83 1 ft 2
2
Total area to be painted:

Atotal = A4 − A8

Atotal = 494 ft 2 − 83 1 ft 2
2
Atotal = 410 1 ft 2
2
The total area to be painted is 410 1 ft2.
2

BUILD YOUR SKILLS

7. Geneviève plans to apply two coats of paint to the walls of her garden shed. The
shed is 8 feet long by 6 feet wide by 7 feet tall. If there are 3 windows that are 2 feet
by 18 inches each, what will be the total area she paints?
 Chapter 3 Length, Area, and Volume 151

8. A metal cylindrical canister is 1′ 3″ long and has a diameter of 4 inches. What is the
total surface area of the cylinder?

9. Jerg is finishing the sides and bottom of a hot tub. The hot tub is 6′ 11″ long,
5′ 6″ wide, and 3′ deep. It has a bench going all the way around the inside that is
16″ high and 15″ wide that will not be finished. How much finishing material will
Jerg need?
152 MathWorks 10 Workbook

Example 4

The wood that Terrance wants to use to make a shelving unit costs $6.49/ft2. How much
will it cost him (assuming no wastage) to make a shelving unit that is 4 ft wide by 12
inches deep by 5 ft tall if there are 4 shelves (plus the top and bottom)?

SOL U T ION

Sketch the unit.

5 ft

4 ft

12″

There will be 6 pieces (shelves) that are 12″ (or 1′) by 4′.

There will be 2 pieces (ends) that are 1′ by 5′.

There will be 1 piece (back) that is 4′ by 5′.

Find the areas of each piece and add.

A1 = 6 × 1 ft × 4 ft
A1 = 24 ft 2
A2 = 2 × 1 ft × 5 ft
A2 = 10 ft 2
A3 = 1 × 4 ft × 5 ft
A3 = 20 ft 2

Total area = 24 ft2 + 10 ft2 + 20 ft2

Total area = 54 ft2

Cost:

$6.49 × 54 = $350.46

The total cost of the wood will be $350.46.

 Chapter 3 Length, Area, and Volume 153

BUILD YOUR SKILLS

10. Randi is installing flooring in her den. The room is 12 feet by 19 feet. A fireplace
that is 6 feet wide juts out 2 feet into the room. Also, there are 2 built-in bookcases,
each 1 foot deep and 3 feet wide. She needs to order 12% more flooring than
required because of wastage and cutting. How much will it cost if the wood costs
$5.25/ft2?

11. Sheet metal costs $54.25/yd2. How much will it cost Hamish to cover a conical roof
if it has a radius of 2.2 yards and a slant height of 3.5 yards?
154 MathWorks 10 Workbook

12. Ted is wallpapering his bedroom. The room is 10 ft long by 8 ft wide by 9 1 ft

2
tall. There is a door measuring 3 ft by 7 ft and a window measuring 3 ft by 4 ft.
Each double roll covers approximately 56 ft2 and costs $29.95. How much will the
wallpaper cost?
 Chapter 3 Length, Area, and Volume 155

PRACTISE YOUR NEW SKILLS

1. A storage bin is a rectangular prism that measuring 114 cm long by 56 cm high by

56 cm deep. What is the surface area of the exterior, in square feet?

2. How much glass is needed to construct a hexagonal (6-sided) fish tank? The tank is
4 feet tall and each pane is 1 1 feet wide. (The tank’s bottom is not made of glass.)
2
156 MathWorks 10 Workbook

3. Stan is deciding between patio tiles that are 39 cm by 39 cm and tiles that are 18 cm
by 27 cm. His patio is 3 m by 2.5 m. Considering area only, how many of each type
of tile would he need?

4. A chocolate bar is in the shape of a triangular prism. The box is 8 1 inches long
2
and the ends are equilateral triangles with sides measuring 1 3 inches. What is the
4
surface area of the box? (Hint: You need to find the height of the triangle. Use the
following formula, A = 1 bh )
2
 Chapter 3 Length, Area, and Volume 157

5. Jocelyne is designing a logo for the outside of a cylindrical water bottle. She knows
that the bottle is 7 1 in tall and has a diameter of 3 3 in. How large can the label
2 4
on the water bottle be?

6. A conical paper cup has a slant height of 3 1 inches and a diameter of 3 inches.
8
How much paper is needed to make the cup?

7. The Great Pyramid of Giza has a square base that, when built, was approximately
756 ft on each side. If the slant height of each triangular side was approximately
610 ft, what was the original surface area of the pyramid?
158 MathWorks 10 Workbook

3.4 Volume

NEW SKILLS: WORKING WITH VOLUME

volume: the amount of The volume of a solid is a measure of how much space it occupies. Volume is measured
space an object occupies in cubic units.

The volume of a prism is calculated using the following formula.

V = Abase × h

Volume is the product of the area of the base times the height of the object.

For more details, see page 124 of MathWorks 10.

Example 1

Alfred has a bulk container that holds 2000 cubic inches of dog biscuits. He plans to
sell the biscuits in small boxes that measure 5″ by 8″ by 6″. How many boxes will he
need to sell all the dog biscuits?

SOL U T ION

Find the volume of the small box.

V = ℓwh

V = 5″ × 8″ × 6″

V = 240 cu in

Divide 2000 by 240.

2000 + 240 = 8.3

Round up, so that all of the biscuits fit in boxes. Alfred would need 9 small boxes.
 Chapter 3 Length, Area, and Volume 159

BUILD YOUR SKILLS

1. A fish tank is a rectangular prism that is 30 inches long, 24 inches deep, and
18 inches high. How much water will it hold:

a) in cubic inches?

b) in cubic feet?

2. Petra must stack boxes that are 3 ft by 2 1 ft by 1 1 ft onto a truck. What is the
2 2
volume of each box?

3. Will the contents of a box that is 3 inches by 4 inches by 6 inches fit into a cube
with sides of 4 inches?
160 MathWorks 10 Workbook

Example 2

Paulino runs a landscaping business. He needs to cover an area that is 10.8 m by 9.5 m
with 10 cm of topsoil. How much will it cost if the soil costs $18.75/yd3, and soil is
available in multiples of 1 yd3?
2

SOL U T ION

Remember: Find the volume of topsoil needed.

10 cm = 0.1 m
V = ℓwh

V = 10.8 m × 9.5 m × 0.1 m

V = 10.26 m3

Calculate how many yards are in 1 metre.

1 cm = 1 in
2.54
100 cm = 100 in
2.54
100 cm = 39.37 in
1 m = 39.37 in

1 in = 1 yd
36
39.37 in = 39.37 yd
36
1 m ≈ 1.0
09 yd

Calculate how many cubic yards are in a cubic metre by cubing both sides.

1 m3 = 1.09 yd3

1 m3 ≈ 1.3 yd3

Since he needs 10.26 cubic metres, change cubic metres to cubic yards.

1.3 yd 3
10.26 m 3 = 10.26 m 3 ×
1 m3
10.26 m 3 ≈ 13.3 yd 3
 Chapter 3 Length, Area, and Volume 161

He needs 13.3 yd3, but Paulino will have to round up to the nearest 1 yd3, to
2
13.5 yd3. Since each cubic yard costs $18.75, multiply the number of cubic yards by
the cost per cubic yard.

13.5 × $18.75 = $253.13

The topsoil will cost $253.13.

A LT ERN AT I V E SOL U T ION

Begin by changing the measurements to yards.

Using the calculations above, 1 m is approximately 1.09 yd.

10.8 m = 10.8 × 1.09 yd

10.8 m ≈ 11.8 yd

9.5 m = 9.5 × 1.09 yd

9.5 m ≈ 10.4 yd

0.1 m = 0.1 × 1.09 yd

0.1 m ≈ 0.1 yd

Find the volume in cubic yards.

V = ℓwh

V = 11.8 × 10.4 × 0.1

V = 12.3 yd3

Paulino will need to round up to the nearest 0.5 cubic yard, so he will need
12.5 yd3.

Find the cost by multiplying the number of cubic yards by the cost per cubic yard.

12.5 × $18.75 = $234.38

It will cost approximately $234.38 for the topsoil. This answer is different from the
first answer because you round at a different point in the calculation.
162 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. A garden bed is 4′ by 3′, and a 6″ layer of soil will be spread over the garden. A bag
of soil contains 2 ft3 of soil. How many bags are needed to cover the garden?

5. Karl buys bales of hay that measure 15″ × 24″ × 36″. He needs to buy 250 bales,
and he needs to know if they will fit in his barn. What is the total volume of hay in
cubic feet?

6. A computer measures 9 inches by 16 inches by 16 1 inches. It requires 1 1 inches

2 2
of padding on each side to protect it during shipping. What is the volume of the
packed box?
 Chapter 3 Length, Area, and Volume 163

NEW SKILLS: WORKING WITH CAPACITY

The capacity of a container is the amount it can hold. Capacity is the volume of a capacity: the maximum
container. Capacity is often used with liquid measures. amount that a container
can hold
In the SI, the basic unit of capacity is the litre. A litre is one one-thousandth of a cubic
metre, or 1000 cubic centimetres.

In imperial units, capacity is measured in gallons.

4 quarts = 1 gallon

2 pints = 1 quart

2 cups = 1 pint

However, the gallon has two different sizes:

• The British gallon is approximately 4.5 litres and 1 pint is 20 fluid ounces.

• The American (US) gallon is approximately 3.8 litres and 1 pint is 16 fluid ounces.

In measuring liquids for recipes, the US system is often used. 1 cup is actually
237 mL, but in
1 teaspoon (tsp) = 5 millilitres (mL)
cooking it is rounded
1 tablespoon (tbsp) = 15 mL to 250 mL for easier
measuring.
1 cup = 250 mL

For more information, see page 124 of MathWorks 10.

164 MathWorks 10 Workbook

Example 3

Paula is opening a French bakery and wants to make authentic French recipes. All the
recipes are given in metric units, but she has imperial measuring devices. The crème
brulée recipe requires 500 mL of cream and 1.25 mL of vanilla.

a) How much cream will she need, in cups?

b) How much vanilla will she need, in teaspoons?

c) How much cream will she need, in fluid ounces?

SOL U T ION

a) Convert 500 mL to cups.

1 cup = 250 mL
1 cup
500 mL = 500 mL ×
250 mL
500 mL = 2 cups

She will need 2 cups of cream.

b) Convert 1.25 mL to teaspoons.

1 tsp = 5 mL
1 tsp
1.25 mL = 1.25 mL ×
5 mL

1.25 mL = 0.25 tsp or 1 tsp

4
She will need 1 tsp of vanilla.
4
c) Convert 500 mL to fluid ounces.

500 mL = 2 cups
2 cups = 1 pint
1 pint = 16 fl oz

She will need 16 fl oz of cream.

 Chapter 3 Length, Area, and Volume 165

BUILD YOUR SKILLS

7. Serina is travelling through the US and her car’s gas tank has a capacity of 55 litres.

a) How much is this in American gallons?

b) If gas costs $2.99/gal in Bellingham, WA, how much will it cost to fill her car
(assuming that it is totally empty)?

c) Assuming she has the same car in London, England, where gas costs $9.86/gal
(converted from pounds), how much will it cost to fill her tank? (Remember that
the British gallon is a different size.)

8. The box of Jakob’s cube van has inside dimensions of 20 feet (length), 10 feet 8
inches (width), and 12 feet 6 inches (height). Calculate the volume of the interior.
166 MathWorks 10 Workbook

9. Bev has two storage bins for grain. The first bin is 12 feet 8 inches by 8 feet 9 inches
and is filled to a level height of 4 feet 6 inches. If she has to move the grain to a bin
with a base measuring 9 feet by 9 feet, what will be the level height of the grain in
the second bin?

PRACTISE YOUR NEW SKILLS

1. A storage container measures 6 feet by 3 feet by 4 feet.

a) What is the volume in cubic feet?

b) What is the volume in cubic yards?

2. A recipe calls for 2 3 cups of milk. How much is this in mL?

4
 Chapter 3 Length, Area, and Volume 167

3. What is the capacity of a 5-fl oz jar, in mL? (Hint: 1 fl oz = 30 mL.)

4. If your car’s fuel consumption rate is 8.8 L/100 km, how many US gallons will you
need for a trip of approximately 450 km?

5. In spring, a tree needs 10.5 fl oz of fertilizer. If fertilizer is sold in a 4.1-L bottles,

how many bottles are needed to fertilize 15 trees?
168 MathWorks 10 Workbook

6. The exterior of a concrete container will be 10 feet by 8 feet by 4 feet tall. The walls
and the bottom are 6 inches thick. What will it cost to construct it if concrete is
$98.95/cubic yard?
 Chapter 3 Length, Area, and Volume 169

CHAPTER TEST

1. The tallest person in the world was Robert Pershing Wadlow. At the time of his
death in 1940, he was 8′ 11.1″ tall. The record for the world’s shortest adult was held
by He Pingping; at the time of his death in 2010, he was 2′ 4.7″.

a) What is the difference between their heights in feet and inches?

b) Find the height of each man in metres.

c) What is the difference between their heights in metres?

170 MathWorks 10 Workbook

2. While driving in the United States, Franklin sees that the height of a tunnel is marked
as 10′6″. He knows that his truck is 3.3 m tall. Can he drive through the tunnel?

3. Rachelle is buying panelling for wainscotting for her hall. The panels are 4′ by 8′.
The wainscotting will be 4′ high and the room is 19′ by 13′. There are two doors
measuring 30″ wide, and one 12′ window that will only need a 2′ panel of
wainscotting under it. How many panels will she need, assuming no wastage?
 Chapter 3 Length, Area, and Volume 171

4. Mario is laying tiles for the patio below and planting daffodils around the perimeter.
12ʹ 8″

9ʹ 4″

3ʹ 4″

3ʹ 4″

a) Assuming he needs to buy 10% more than the area due to wastage, how many
12″ by 12″ tiles will he need?

b) How many daffodils will he plant around the perimeter if there are no daffodils
along the entrance and he plants them approximately 1 foot apart?
172 MathWorks 10 Workbook

5. Louise needs to give the exterior of a cylindrical granary 2 coats of paint. If the
granary is 10 feet tall and has a diameter of 14 feet, and paint covers approximately
375 square feet per gallon, how many gallons of paint will she need to buy? Assume
that she can only buy full gallons, and will not be painting the roof.

6. Roberto is painting the exterior of a rectangular storage unit to protect it from

rusting. If the unit is 7′ 6″ wide, 9′ 8″ long, and 8′ 2″ tall, what is the surface area in
square feet? Roberto will be painting the sides and the roof.
 Chapter 3 Length, Area, and Volume 173

7. A soccer field is 109 m long and 73 m wide. American soccer league rules state that
a field should be no more than 120 yards long and 80 yards wide. Is the field within
the specified dimensions?

8. What is the volume of water in a fish tank that is 90 cm by 55 cm if it is filled to a

height of 32 cm?

9. A driveway is 36 ft long and 10 ft wide, and will be covered in gravel that is 2 in

deep. How many cubic yards of gravel will be needed?
174 MathWorks 10 Workbook

10. In the US, milk is commonly sold in jugs of 1 gal, 1 gal, 1 quart, and 1 pint. What
2 2
are the equivalent sizes in millilitres?

11. A recipe for pumpkin cheesecake calls for a 5-US fl oz can of evaporated milk.

a) What is this in cups?

b) What is this in mL?

 175

Chapter 4Mass, Temperature, and Volume

To make good bread, the ingredients must be measured accurately and the dough stored and baked at the correct temperature. Cam
McCaw, a Red Seal pastry chef, turns out hundreds of loaves of bread daily.

Temperature Conversions
4.1
NEW SKILLS: WORKING WITH TEMPERATURE

I f you travel to the United States, you will notice that a different temperature scale is The Celsius scale
used there. The US uses the Fahrenheit scale (°F) of the imperial system, while Canada used to be called the
uses the Celsius scale (°C) of the SI. centigrade scale,
and it is sometimes
In the SI, water freezes at 0°C and boils at 100°C. In the imperial system, water freezes
referred to this way.
at 32°F and boils at 212°F. Since water freezes at 0°C and 32°F, the relationship between
the two temperature systems can be calculated with the following formulas, where C
represents degrees Celsius and F represents degrees Fahrenheit.

C = 5 (F − 32) or F = 9 C + 32
9 5
For more details, see page 138 of MathWorks 10.
176 MathWorks 10 Workbook

Example 1

When working  hile visiting Florida, Kathy heard a local person say that it had been very cold
W
with temperatures, overnight, as it was only 42°. At first, she thought this was not cold, but then
convert them to the Kathy realized the person meant degrees Fahrenheit. What was the temperature in
nearest tenth of a degrees Celsius?
degree.

SOL U T ION

Use the following formula, and substitute 42 for F.

C = 5 (F − 32)
9
C = 5 (42 − 32)
9
C = 5 (10)
9
C = 50
9

( )
C = 55 °
9

Convert this mixed numeral to a decimal.

55 = 5+ 5
9 9
5 =5÷9
9
5 ≈ 0.6
9
55 ≈ 5.6
9
The temperature is about 5.6°C, which would be very cold in Florida.
 Chapter 4 Mass, Temperature, and Volume 177

BUILD YOUR SKILLS

1. A cake recipe says to bake at 350°F. Your oven only shows temperatures in degrees
Celsius. At what temperature should you set your oven?

2. Sophie is making fudge in France, using an American cookbook. She needs to

cook the chocolate until the temperature is 238°F, but her thermometer only
shows temperatures in degrees Celsius. What temperature does her fudge mixture
need to reach?

3. Firefighters can estimate the temperature of a burning fire by the colour of its
flame. A clear orange flame has a temperature of about 2190°F. How hot is this in
degrees Celsius?
178 MathWorks 10 Workbook

Example 2

Sverre was paving a road with heated tar during a hot summer day. He noted that the
external temperature of the tar was 48°C. What was this in degrees Fahrenheit?

SOL U T ION

Use the formula for converting degrees Fahrenheit to degrees Celsius, and
substitute 48 for C.

F = 9 C + 32
5
F = 9 (48) + 32
5
F = 432 + 32
5
F ≈ 86.4 + 32
F ≈ 118.4

The temperature was approximately 118.4°F.

BUILD YOUR SKILLS

4. The normal temperature for a dog is from 99°F to 102°F. Ashley’s dog has a
temperature of 40°C. Convert the temperature to Fahrenheit to calculate if it falls
within the normal range.

5. Roger is painting the exterior of a house. He should not apply the paint if the
temperature is below 45°F. The temperature is 9°C. Is it safe to apply the paint?
 Chapter 4 Mass, Temperature, and Volume 179

6. C
 hinook winds are known to cause great changes in temperature over a short
period of time. The most extreme temperature change in a 24-hour period occurred A chinook wind
in Loma, Montana, on January 15, 1972. The temperature rose from –54°F to 49°F. is a warm, dry
wind that blows
a) What was the change in temperature in degrees Fahrenheit?
east of the Rocky
Mountains, often
causing significant
temperature
b) What were the minimum and maximum temperatures in degrees Celsius?
increases in a short
time in winter.

c) What was the change in temperature in degrees Celsius?

PRACTISE YOUR NEW SKILLS

1. Convert the following temperatures to degrees Fahrenheit.

a) 35°C b) −8°C

c) 165°C d) 21°C

e) −40°C f) 202°C
180 MathWorks 10 Workbook

2. Convert the following temperatures to degrees Celsius.

a) −20°F b) 80°F

c) 375°F d) 2°F

e) 0°F f) −2°F

3. Which is hotter: a blowtorch flame at 1300°C or a candle flame at 1830°F? By how

much is one flame hotter than the other in each scale?
 Chapter 4 Mass, Temperature, and Volume 181

4. When Harry mixes different materials to pave a road, he knows that they must
be kept at the following temperatures in degrees Fahrenheit. Calculate the
temperatures in degrees Celsius.

a) Bituminous material must be between 200°F and 260°F.

b) Water solution must be between 65°F and 100°F.

c) The mixing gel must be between 160°F and 210°F.

5. In 1992, the temperature in Pincher Creek, Alberta, rose from –19°C to 22°C in
just one hour due to a chinook wind. What were these temperatures in degrees
Fahrenheit?
182 MathWorks 10 Workbook

6. When the human body reaches a temperature of 41°F, it is said to be in a state of

“medical emergency.” What is this temperature in degrees Celsius?

7. On May 26, 1991, Mount Logan, YT, recorded the coldest temperature outside of
Antarctica at –106.6°F. What is this temperature in degrees Celsius?

8. Some of the tiles on the outside of a space shuttle are able to withstand temperatures
of 2300°F. What is this in degrees Celsius?
 Chapter 4 Mass, Temperature, and Volume 183

Mass in the Imperial System

4.2
NEW SKILLS: WORKING WITH WEIGHT

The mass of an object refers to the quantity of matter in it, and it remains constant, no mass: a measure of the
matter where the object is located. The weight of an object is a measure of the force of quantity of matter in an
gravity on the object. On earth, the mass and the weight of an object are essentially the object
same; on other planets where the pull of gravity is different, weight and mass will not weight: a measure of
be the same. the force of gravity on an

The basic units of weight in the imperial system are ton (tn), pound (lb), and ounce (oz). object

1 ton (tn) = 2000 pounds

1 pound (lb) = 16 ounces (oz)

For more details, see page 146 of MathWorks 10.

Example 1

Manuela needs 1 pound 2 ounces of Gruyère cheese, 12 ounces of cheddar cheese, and
11 ounces of Swiss cheese for a fondue recipe. How many pounds of cheese does she
need in all?

SOL U T ION

Add pounds to pounds and ounces to ounces.

 1 pound + 2 ounces

 + 12 ounces

 + 11 ounces

 1 pound 25 ounces

Change 25 ounces to pounds.

16 ounces = 1 pound
25 ounces = (16 + 9) ounces

25 ounces = 1 pound 9 ounces

184 MathWorks 10 Workbook

Total weight = 1 pound 25 ounces

Total weight = 1 pound + 1 pound 9 ounces
Total weight = 2 pounds 9 ounces

Manuela needs 2 lb 9 oz of cheese for her recipe.

A LT ERN AT I V E SOL U T ION 1

Change 1 pound 2 ounces to ounces.

1 lb 2 oz = 16 oz + 2 oz
1 lb 2 oz = 18 oz

Add all the weights.

18 oz + 12 oz + 11 oz = 41 oz

Change to pounds by dividing by 16.

41 oz = 41 oz × 1 lb
16 oz

( )
41 oz = 41 lb
16

41 oz = ( 2 9 ) lb
16
41 oz = 2 lb 9 oz

A LT ERN AT I V E SOL U T ION 2

You could also change the ounces to pounds and work with decimals or fractions.

1 lb 2 oz of Gruyère equals 1 plus 2 lb, or 1.125 lb.

16
12 oz of cheddar equals 12 lb, or 0.75 lb.
16
11 oz of Swiss equals 11 lb, or 0.6875 lb.
16
Total weight = 1.125 + 0.75 + 0.6875

Total weight ≈ 2.56 lb

 Chapter 4 Mass, Temperature, and Volume 185

BUILD YOUR SKILLS

1. Rochelle gave birth to twin boys weighing 6 lb 5 oz and 5 lb 14 oz. What was their
total weight?

2. The weight of water is approximately 2 pounds 3 ounces per litre. How much will
8 litres of water weigh?

3. If a basket of raspberries weighs 12 ounces and you need 4 pounds to make jam,
how many baskets do you need to buy?
186 MathWorks 10 Workbook

Example 2

The cab of Arthur’s semi-trailer truck weighs 8.7 tons and the trailer weighs 6.4 tons. If
the loaded gross weight of the truck is 21.3 tons, what is the weight of the load:

a) in tons?

b) in pounds?

SOL U T ION

a) The total weight of the truck is found by adding the weight of the cab and
the trailer.

8.7 tn + 6.4 tn = 15.1 tn

Subtract this amount from the gross weight to get the weight of the load.

21.3 tn − 15.1 tn = 6.2 tn

The weight of the load is 6.2 tons.

b) Since 1 ton equals 2000 pounds, find the weight in pounds by multiplying.

6.2 tn = 6.2 tn × 2000 lb

1 tn
6.2 tn = 12 400 lb

The weight of the load is 12 400 pounds.

 Chapter 4 Mass, Temperature, and Volume 187

BUILD YOUR SKILLS

4. An elevator has a maximum load restriction of 1.5 tons. Is it safe for two tile layers
weighing 195 lb and 210 lb to load it with 65 boxes of tile weighing 42 lb each?

5. A small truck weighs approximately 1300 lb. It is loaded with cement slabs that
weigh 150 lb each. If the maximum loaded weight of the truck is 2.75 tons, how
many slabs can be loaded?

6. Kurt is planting wheat at the rate of 90 pounds per acre. If he plans to plant
320 acres of wheat, how many tons of wheat will he use?
188 MathWorks 10 Workbook

Example 3

A 12-ounce can of vegetables costs $1.49. A 1 lb 2-oz can of the same vegetables costs
$2.19. Which is the better buy?

SOL U T ION

Use unit pricing to find the cost per ounce.

12 ounces cost $1.49.

12 oz = $1.49

( 1212 ) oz = $112.49
1 oz = $1.49
12
1 oz = $0.1242

1 lb 2 oz costs $2.19.

1 lb 2 oz = 16 oz + 2 oz

1 lb 2 oz = 18 oz

18 oz = $2.19

( 1818 ) oz = $218.19
1 oz = $2.19
18
1 oz ≈ $0.1217

The 1 lb 2-oz can is the better buy.

 Chapter 4 Mass, Temperature, and Volume 189

BUILD YOUR SKILLS

7. An 18-oz jar of peanut butter costs $3.29, a 28-oz jar costs $4.79, and a 2.5-lb jar
costs $5.99. Which is the best buy?

8. If knitting yarn costs $6.24 per 3-oz skein, how much will it cost to knit a sweater
that requires 1 pound of yarn? (You cannot buy partial skeins.)
190 MathWorks 10 Workbook

9. About 200 cocoa beans are used to make 1 lb of chocolate. Beans are shipped in
200-lb sacks, which contain about 88 000 beans. How many 1.5-oz chocolate bars
can be made from one sack of beans?

Example 4

Valérie bought 4 pounds 6 ounces of steak for dinner at $2.74/lb. After removing the
excess fat, she had only 4 pounds of meat. What was her true cost per pound?

SOL U T ION

Change 6 ounces to pounds.

1 lb = 16 ounces
6 oz = (6 ÷ 16) lb
6 oz = 0.375 lb

She bought 4.375 pounds of meat at $2.74/lb.

Total cost:

4.375 × $2.74 = $11.99

Since only 4 pounds were usable, calculate how much each pound cost her.

$11.99 ÷ 4 = $3.00

Valérie paid approximately $3.00/lb for the steaks.

 Chapter 4 Mass, Temperature, and Volume 191

BUILD YOUR SKILLS

10. Zara buys 8 pounds 12 ounces of strawberries at $1.98/lb. What is her true cost per
pound if 10% of the berries rot before she uses them?

11. Mark bought 8 bags of sand, each weighing 25 lb, for $1.68/bag. One bag ripped
and he lost all the sand. What was his true price per pound of sand?

12. Alyson paid $28.45 for 24 ounces of coffee beans, but when she checked, the actual
weight was 22 ounces. What was her true cost per ounce?
192 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. Calculate the conversions.

a) 24 oz = _____________ lb b) 7890 lb = _____________ tn

c) 54 oz = _______ lb ________ oz d) 6 lb 2 oz = _____________ oz

e) 4.54 tn = _____________ lb f) 654 oz = _______ lb ________ oz

2. What is the total weight, in pounds and ounces, of six books on a shelf if they weigh
12 oz, 1 lb 7 oz, 1 lb 2 oz, 15 oz, 9 oz, and 1 lb 3 oz?
 Chapter 4 Mass, Temperature, and Volume 193

3. A bakery uses a recipe for oatmeal cookies that calls for 1 lb 4 oz of flour to make
9 dozen cookies. How many ounces of flour are needed to make 3 dozen cookies?

4. Kris needs to transport 5 slabs of concrete to an apartment work site. If each slab
weighs 46 pounds, Kris weighs 195 pounds, and the truck weighs 1.5 tons, what is
the total weight of the loaded truck in pounds?

5. Harinder is concerned about the weight that paint might add to a delicate structure
he built. He estimates that he needs 1.5 gal of paint and that the structure can
withstand 15 lb of weight. The weight of a particular paint is 9 lb/gal. When it dries,
the weight is only 5.4 lb/gal. Can Harinder paint his structure without having it
collapse?
194 MathWorks 10 Workbook

6. U-pick organic blueberries sell for $20.00 for a 12-pound box.

a) How much would 1 pound cost?

b) How much would 12 ounces cost?

7. What is the true cost per pound of a 10-pound box of oranges if the original
price of the box was $12.99 and 1 of them had to be thrown away because they
4
were mouldy?
 Chapter 4 Mass, Temperature, and Volume 195

Mass in the Système International

4.3
NEW SKILLS: WORKING WITH SI UNITS OF MASS

In the Système International (SI), the kilogram is the basic unit of mass, but it is kilogram: the mass of
commonly used for weight as well. one litre of water at 4°C

1000 grams (g) = 1 kilogram (kg)

1000 milligrams (mg) = 1 gram

1 tonne (t) = 1000 kilograms Note that a tonne (t)

is not the same as a
For more details, see page 154 of MathWorks 10.
ton (tn). A tonne is

Example 1 sometimes referred

to as a metric ton.
A recipe for cornbread calls for 120 g of flour, 170 g of cornmeal, and 50 g of sugar. If
you double the recipe, what is the total weight of the dry ingredients?

SOL U T ION

You can either add the weights together and then double the sum, or double the
weights and then add them together.

Flour: 2 × 120 g = 240 g

Cornmeal: 2 × 170 g = 340 g

Sugar: 2 × 50 g = 100 g

Add the weights of all the ingredients.

240 g + 340 g + 100 g = 680 g

There are 680 g of dry ingredients in the doubled recipe.

196 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. What is the total weight of a loaded truck if the truck weighs 2.6 tonnes and it is
loaded with 15 skids of boxes that weigh 210 kilograms each? Give your answer
in tonnes.

2. Irène needs 1.6 kg of tomatoes to make her grandmother’s recipe for ratatouille. She
has baskets of tomatoes that weigh 256 g, 452 g, 158 g, and 320 g. How many more
grams of tomatoes does she need?

3. Genoa salami sells for $1.79/100 g at the deli.

a) How much will 350 g cost?

b) What is the price per kilogram?

 Chapter 4 Mass, Temperature, and Volume 197

NEW SKILLS: WORKING WITH MASS/WEIGHT

CONVERSION BETWEEN IMPERIAL AND SI

I n this section, you will need to work with the relationship between the SI and the To estimate a
imperial units of weight. One kilogram weighs about 2.2 lb. You can also use this conversion from
information to convert from grams to ounces and tonnes to tons. pounds to kilograms,
you can think of
For more details, see page 157 of MathWorks 10.
a pound as being

Example 2 about 21 kg.

Lorinda is baking apple pies. According to her recipe, she needs 6 pounds of apples. The
bag of apples she bought only shows the weight in kilograms. How many kilograms of
apples does she need?

SOL U T ION

2.2 lb = 1 kg
1 kg
1 lb =
2.2
1 kg
6 lb = 6 ×
2.2
6 lb ≈ 2.7 kg

She will need approximately 2.7 kg of apples.

198 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. A recipe calls for 180 g of flour. How much is this in ounces?

5. A baby weighed 7 pounds 12 ounces at birth. How much did it weigh in grams?

6. Chen weighs 68 kg. How much does he weigh in pounds?

 Chapter 4 Mass, Temperature, and Volume 199

Example 3

The cost of bananas is $0.49/lb at one store, but you see an advertisement for bananas
on sale at another store for $1.05/kg. Which is the better buy?

SOL U T ION

Convert the price of bananas at the first store to kilograms.

1 lb costs $0.49.

1 kg = 2.2 lb

Therefore, 1 kg costs 2.2 times the cost of 1 lb.

2.2 × $0.49 = $1.08

One kilogram of bananas at the first store costs about $1.08. The sale at the second
store is a better buy.

BUILD YOUR SKILLS

7. How much does 1 pound of beef cost if the butcher shop sells it for $9.74/kg?

8. Which is the better buy: 200 g of coffee beans at $3.85 or 1 pound at $9.60?
200 MathWorks 10 Workbook

9. The dosage of a certain medicine is 0.05 mg/kg of weight. Tom weighs 185 lbs.

a) How many milligrams of the medicine should he take?

b) If the medicine costs $1.95/mg, what will his dosage cost?

PRACTISE YOUR NEW SKILLS

1. Convert the following weights.

a) 2.5 t = _____________ kg

b) 2.8 kg = _____________ g

c) 125 g = _____________ kg
 Chapter 4 Mass, Temperature, and Volume 201

d) 2.4 g = _____________ kg

e) 1 t = _____________ lb

f) 3.6 tn = _____________ kg

2. How many tons are in 1 tonne?

3. What is the total weight in grams of 3 packages of nuts weighing 1.2 kg, 0.75 kg,
and 1.5 kg?

4. Win weighs 78 kg and his dog weighs 18 kg. If his truck weighs 1.9 t and there are
5 boxes of books each weighing 9.8 kg in the truck, what is the total weight of the
truck, including Win, his dog, and the books?
202 MathWorks 10 Workbook

5. Karen is making a batch of potato soup. She needs 8 potatoes, and each potato
weighs about 375 g. How many pounds of potatoes does she need?

6. If a 10-lb bag of grass seed costs $75.45, how much does the seed cost per kilogram?

7. How many quarter-pound (before cooking) hamburgers can you make from 1.9 kg
of ground beef?
 Chapter 4 Mass, Temperature, and Volume 203

Making Conversions
4.4
NEW SKILLS: WORKING WITH CONVERSIONS BETWEEN
MEASURES OF VOLUME AND WEIGHT

You have converted measures from one unit to another, within the SI or imperial
system, or between them.

 ere, you will convert from a unit of volume to a unit of weight. For example, grain is
H 1 bushel = 2220 in3
often measured in bushels, a volume measure, but its weight may be needed to judge or approximately
whether it is a safe load for a truck. Each grain has a different weight, so conversions 8 gallons.
between bushels and weight depend on knowing the conversion factor.

For more details, see page 162 of MathWorks 10.

Example 1

How many bushels (bu) of flax seed are there in 2.4 tonnes, if the conversion factor is
39.368 bushels/tonne?

SOL U T ION

A conversion factor of 39.368 means that there are 39.368 bushels of flax seed per
tonne. To find the number of bushels, multiply.

39.368 bu
2.4 t × = x bu
1t
2.4 × 39.368 = 94.5

There are approximately 94.5 bushels of flax seed.

204 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Laila bought 5 bushels of sunflower seeds. If the conversion factor is 73.487 bu/t,
what is the weight of the sunflower seeds:

a) in kilograms?

b) in pounds?

2. The conversion factor for white beans is 36.744, and for corn it is 39.368. Which
weighs more per unit volume?

3. If Jore gets $195.76 per metric ton for wheat, how much does he earn per bushel
(conversion factor 36.744 bu/t)?
 Chapter 4 Mass, Temperature, and Volume 205

4. If one bushel of triticale grain is about 2220 cubic inches, how many bushels are in
a pile that measures approximately 8 feet by 6 feet by 5 feet?

5. How many tonnes of rye are there is 900 bushels if there are 39.368 bushels/tonne?

6. Wheat is 36.744 bushels/tonne and sunflower seeds are 73.487 bushels/tonne. What
does this tell you about the relative weights of wheat and sunflower seeds?
206 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH CONVERSION BETWEEN

SI AND IMPERIAL UNITS OF WEIGHT

You can use the following equivalencies to convert between SI and imperial units.

1 lb ≈ 0.45 kg

1 oz ≈ 28.3 g

1 tn ≈ 0.9 t

BUILD YOUR SKILLS

7. Alphonse is making chicken kebabs for 14 people. His recipe suggests about 7 oz
of chicken per person. At the grocery store, the weight of chicken is labelled in
kilograms. How much chicken does Alphonse need to buy?

8. A crane can lift a maximum of 5 t. Sandstone weighs about 150 lb per cubic foot,
and a container contains 70 cubic feet of sandstone. Can the crane be used to load
the container onto a train?
 Chapter 4 Mass, Temperature, and Volume 207

9. Josephine is sending a gift of a bottle of maple syrup that weighs 3 lb, and
3 packages of smoked salmon jerky that each weigh 100 g. If the package’s total
weight is less than 2 kg, she can ship it at a cheaper rate. Will she be able to do so?

PRACTISE YOUR NEW SKILLS

1. If 1 bushel is approximately 2220 cubic inches, approximately how many bushels of

grain are there in a bin that is 8 feet by 8 feet by 4 feet?
208 MathWorks 10 Workbook

2. A truck has a maximum load limit of 5000 kg. Can it safely carry 230 bushels of
canola, if the conversion factor is 44.092 bushels/tonne?

3. How many kilograms are in 1 ton?

4. A sign posted in an elevator says “Maximum capacity 1400 lb.” If the average weight
of an adult is 80 kg, how many average-weight adults can the elevator carry?
 Chapter 4 Mass, Temperature, and Volume 209

5. Recall that 1000 cubic centimetres equal 1 litre. How many millilitres are in a box
that is 10 cm by 5 cm by 3 cm?

6. A hectare (ha) is an area measure of 10 000 square metres. How many hectares are
there in a field that is 620 m by 380 m?
210 MathWorks 10 Workbook

CHAPTER TEST

1. Convert the following temperatures.

a) 25°C = ______ °F b) 25°F = ______ °C

c) –40°C = ______ °F d) –25°F = ______ °C

e) 405°F = ______ °C f) 45°C = ______ °F

2. A welder’s electrical arc has a temperature ranging from 500°C to 20 000°C. What
is this in degrees Fahrenheit?
 Chapter 4 Mass, Temperature, and Volume 211

3. Convert the following weights.

a) 12 lb 4 oz = _____________ oz b) 2.3 tn = _____________ lb

c) 5284 lb = _______ tn ________ lb d) 3 3 lb = _____________ oz

4

e) 165 oz = _____________ lb f) 454 g = _____________ lb

4. It is estimated that recycling 1 ton of paper saves about 17 trees. About how many
trees are saved if 8254 tons of paper are recycled?
212 MathWorks 10 Workbook

5. Eva bought 3 rainbow trout for dinner. They weighed 3 lb 5 oz, 2 lb 12 oz, and 3 lb
8 oz. She cut off the heads and the tails and was left with 8 lb 2 oz. What was the
amount of waste?

6. The weight of a piece of raw silk that is 100 yards long by 1.25 yards wide (standard
width) is about 38 pounds. The weight of an equal amount of habutai silk is about
12 pounds. If Katharine bought pieces of raw silk and habutai silk that were both
12.5 yards long, how much would they weigh together?

7. A robin’s egg weighs about 70 g. How many eggs would it take to make 1 kilogram?
 Chapter 4 Mass, Temperature, and Volume 213

8. Huang bought 12 boxes of floor tiles that weigh 288 pounds each. How much is this
in kilograms?

9. Soybeans have a conversion factor of 36.744 (bu/t). How much do 45 bushels weigh?
214

Chapter 5Angles and Parallel Lines

Angles and parallel lines provide BC’s Liard River Bridge with strength, stability, and visual appeal. The bridge was built in 1942.

5.1 Measuring, Drawing, and Estimating Angles

REVIEW: WORKING WITH ANGLES

In this section, you will review types of angles and how to classify them.

An angle is formed when two rays meet at a point called the vertex. Angles are usually
measured in degrees using a protractor. Angle measures range from 0° to 360°.

Angles are:

•• acute, if their measure is between 0° and 90°;

•• right, if their measure is 90°; the two rays are perpendicular to each other;

•• obtuse, if their measure is between 90° and 180°;

•• straight, if their measure is 180°; and

•• reflex, if their measure is between 180° and 360°.

 Chapter 5 Angles and Parallel Lines 215

Example 1

Identify the type of angle: acute, right, obtuse, straight, or reflex.

a) b) c)

d) e)

SOL U T ION

a) This is a right angle.

b) This is a straight angle.

c) This is an obtuse angle.

d) This is an acute angle.

e) This is a reflex angle.

BUILD YOUR SKILLS

1. Identify the type of angle: acute, right, obtuse, straight, or reflex.

a) 68° b) 215° c) 91° d) 32°

e) 180° f) 99° g) 195° h) 265°

216 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH REFERENT ANGLES

In many jobs, people have to draw angles or estimate their measure. To estimate the size
of an angle, you can use referent angles, which are angles that are easy to visualize. You
can use these referents to determine the approximate size of a given angle.

90°
45° 30° 60°

For more details, see page 174 of MathWorks 10.

Example 2

Use the referents above to estimate the size of each of the following angles. Use a
protractor to check your answers.
B


C

SOL U T ION

∠ is a symbol used ∠A is slightly bigger than the 45° referent. It is probably about 50°.
to indicate an angle.
∠B is less than 30°, so it is probably between 15° and 20°.

∠C is close to a straight angle but it is probably about 10° less. Therefore, ∠C is

approximately 170°.

Using a protractor, measure the angles.

∠A is 52°.

∠B is 18°.

∠C is 172°.
 Chapter 5 Angles and Parallel Lines 217

BUILD YOUR SKILLS

2. Use the referents to determine the approximate size of the following angles.

A B C D

3. What is the approximate angle of the railing on the stairs?

4. Use the referents in the New Skills section to determine the approximate size of the
following angles.

C D
218 MathWorks 10 Workbook

5. Jason is doing a survey of a city block. What is the approximate angle between his
sightings of the two buildings?

Example 3

complementary angles: Given each of the following angles, determine the size of the complement and/or the
two angles that have size of the supplement (if they exist).
measures that add up
a) 75°
to 90°

supplementary angles: b) 43°

two angles that have c) 103°
measures that add up
to 180° d) 87°

e) 300°

Two angles with

SOL U T ION
the same measure
are referred to as To find the complement, subtract the angle measure from 90°.
congruent.
To find the supplement, subtract the angle measure from 180°.
 Chapter 5 Angles and Parallel Lines 219

a) Complement:

90° − 75° = 15°

Supplement:

180° − 75° = 105°

b) Complement:

90° − 43° = 47°

Supplement:

180° − 43° = 137°

c) Complement: The angle is already greater than 90° so there is no complement.

Supplement:

180° − 103° = 77°

d) Complement:

90° − 87° = 3°

Supplement:

180° − 87° = 93°

e) The angle is greater than 180°, so it has no complement or supplement.

BUILD YOUR SKILLS

6. Fill in the chart with the complement and the supplement of each angle, if they
exist. If they don’t exist, state why.

ANGLE COMPLEMENTS AND SUPPLEMENTS

Angle Complement Supplement
45°

78°

112°

160°

220°
220 MathWorks 10 Workbook

7. The complement of an angle is 58°.

a) What is the size of the angle?

b) What is the supplement of the angle?

8. The complement of an angle is 0°.

a) What is the size of the angle?

b) What is the size of the supplement of the angle?

 Chapter 5 Angles and Parallel Lines 221

NEW SKILLS: WORKING WITH TRUE BEARING

In navigation and map-making, people often measure angles from the vertical, or north. true bearing: the angle
The angle, measured in a clockwise direction from a line pointing north, is referred to measured clockwise
as the true bearing. Straight north has a bearing of 0°. between true north and
an intended path or
For more details, see page 182 of MathWorks 10.
direction, expressed in
N degrees

NNW NNE

NW NE

WNW ENE

W E

WSW ESE

SW SE
SSW SSE

Example 4

A boat is heading directly southwest. What is its true bearing?

SOL U T ION
N

W E

If the boat is heading southwest, measuring from the vertical will give you an
obtuse angle of 225° (45° beyond a straight angle).
222 MathWorks 10 Workbook

BUILD YOUR SKILLS

9. If a boat is travelling 25° south of straight east, what is its true bearing?

10. What is the true bearing of a boat travelling south?

11. What is the true bearing of a boat travelling north-northwest?

PRACTISE YOUR NEW SKILLS

1. Identify the type of angle: acute, right, obtuse, straight, or reflex.

a) 56°

b) 91°

c) 270°

d) 170°

e) 43°

f) 192°
 Chapter 5 Angles and Parallel Lines 223

2. Estimate, using referents, the size of the angles indicated in the diagrams.

a) b)

x x

c) d) y

3. If Renata cuts a rectangular tile diagonally, one of the acute angles formed is 65°.
What is the size of the other acute angle?

4. Pete is laying irregularly shaped paving stones. He needs to find one to fit in
position A. Approximately what size of angle will it have?


224 MathWorks 10 Workbook

5. On the map below, what is the bearing from the following points?

Point A

Point B

Point C

a) A to B

b) B to C
 Chapter 5 Angles and Parallel Lines 225

Angle Bisectors and Perpendicular Lines

5.2
NEW SKILLS: WORKING WITH ANGLE BISECTORS

To bisect something is to cut it into two equal parts. An angle is bisected by a ray that angle bisector: a
divides it into two angles of equal measure. The ray that divides the angle is called an segment, ray, or line that
angle bisector. separates two halves of a
bisected angle
Perpendicular lines are two lines that form a right angle. A right angle (90°) can be
thought of as a bisected straight (180°) angle. The process used to draw perpendicular
lines is the same as drawing angle bisectors because a perpendicular line bisects a
straight angle.

For more details, see page 187 of MathWorks 10.

Point C bisects AB if AC is equal to CB.

A C B

B D Ray BD bisects ∠ABC if ∠ABD is equal to ∠DBC.

Example 1

Bisect ∠ABC using a straight edge and compass.

A

B C
226 MathWorks 10 Workbook

SOL U T ION

To draw a ray, BD, that bisects ∠ABC , follow these steps:

•• With the compass point at B, draw an arc to intersect BA and BC at X and Y,

respectively.

•• With the compass point at X and the radius more than half of XY, draw a small
arc in the interior of ∠ABC .

•• With the same radius and compass point at Y, draw a small arc to intersect
this arc at D.

•• Join B and D.

BD is the bisector of ∠ABC.

A

X
D

B C
Y

BUILD YOUR SKILLS

1. If a right angle is bisected, what is the size of each angle?

2. Bisect the given angles using a straight edge and compass.

a) b) c)
 Chapter 5 Angles and Parallel Lines 227

3. An angle is bisected. Each resulting angle is 78°. How big was the original angle?

4. The size of one resulting angle after the original angle is bisected is equal to the
supplement of the original angle. What is the measure of the original angle?

Example 2

Using a protractor, determine which of the following lines are perpendicular. In the workplace,

ℓ1 ℓ2 carpenters often
use framing squares
and levels to ensure
that they have right
ℓ3 angles. A framing
square is a tool that
is a right angle.

ℓ4

ℓ5

SOL U T ION

The angles formed between ℓ1 and ℓ3 are each 90°, so ℓ1 and ℓ3 are perpendicular.

The angles formed between ℓ1 and ℓ4 are not 90°, so ℓ1 and ℓ4 are not perpendicular.

The angles formed between ℓ2 and ℓ3 are not 90°, so ℓ2 and ℓ3 are not perpendicular.

The angles formed between ℓ2 and ℓ4 are not 90°, so ℓ2 and ℓ4 are not perpendicular.

The angles formed between ℓ5 and ℓ3 are not 90°, so ℓ5 and ℓ3 are not perpendicular.

The angles formed between ℓ5 and ℓ4 are each 90°, so ℓ5 and ℓ4 are perpendicular.
228 MathWorks 10 Workbook

BUILD YOUR SKILLS

5. A crooked table leg makes an angle of 86.7° with the tabletop. How much must the
carpenter move the leg so that it is perpendicular to the tabletop?

6. At what approximate angle does the hill incline from the horizontal?

7. A carpenter is inlaying different types of wood on a tabletop. What must be the size
of angles a, b, c, and d?
60° 75° 75° 55°
a b

d
60° 75° c
 Chapter 5 Angles and Parallel Lines 229

PRACTISE YOUR NEW SKILLS

1. Use a protractor to determine whether these lines are perpendicular.

a) b)

c) d)

2. Complete the following table.

ANGLE CALCULATIONS
Angle Complement Supplement Resulting angle
measure after the
angle is bisected
73°

12°

15°

132°

90°

34°

49°

68°

100°

127°
230 MathWorks 10 Workbook

3. Kaleb is edging a garden bed with square tiles. In the corner shown below, he wants
two congruent tiles. At what angle must he cut the tiles so that they fit?

x
y

135°

4. Calculate the size of the indicated angles. Name as many pairs of complementary
and supplementary angles as possible.

a) b) 144°

70° x
y
x

c) d)

x 81° 115°
x

5. The angle at the peak of a roof is 135°. Calculate the measure of the angle formed by
the rafter and the king post.
135°
rafter

king post
 Chapter 5 Angles and Parallel Lines 231

Non-Parallel Lines and Transversals

5.3
NEW SKILLS: WORKING WITH ANGLES FORMED BY INTERSECTING LINES

A line that intersects two other lines at two distinct points is a transversal. When two transversal: a line that
non-parallel lines are intersected by a transversal, they form angles of varying sizes. intersects two or more
lines
Consider the diagram below: t is a transversal that intersects ℓ1 and ℓ2.
corresponding angles:
t
two angles that occupy
the same relative
2 1 position at two different
3 4
ℓ1 intersections

vertically opposite
6 5
ℓ2 angles: angles created
7 8
by intersecting lines that
share only a vertex

alternate interior
Eight angles are formed.
angles: angles in
• ∠1 and ∠5, and ∠4 and ∠8, ∠2 and ∠6, and ∠3 and ∠7 are pairs of opposite positions
corresponding angles. between two lines
intersected by a
• ∠1 and ∠3, and ∠2 and ∠4, ∠5 and ∠7, and ∠6 and ∠8 are pairs of vertically
transversal and also on
opposite angles.
alternate sides of the
• ∠3 and ∠5, and ∠4 and ∠6 are pairs of alternate interior angles. same transversal

• ∠2 and ∠8, and ∠1 and ∠7 are pairs of alternate exterior angles. alternate exterior
angles: angles in
• ∠3 and ∠6, and ∠4 and ∠5 are pairs of interior angles on the same side of the opposite positions outside
transversal. two lines intersected by a

• ∠1 and ∠8, and ∠2 and ∠7 are pairs of exterior angles on the same side of the transversal

transversal.

For more details, see page 198 of MathWorks 10.

232 MathWorks 10 Workbook

Example 1

A transversal is I n the following diagram, identify each of the following, and specify which lines and
not necessarily one transversals you are using.
line segment in a ℓ1 ℓ2
specific drawing.
In this figure, there
are several lines
1 6
that intersect other 2 5
8 ℓ3
9
lines. These can
be considered
7
transversals.
4

3
ℓ4

a) an interior angle on the same side of the transversal as ∠6

b) an angle corresponding to ∠2

c) an angle corresponding to ∠4

d) an alternate interior angle to ∠4

SOL U T ION

a) Using ℓ1 and ℓ2, with transversal ℓ3, ∠2 and ∠6 are interior angles on the same
side of the transversal.

b) Using ℓ1 and ℓ2, with transversal ℓ3, ∠1 corresponds to ∠2.

c) Using ℓ1 and ℓ2, with transversal ℓ4, ∠7 corresponds to ∠4.

d) Using ℓ3 and ℓ4, with transversal ℓ2, ∠4 and ∠9 are alternate interior angles.
 Chapter 5 Angles and Parallel Lines 233

BUILD YOUR SKILLS

1. In the diagram below, identify the relationship between each pair of angles.

2 1
5 8

7 3
4 6

a) ∠7 and ∠8

b) ∠2 and ∠7

c) ∠1 and ∠6

d) ∠5 and ∠7

2. Given the diagram below, identify the following angles.

2 1
3

7 4
5 6

a) an alternate exterior angle to ∠2

b) an interior angle on the same side of the transversal as ∠7

c) an alternate interior angle to ∠4

d) an angle corresponding to ∠5
234 MathWorks 10 Workbook

3. Identify each of the following angles. Name the two lines and the transversal you
are using.
ℓ2
ℓ1

1 2 3
ℓ3
5 4

8 10 ℓ4
6 7
9

a) two angles corresponding to ∠1

b) an interior angle on the same side of the transversal as ∠10

c) an alterate interior angle to ∠5

d) two interior angles on the same side of the transversal as ∠8

Example 2

In the diagram below, measure and record the sizes of the angles. Identify pairs of equal
angles and state why they are equal.

2 1
3 4

6 5
7 8

SOL U T ION

Use a protractor to measure the angles.

∠1 and ∠3 measure 125°

∠2 and ∠4 measure 55°
∠5 and ∠7 measure 120°
∠6 and ∠8 measure 60°

These are each pairs of vertically opposite angles.

 Chapter 5 Angles and Parallel Lines 235

BUILD YOUR SKILLS

4. Look at the diagram below. Identify two transversals that intersect both AB and AD.

B
A

D
E

5. In the diagram below, can t be a transversal that intersects ℓ1 and ℓ2? State why
or why not.
t

ℓ1
ℓ2
236 MathWorks 10 Workbook

6. In the diagram below, t is a transversal that intersects ℓ1 and ℓ2. Name another pair
of lines and their transversal.
t ℓ3

ℓ1

ℓ2

PRACTISE YOUR NEW SKILLS

1. In the diagram below, where t is the transversal, identify two pairs of each of the
following angles.

1 2
5 6
4 3 t
8 7

a) alternate interior angles

b) corresponding angles

c) interior angles on the same side of the transversal

 Chapter 5 Angles and Parallel Lines 237

2. A flashlight shines down onto a floor as shown in the diagram below. If the outer
rays are considered to be two lines and the floor is a transversal, name a pair of
corresponding angles.

2
1

6 5 4 3

3. In the diagram below, identify which line is a transversal that intersects ℓ1 and ℓ2
that makes the following relationships between the pairs of angles.

a) ∠1 and ∠2 a pair of corresponding angles

b) ∠3 and ∠4 a pair of alternate interior angles

2
4 ℓ1

ℓ2
1

ℓ4 ℓ3
238 MathWorks 10 Workbook

4. In the diagram below, calculate the sizes of each of the interior angles. What is
their sum?
t

2
1
3
85°

ℓ1

5 4 ℓ2
6 112°

5. Calculate the sizes of the six angles indicated in the figure.

t

1 120°
2 3 ℓ1

70° 4 ℓ2
6 5
 Chapter 5 Angles and Parallel Lines 239

Parallel Lines and Transversals 5.4

NEW SKILLS: WORKING WITH ANGLES FORMED BY PARALLEL LINES
INTERSECTED BY A TRANSVERSAL

If two parallel lines are intersected by a transversal:

• the alternate interior angles are equal;

• the corresponding angles are equal; and

• the interior angles on the same side of the transversal are supplementary.

If you know that, given two lines cut by a transversal:

• alternate interior angles are equal; or

• corresponding angles are equal; or

• interior angles on the same side of the transversal are supplementary;

then you can conclude that the lines are parallel.

For more details, see page 209 of MathWorks 10.

Example 1

Consider the diagram below, in which ℓ1 is parallel to ℓ2. What are the measures of the
three indicated angles? Explain how you reached your answers.
ℓ1
ℓ2
t
4 = 122°
3
1 2

SOL U T ION

∠1 measures 122° because it corresponds to ∠4 .

∠2 measures 58° because it forms a straight angle with ∠4.

∠3 measures 58° because it is vertically opposite ∠2.

240 MathWorks 10 Workbook

A LT ERN AT I V E SOL U T ION

The order in There may be more than one reason for stating why two angles are equal.
which you find the
∠3 is 58° because it forms a straight angle with ∠4.
angle measures
is important in ∠2 is 58° because it is vertically opposite ∠3.
explaining your
∠1 is 122° because it is an interior angle on the same side of a transversal as ∠3.
reasoning.

BUILD YOUR SKILLS

1. In the diagram below, ℓ1 is parallel to ℓ2. State the measures of the indicated angles
and explain your reasoning.

ℓ1
71° 4 3

1
ℓ2
2 118°

2. What are the measures of the interior angles in the trapezoid shown below? (Hint:
Be careful of the order in which you calculate the angles.)
D C
2 3

68° 1 4
A B
 Chapter 5 Angles and Parallel Lines 241

3. Quadrilateral ABCD is a parallelogram in which ∠B measures 74°. Determine the

measures of the other angles and state your reasons.
B
74° A

C
D

Example 2

Given the diagram below, identify all the pairs of parallel lines and explain your
selection.

ℓ1
100° 80°

110°
ℓ2
100°

70°
95°
ℓ3

ℓ6
ℓ5 ℓ4

SOL U T ION

ℓ6 is parallel to ℓ4. If you consider ℓ1 to be a transversal, 100° and 80° are interior
angles on the same side of the transversal.

ℓ1 is parallel to ℓ2. If ℓ6 is a transversal, the two 100° angles are corresponding angles.
242 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. Find a pair of parallel lines in the following diagram. On the diagram, mark all the
angles necessary to determine this.
ℓ1 ℓ2
ℓ3

62° ℓ4

103°
ℓ5
77°

ℓ6

68°

5. What size must ∠1 be if ℓ1 is parallel to ℓ2?

ℓ1

123°

1
 Chapter 5 Angles and Parallel Lines 243

6. The two vertical pipes in the diagram need to be moved to be parallel to each other.
By what angle must the plumber move the second pipe?

78°

106°

Pipe 2

Pipe 1

Example 3

Given parallelogram ABCD, determine the values of ∠B, ∠C, and ∠D in that order,
stating your reason for each measure.
A B
130°

D C

SOL U T ION

In a parallelogram, opposite sides are parallel.

AD is parallel to BC, and they are intersected by transversal AB. ∠B is an

interior angle on the same side of the transversal as ∠A. Therefore, it is
complementary to ∠A.

180° – 130° = 50°

∠B is 50°.
244 MathWorks 10 Workbook

 B is parallel to DC, and they are intersected by transversal BC. You know that ∠B
A
You can use the is 50°. ∠C is an interior angle on the same side of the transversal as ∠B, so they are
notation AD || BC to complementary. ∠C measures 130°.
indicate that AD and
AB is parallel to DC, and they are intersected by transversal AD. You know that ∠A
BC are parallel.
measures 130° and is complementary to ∠D. Therefore, ∠D measures 50°.

BUILD YOUR SKILLS

7. If ℓ1 and ℓ2 are parallel and are intersected by transversals t1 and t2, what are the
measures of the indicated angles? Solve for the measures in the given order, stating
your reasoning.
ℓ1 ℓ2

2 1
54°
t
t2

83°
3

SOLVING ANGLE MEASURES

Angle Measure Reason

∠1 =

∠2 =

∠3 =

∠4 =
 Chapter 5 Angles and Parallel Lines 245

8. In the diagram below, if the side of the house and the side of the shed are parallel, what are
the measures of ∠1 and ∠2?

58°

9. A plumber must install pipe 2 parallel to pipe 1. He knows that ∠1 is 53°. What is the
measure of ∠2?

Pipe 2

Pipe 1 1
246 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. Given the diagram below, where ℓ1 is parallel to ℓ2, find the measures of the
indicated angles and state your reasons.

1 ℓ1
4 2 112°
60°

ℓ2
3
5

2. In the diagram below, the top of the bridge is parallel to the deck, and the brace in
the middle is vertical, perpendicular to the deck, determine the size of ∠1 and ∠2.

57°

∠2

∠1
 Chapter 5 Angles and Parallel Lines 247

3. Identify the pairs of parallel lines in the following diagram. (Hint: The lines can be
extended.)
ℓ1 ℓ2

48° 132°
ℓ3

ℓ4
46°

46° ℓ5

132°
ℓ6

4. Examine the following diagram. By how many degrees do the studs need to be
moved in order to be parallel to each other? What direction do they need to move
in? (The studs are indicated by the capital letters.)
A B C

89° 91°

134°

D
135°

F
248 MathWorks 10 Workbook

CHAPTER TEST

1. Classify each of the following as acute, right, obtuse, straight, or reflex angles.

a) b)

c) d)

e) f)

2. Fill in the missing parts in the table. If no such angle exists, explain why.

ANGLE CALCULATIONS
Angle Complement Supplement Resulting angle
measure after the
angle is bisected
58°

47°

93°

153°

25°
 Chapter 5 Angles and Parallel Lines 249

3. Name the relationship between the indicated pairs of angles.

2 1
3 4

6 5
7 8

a) ∠3 and ∠5

b) ∠4 and ∠5

c) ∠1 and ∠3

d) ∠2 and ∠6

4. In the diagram below, ℓ1 is parallel to ℓ2. Determine the measures of the indicated
angles and explain your reasons. Write the answers in the order that you
calculated them.

1 67°

4
2
62°

3
ℓ1

ℓ2
250 MathWorks 10 Workbook

5. Given the following diagram, what must be the measures of ∠1 and ∠2 if BE is

parallel to CD? State your reasons.
A E D
1 68°

2
B

75°

6. In trapezoid PQRS, PS is parallel to QR. What are the measures of ∠1 and ∠2?
P S
133° 1

24°
2 67°
Q R
 Chapter 5 Angles and Parallel Lines 251

7. If ℓ1 is parallel to ℓ2, and ℓ3 is parallel to ℓ5, what are the following angle measures?
ℓ2
ℓ3
ℓ1
1 ℓ4

108° ℓ5
2

a) the value of ∠1

b) the value of ∠2 that will make ℓ4 perpendicular to ℓ2

8. On the map below, what is the true bearing from the following points?

Point B
Point A

Point C

a) A to B b) B to C
252 MathWorks 10 Workbook

9. Fred states that if ℓ1 is parallel to ℓ2, and ℓ2 is parallel to ℓ3, then it follows that ℓ1 is
parallel to ℓ3. Is Fred right? Show your answer using a diagram.

10. In the diagram below, ℓ1 is parallel to ℓ2, and ℓ2 is parallel to ℓ3. State two angles
whose measures are the same as ∠7. Explain your reasoning.
ℓ3
ℓ2

ℓ1

1 2 3 4
t1
7 6 5
t2
11
10

9
8
 253

Chapter 6Similarity of Figures

These nesting toolboxes were designed to be similar figures.

Similar Polygons
6.1
REVIEW: WORKING WITH RATIO, RATE, AND PROPORTIONAL REASONING

In this chapter, you will apply ratio, rate, and proportional reasoning.

See chapter 1 of this workbook for a review of these concepts.

254 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH SIMILAR FIGURES

Two figures are similar if they have the same shape but are different sizes. A diagram
drawn to scale to another diagram creates a similar figure. Likewise, an enlargement of
a photograph, when reproduced to scale, yields a similar figure.

For two figures to be similar:

corresponding sides: • corresponding angles must be the same size; and

two sides that occupy the
• corresponding sides must be in the same proportion.
same relative position in
similar figures A D W Z

B Y

In quadrilateral ABCD and quadrilateral WXYZ, the following equivalencies are true.

∠A = ∠W
∠B = ∠X
∠C = ∠Y
∠D = ∠Z

AB = BC = YZ = DA
WX XY DC ZW

If two figures have The two quadrilaterals are similar. This can be written as ABCD ~ WXYZ.
the same size and
For more details, see page 227 of MathWorks 10.
shape, they are said
to be congruent, or
similar. Similarity
is shown by the
symbol ~.
 Chapter 6 Similarity of Figures 255

Example 1

Tara has drawn a scale diagram of her bedroom so that she can sketch different
arrangements of her furniture. On her diagram, the walls have the following lengths:
a = 8.5″

f = 3.4″

b = 6″

e = 2″

d = 2.6″

c = 6.5″

If the longest wall in her room is actually 12.75ʹ, how long are the other walls?

SOL U T ION

Set up proportions between the longest wall and each of the other walls. Label the
walls of the actual room with the same letters as those in the scale drawing, but use
upper case letters. (For example, wall a is 8.5ʺ on the drawing, and wall A is 12.75ʹ
in the actual room.)

Start with wall b.

a = b
A B
8.5 = 6 Substitute the known values.
12.75 B
8.5 = 6
Work without units of measurement.
12.75 B
12.75 × B × 8.5 = 6 × B × 12.75 Multiply by 12.75B.
12.75 B
8.5B = 6 × 12.75
8.5B = 76.5
8.5B = 76.5 Divide to isoolate B.
8.5 8.5
B=9
256 MathWorks 10 Workbook

Use the same method to solve the following proportions:

a = c
A C
8.5 = 6.5
12.75 C
a = d
A D
8.5 = 2.6
12.75 D
a = e
A E
8.5 = 2
75 E
12.7

a = f
A F
8.5 = 3.4
12.75 F
The lengths of the walls in Tara’s bedroom are:

B = 9.0ʹ

C = 9.75ʹ

D = 3.9ʹ

E = 3ʹ

F = 5.1ʹ

A LT ERN AT I V E SOL U T ION

Determine the unit scale. If 8.5ʺ represents 12.75ʹ, then calculate how many feet
1 inch represents.

12.75ʹ ÷ 8.5 = 1.5ʹ

The remaining side lengths can be calculated by multiplying each scaled unit by
1.5 ft per inch.

B = 6 × 1.5
B = 9ʹ
 Chapter 6 Similarity of Figures 257

C = 6.5 × 1.5
C = 9.75ʹ

D = 2.6 × 1.5
D = 3.9ʹ

E = 2 × 1.5
E = 3ʹ

F = 3.4 × 1.5
F = 5.1ʹ

BUILD YOUR SKILLS

1. The two figures shown below are similar. Find the lengths of the sides of the smaller
figure. (The diagrams are not drawn to scale.)
A 21 cm E

18 cm
L P
15 cm
D
5 cm
12 cm O

B 18 cm C M N

2. On a blueprint, a room measures 2.75 inches by 1.5 inches. If 1 inch represents

8 feet, what will be the dimensions of the room?
258 MathWorks 10 Workbook

3. Identify the pairs of similar polygons below.

A D Q R M P

B C N O
H W Z
I
E L

J
K
G

F X Y

Example 2

If ∆RST is similar to ∆LMN and angle measures of ∆LMN are as follows, what are the
angle measures of ∆RST?

∠L = 85°
N
∠M = 78°
T
∠N = 17°

S R M L
 Chapter 6 Similarity of Figures 259

SOL U T ION

Since the triangles are similar, corresponding angles must be equal.

∠R = ∠L
∠R = 85°

∠S = ∠M
∠S = 78°

∠T = ∠N
∠T = 17°

Could you have answered this question without being given ∠N? Explain your
reasoning.

BUILD YOUR SKILLS

4. If two polygons ABCDEF and GHIJKL are similar, and the following angle measures
are given, state the corresponding angles and their measures.

∠J = 73°

∠B = 21°

∠K = 40°

5. If ∆ABC is similar to ∆XYZ and the following angle measures are known, what are
the values of the remaining angles?

∠A = 32°

∠C = 48°

∠Y = 100°
260 MathWorks 10 Workbook

6. If trapezoid PQRS is similar to trapezoid LMNO, what are the values of w,

x, y, and z?
S N
40 cm
42°
P x

w y M
35 cm R

O z
90 cm 20 cm
70°
Q L

Example 3

A true model must J ason wants to build a model of his house. He will build the model using a scale where
be mathematically 5 cm represents 2 m. If one room is 6.5 m long, 4.8 m wide, and 2.8 m tall, what will its
similar in shape to dimensions be in the model?
the original.

SOL U T ION

Similarity can be applied to three-dimensional objects.

The ratio of the model to the actual room is 5 cm:2 m. Write this ratio without units
as the fraction 5 .
2
Represent length, width, and height with ℓ, w, and h, respectively, and set up
proportions to calculate their measures.

5 = 
2 6.5

5 = w
2 4.8

5 = h
2 2.8
 Chapter 6 Similarity of Figures 261

Solve for ℓ.

5 = 
2 6.5 Substitute the known values.

2 × 6.5 × 5 =  × 6.5 × 2 Multiply by the two denominators.

2 6.5
6.5 × 5 = 2
32.5 = 2 Divide to isolate .
32.5 = 
2
16.25 = 

The length of the room in the model will be 16.25 cm.

Use the same method to solve for w and h.

5 = w
2 4.8
12 = w

5 = h
2 2.8
7=h

The width in the model is 12 cm and the height is 7 cm.

BUILD YOUR SKILLS

7. Redo Example 3 by first determining a unit scale (the number of centimetres that
represent 1 metre), then calculating length, width, and height for the model.
262 MathWorks 10 Workbook

8. If a house is 40 feet long, 35 feet wide, and the top of the roof is 27 feet above
ground level, what will the corresponding dimensions be of a model built so that
1 foot is represented by 1 inch?
2

9. Theresa folds origami paper to make stacked boxes. The outer box is 12 cm by 8 cm
by 4 cm. Theresa would like to make three smaller, similar boxes, each scaled down
by 1 of the previous box. What are the dimensions of the three smaller boxes?
4

PRACTISE YOUR NEW SKILLS

1. The scale of a model airplane to the actual airplane is 2:45. If the model is
38 centimetres long, how long is the actual airplane?
 Chapter 6 Similarity of Figures 263

2. Two triangles are similar. One has sides of 8 m, 5 m, and 6 m. If the longest side of
the second triangle is 5 m, what are the lengths of the other two sides?

3. A pentagon has interior angles of 108°, 204°, 63°, 120°, and 45°. Rudy wants to
draw a similar pentagon with sides twice as long as the original. What size will the
angles be?

4. Michaela has a microscope that enlarges images between 40 and 1600 times.
How large will an object that is 1.2 mm by 0.5 mm appear to be at each of these
extremes?
264 MathWorks 10 Workbook

5. Marie-Claude has a series of four nested funnels in her kitchen that are similar
to the one shown in the diagram. If the other three funnels have top diameters
of 10 cm, 8 cm, and 6 cm, find the measures of the remaining parts for all
three funnels.
12 cm

16 cm

10 cm

 2 cm
 Chapter 6 Similarity of Figures 265

Determining if Two Polygons Are Similar

6.2
NEW SKILLS: WORKING WITH SIMILAR POLYGONS

I n the last section, you considered two or more figures that were similar and found their Figures are similar
corresponding sides and angles. How do you determine if two figures are similar, and if corresponding
what changes can you make to a given shape and keep it similar to the original? angles are equal and
corresponding sides
Example 1 are proportional.

Are the two pentagons shown below similar? If so, explain how you know. If not,
explain what you would need to know. (Angles marked with the same symbol
are equal.)
R V

A E
6 cm
S
B 3 cm

C
2.5 cm D
T 5 cm U

SOL U T ION

You know that three angles of one pentagon are equal to the three corresponding
angles of the second pentagon.

∠R = ∠A

∠T = ∠C

∠V = ∠E

However, you cannot determine if the other two pairs of corresponding angles (∠S
and ∠B, ∠U and ∠D) are equal. Therefore, you cannot state for certain that the two
pentagons are similar.

For more information, see page 239 of MathWorks 10.

266 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Pierre drew two regular hexagons (6-sided figures with all sides equal in length).
Are the two hexagons similar? Why or why not?

2. Frank enlarges a photo to poster size. The original photo is 4 inches by 6 inches. If
Frank enlarges it to 1 m by 1.5 m, will it be similar in shape to the original?

3. Zora says that the two rectangles below are not similar because 60 does not equal
100 . Is Zora right? Explain. 50
30
A 60 cm B

P 50 cm Q

100 cm 30 cm

S R

D C
 Chapter 6 Similarity of Figures 267

Example 2

Determine if the two given parallelograms, ABCD and WXYZ, are similar.
A D

W Z

8 in
6 in

70° 70°
B 12 in C X 8 in Y

SOL U T ION

In a parallelogram, opposite angles are always equal in measure.

∠A = ∠C

∠B = ∠D = 70°

∠X = ∠Z

∠X = ∠Z = 70°

Also, interior angles in a parallelogram always add up to 360°. Because the 70°
angles in the two parallelograms correspond, the other angles must also correspond.

∠A = ∠C = ∠W = ∠Y

Therefore, all corresponding angles are equal.

For the parallelograms to be similar, the sides would have to be proportional and
the following would have to be true.

AB = WX
BC XY

Calculate to determine if this is true.

AB = 8
BC 12
AB = 8 ÷ 4
BC 12 ÷ 4
AB = 2
BC 3
268 MathWorks 10 Workbook

WX = 6
XY 8
WX = 6 ÷ 2
XY 8÷2
WX = 3
XY 4

2 ↑ 3
3 4

The sides are not proportional, so the parallelograms are not similar.

BUILD YOUR SKILLS

4. Janelle stated that increasing or decreasing the sides of a given figure by the same
factor will always produce a figure similar to the original. Is this true? Give one
example that illustrates your answer.

5. Aidan frames a 24-inch by 36-inch picture with a 4-inch frame. Is the framed
picture similar in shape to the unframed picture? Show your calculations.
 Chapter 6 Similarity of Figures 269

6. One cylinder has a radius of 25 cm and a height of 35 cm. Another cylinder has
a radius of 30 cm and a height of 40 cm. Are the cylinders similar? Show your
calculations.

PRACTISE YOUR NEW SKILLS

1. The scale on a map is 2.5 cm:500 m.

a) What distance is represented by a 12.5-cm segment on the map?

b) How long would a segment on the map be if it represented 1.5 km?

2. Show whether a rectangular prism that is 6 m by 10 m by 8 m is similar to one that

is 4 m by 7 m by 5 m.
270 MathWorks 10 Workbook

3. Colin states that the following two figures are similar, but Tai disagrees, saying that
they don’t have enough information. Who is right? Show your calculations.
10 in

8 in
8 in 5 cm
18 in
4 cm
4 cm 9 cm
5 cm

4. While he was at the pet food store, Jeremy saw three different sized dog mats.
They measured 36 inches by 28 inches, 27 inches by 21 inches, and 24 inches by
18 inches. Are all the mats similar? Show your calculations.

5. Using two similar rectangles, show whether their areas are in the same proportion
as the sides.
 Chapter 6 Similarity of Figures 271

Drawing Similar Polygons

6.3
NEW SKILLS: WORKING WITH SCALE DIAGRAMS

Artists, architects, and planners use scale diagrams or models in their work. The
diagrams or models should be in proportion to the actual objects so that others can
visualize what the real objects look like.

In this section, you will practise drawing scale diagrams.

Example 1

Use graph paper to construct a figure similar to the one given, with sides that are
1 1 times the length of the original. Explain how you know that the corresponding
2
angles are equal.

SOL U T ION

The sizes of the angles will not change by increasing the lengths of the sides, so the
corresponding angles must be equal.

For more information, see pages 248–250 of MathWorks 10.

272 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Draw and label the lengths of the sides of a rectangle that has a length of 8 cm and
is similar to a rectangle that has a width of 10 cm and a length of 20 cm.

2. Maurice has drawn the plan below for his backyard. However, he finds that the
diagram is too small to fit in all the details. Redraw the diagram at 2.5 times the size
of the original. Bench

Table

Planters
Pool
 Chapter 6 Similarity of Figures 273

3. Barnie’s house is 55 ft wide and 40 ft deep. A drawing of his property shows the
house is 10 in wide and 7.3 in deep. What scale was used on the drawing?

Example 2

Xavier is building a staircase using scale drawings. On the drawing, the height of one stair
is 0.5 cm and the depth is 0.9 cm. Xavier will use a scale factor of 40 to build the stairs.
Calculate the height and depth of the stairs he will build.

SOL U T ION

A scale factor of 40 means that the actual measure is 40 times the measure on the
drawing. This can be written as a ratio of 1:40 or 1 .
40
To calculate the height of one stair, set up a proportion.

0.5 = 1
x 40
40 × x × 0.5 = 1 × x × 40
x 40
40 × 0.5 = x
20 = x

The height of the stair will be 20 cm.

Use the same method to calculate the depth of the stair.

0.9 = 1
x 40
40 × x × 0.9 = 1 × x × 40
x 40
40 × 0.9 = x
36 = x

The depth of the stair will be 36 cm.

274 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. Simrin has built two end tables. The second table is a slightly larger version of the
first. Given the dimensions below, calculate what scale factor Simrin used to make
the larger table.

30 in
37.5 in
20 in
25 in

18 in
22.5 in

5. A craft store uses small gift boxes to wrap purchases. They have one box that is
20 cm by 12 cm by 5 cm. Another box is larger by a scale factor of 1.3. What are the
dimensions of the larger box?

6. Hazuki made a kite with the dimensions shown below. She decided it would work
better if it were bigger. If her new kite tail has a length of 49 cm, what scale factor
did she use, and what are the kite’s other dimensions?
20 cm

8 cm

40 cm

tail length = 28 cm
 Chapter 6 Similarity of Figures 275

PRACTISE YOUR NEW SKILLS

1. The scale ratio (model:original) between two diagrams is 3:5. If one measure on the
model is 45 mm, what was the measure on the original?

2. Draw a square with sides of 2 cm, and a second square with sides that are 3 cm.

a) Is the second square similar to the first? Explain your reasoning.

b) If you draw a rectangle whose sides are 5 cm and 8 cm, and a second rectangle
with sides that are 3 cm longer, will the two be similar? Explain your reasoning.

3. Draw a rectangular prism similar to the one shown below with sides that are 1 the
2
length of the original.

3 cm

2 cm
8 cm
276 MathWorks 10 Workbook

4. A poster shows a photograph of a cruise ship. The actual ship is 310 metres long. In
the photograph, the cruise ship is 1.2 m long.

a) What is the scale factor (to 4 decimal places)?

b) A person 1.8 m tall was standing on the deck of the cruise ship when the
photo was taken. How tall is the person on the photo (to the nearest tenth of a
centimetre)?

5. A sporting goods store has miniature versions of tents on display. A six-person tent
is 12ʹ long by 10ʹ wide. The miniature version has a length of 1 1 .
2
a) What is the width of the miniature version?

b) What is the scale ratio (miniature:actual)?

 Chapter 6 Similarity of Figures 277

Similar Triangles
6.4
NEW SKILLS: WORKING WITH SIMILAR TRIANGLES

Similar triangles are very useful in making calculations and determining measurements.

The sum of the angles of a triangle is always 180°. If two corresponding angles in two
triangles are equal, the third angles will also be equal.

Two triangles are similar if any two of the three corresponding angles are congruent, or
one pair of corresponding angles is congruent and the corresponding sides adjacent to
the angles are proportional.

Two right triangles are similar if one pair of corresponding acute angles is congruent.

Example 1

Given the two triangles below, find the length of n.

C

a = 5 in b
N

ℓ = 2 in m

x x
B A M L
c = 7 in n

SOL U T ION

You know that two of the three corresponding angles are congruent.

∠C = ∠N

∠B = ∠M

This means that ∆ABC is similar to ∆LMN.

278 MathWorks 10 Workbook

To solve for n, set up a proportion.

n = 
c a
n = 2
7 5
5× 7 × n = 2 ×7× 5
7 5
5n = 7 × 2
5n = 14

n = 14
5
n = 2.8

Side n is 2.8 in long.

For more information, see pages 257–258 of MathWorks 10.

BUILD YOUR SKILLS

1. In each of the diagrams below, ∆ABC is similar to ∆XYZ. Find the length of the
indicated side (to one decimal place).
A
a)
Z
x
6.1 cm
Y

4.5 cm
X
B
5.3 cm C

b) X
A 12.7 ft

x Y 8.2 ft Z

18.8 ft C
B
 Chapter 6 Similarity of Figures 279

c) X
B
4m
c C

25 m

Y 16 m Z

2. Given that ∆ABC in similar to ∆RST, AB is 6 cm long, BC is 5 cm long, and RS is

8 cm long, find the length of a second side in ∆RST. Can you find the length of the
third side? Explin your answer.

3. C
 armen thinks that any two isosceles triangles will be similar. Use examples to An isosceles triangle
prove or disprove her belief. has two sides equal
in length, and two
angles of equal
measure.
280 MathWorks 10 Workbook

Example 2

Ravi notices that a 2-m pole casts a shadow of 5 m, and a second pole casts a shadow of
9.4 m. How tall is the second pole?

SOL U T ION

Sketch the situation.

sun
sun pole 2 = x
pole 1 = 2 m

shadow = 5 m shadow = 9.4 m

The angle between the rays of the sun and the pole is the same in both cases, so the
two triangles are similar.

Set up a proportion to solve for the height of the second pole.

height of pole 1 height of pole 2

=
shadow 1 shadow 2
2 = x
5 9.44
5 × 9.4 × 2 = x × 9.4 × 5
5 9.4
9.4 × 2 = 5 x
18.8 = 5 x
3.8 ≈ x

The second pole is approximately 3.8 m tall.

 Chapter 6 Similarity of Figures 281

BUILD YOUR SKILLS

4. Assuming that the slope of a hill is constant, and that a point 100 metres along the
surface of the hill is 4.2 metres higher than the starting point, how high will you be
if you walk 250 metres along the slope of the hill?

5. Maryam is sewing a patchwork quilt. The sketch she has drawn is to a scale
of 1:8. Part of the design consists of right triangles that have legs that are
2.2 cm and 4.6 cm long. What will the lengths of the legs of the triangles in the
finished quilt be?

6. Which of the following triangles are similar?

C

45° 45°

A D B
282 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. In the following diagram, AB is parallel to ED, AB is 8 m, AC is 12 m, and CE is

7 m. Calculate ED to one decimal place. 8m
B A

12 m

C
7m

E D

2. Madge has cut out two triangular shapes from a block of wood, as shown below. She
says that the two shapes are similar. Is she correct? Show your calculations.

16 in

8 in

2 in

10 in

8 in

2 in
5 in

3. Given that ∆FGH ~ ∆XYZ, state which angles are equal and which sides are
proportional.
 Chapter 6 Similarity of Figures 283

4. Julian is visiting the Manitoba Legislative Building in Winnipeg, where he sees the
statue of Louis Riel. Use the information in the diagram to find the height of the
statue. Round your answer to a whole number.

Julian

5 ft 8 in 6 ft

2 ft 8 ft
284 MathWorks 10 Workbook

CHAPTER TEST

1. The lengths of the sides of a pentagon are 2ʺ, 6ʺ, 10ʺ, 14ʺ, and 24ʺ. Calculate the
lengths of the sides of a similar pentagon if the shortest side is 5ʺ.

2. Given that the two figures shown are similar, determine the values of x and y.
8.5 m y

2.5 m
3m
6m

3. To determine the distance across a river (distance AB), Lila took the following
measurements. Assuming the two triangles in the diagram are similar, how wide is
the river?

A Shore

River

18 m
B 32 m Shore
24 m
 Chapter 6 Similarity of Figures 285

4. If a man casts a shadow that is 3.8 m long at the same time that an 8-m flagpole
casts a shadow that is 15 m long, how tall is the man?

5. How does doubling the lengths of the sides of a rectangle to form a similar rectangle
affect the area?

6. Amin cut out two blocks of wood as indicated. Are the two blocks similar in shape?
Round your final calculations to two decimal places.
4.8 in

7 in 5.25 in
2.75 in

4 in

9.2 in
286 MathWorks 10 Workbook

7. Determine if the following statements are true or false and explain your reasoning.

a) All equilateral triangles are similar.

b) All isosceles triangles are similar.

c) Any pair of congruent triangles is similar.

8. A science teacher uses an overhead projector to display a diagram of a pulley system.

The actual diagram is 6ʺ by 7.2ʺ. The image projected onto the wall is 4ʹ 2ʺ by 5ʹ.
What is the scale factor of the projection?
 Chapter 6 Similarity of Figures 287

9. Young-Mee draws a sketch of her new office space to 13ʹ

envision different furniture and equipment placements.

On her drawing, the main area is 6ʺ wide. What are the
other dimensions of the drawing, given the measurements main area
17ʹ
of the actual space shown in the diagram?

7ʹ 2ʺ

photocopier
6ʹ
room

storage
5ʹ 2ʺ
space

10. François is commissioned to paint a portrait based on a photograph. The

photograph is 5ʺ by 7ʺ, and the painting must be enlarged by a factor of 6.25. What
are the dimensions of the painting?

11. Joanne knitted a blanket that measures 174 cm by 230 cm. Her sister asked Joanne
to make a matching one for her son. If Joanne wants to make a similar blanket using
a scale factor of 0.55, what will its dimensions be?
288

Chapter 7Trigonometry of Right Triangles

This man is installing an angle brace. By incorporating right triangles, angle braces provide support and strength to bookshelves,
building roofs, and beyond.

7.1 The Pythagorean Theorem

REVIEW: WORKING WITH TRIANGLES

Each vertex of a triangle is labelled with an upper case letter, and each side is labelled
either with the lower case letter corresponding to the opposite vertex or with the upper
case letters of the vertices it connects.
 Chapter 7 Trigonometry of Right Triangles 289

Example 1

Consider DRST.
R

T
S

a) Label the sides with the appropriate lower case letter.

b) Name the sides using the upper case letters of the vertices they connect.

SOL U T ION

a) Each side is labelled with the lower case letter corresponding to the
opposite vertex.
R

s
t

T
S r

b) The sides can also be named according to the upper case letters of the vertices
they connect.

Side r can be called side ST.

Side s can be called side TR.

Side t can be called side RS.

290 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Label each side of the triangles below using a single lower case letter corresponding
to the opposite vertex.

a) X b) Q
Z

R S

c)
D E d) S T

R
F

2. Label each vertex of the triangles below using a single upper case letter
corresponding to the opposite side.

a) b)
s
c r

b q

c) d) d

x w
e f

y
 Chapter 7 Trigonometry of Right Triangles 291

NEW SKILLS: WORKING WITH THE PYTHAGOREAN THEOREM

A right triangle is a triangle with one right angle. The side opposite the right angle is the hypotenuse: the longest
longest side and is called the hypotenuse. The other two sides are called legs (or, in side of a right triangle,
some cases, arms). opposite the 90° angle

The Pythagorean theorem states the relationship among the sides of a right triangle. Pythagorean theorem:
Given a right triangle ABC with right angle C, the Pythagorean theorem states the in a right triangle, the
following. sum of the squares of
the lengths of the legs is
a2 + b2 = c2
equal to the square of the
For more details, see page 272 of MathWorks 10. length of the hypotenuse

Example 2

Label the sides of the triangles and state the Pythagorean theorem as it applies to them.

R
Q

SOL U T ION

q
r

R
Q p

Since q is the hypotenuse, the Pythagorean theorem is written as follows.

p 2 + r 2 = q2
292 MathWorks 10 Workbook

BUILD YOUR SKILLS

3. Given the following diagram, use the lettering provided to state three Pythagorean
relations that apply.
A

c D
z y

C
B a

4. A ladder, ℓ, is placed against the side of a house, h. The foot of the ladder is a
distance d from the base of the house. Draw a diagram and express the relationship
that exists between ℓ, h, and d.

5. Rearrange the Pythagorean theorem to solve first for x and then for y.

x2 + y2 = z2
 Chapter 7 Trigonometry of Right Triangles 293

Example 3

Use the Pythagorean theorem to find the lengths of the missing sides of the triangles to
the nearest tenth of a unit.

a)
P b) Y

q
r = 3.8 m z

R
Q p = 5.2 m x = 12.8 in

y = 6.9 in

Z
SOL U T ION

a) Write the Pythagorean theorem using the labels on the given triangle.

p2 + r 2 = q2
5.22 + 3.82 = q2 Substitute the known values.
27.04 + 14.44 = q2
41.48 = q2
41.48 = q Take the square roott of both sides.
6.44 ≈ q

Side q is approximately 6.4 m.

b) Write the Pythagorean theorem using the labels on the given triangle.

y2 + z 2 = x 2
6.92 + z 2 = 12.82 Substitute the known values.
2
47.61 + z = 163.84
z 2 = 163.84 − 47.61 Subtract 47.61 from both sides to isolate z.
z 2 = 116.23
z = 116.23 Take the square root of both sides.
z ≈ 10.78

Side z is approximately 10.8 inches.

294 MathWorks 10 Workbook

BUILD YOUR SKILLS

6. Calculate the values of x and y.

10.4 cm

x
4.2 cm

6.8 cm

7. A 40-foot ladder reaches 38 feet up the side of a house. How far from the base of the
house is the foot of the ladder?
 Chapter 7 Trigonometry of Right Triangles 295

8. A field is 120 m by 180 m. How much shorter is your route if you walk diagonally
across the field rather than walking around the edge to the opposite corner?

PRACTISE YOUR NEW SKILLS

1. A stairway rises 6 feet 4 inches over a horizontal distance of 8 feet 6 inches. What is
the diagonal length of the stairway?

2. A 28-metre long guy wire is attached to a point 24 m up the side of a tower. How far
from the base of the tower is the guy wire attached?
296 MathWorks 10 Workbook

3. The construction plans for a ramp show that it rises 3.5 metres over a horizontal
distance of 10.5 metres. How long will the ramp surface be?

4. The advertised size of a TV screen is the distance between opposite corners. Sally
bought a 52-inch TV. If the height of the TV is 32 inches, how wide is it?

5. A boat sailed due north at a rate of 12 km/h for 3 hours, then due east at a rate
of 18 km/h for 2 hours. How far was it from its starting point, measuring the
shortest distance?
 Chapter 7 Trigonometry of Right Triangles 297

The Sine Ratio

7.2
NEW SKILLS: WORKING WITH THE SINE RATIO TO SOLVE TRIANGLES

In chapter 6, you worked with similar triangles to discover that, in triangles with congruent
angles, the ratio between the corresponding sides of the similar triangles is the same.

The following diagram shows similar triangles. DABC ∼ DXYZ. Angles marked with
Y the same symbol are
B equal.

z
c x
a

A X
b C y Z

The ratios between corresponding sides are equal, so we know that the following is true.
a = c
x z
This proportion can be rearranged so that each side of the equation represents a ratio of
sides from the same triangle.

x ×z× a = c × z ×x Multiply both sides by the product

x z of the denomin
nators and simplify.
za = cx
sine ratio: in a right
za cx Divide both sides by the saame number
= triangle, the ratio of
cz cz and simplify.
the length of the side
a = x
c z opposite a given angle
When triangles are similar, the ratio of the length of the side opposite a given angle to the to the length of the
length of the hypotenuse is always the same. This ratio is referred to as the sine ratio. hypotenuse (abbreviated
as sin)
Given any right triangle with acute angle A, the sine ratio can be written as follows.

length of side opposite ∠A

sine ∠A = Use your scientific
length of hypotenuse
calculator to
The ratio is abbreviated as follows.
calculate the values
opp
sin A = hyp of the sines of
angles.
For more details, see page 283 of MathWorks 10.
298 MathWorks 10 Workbook

Example 1

Use your calculator to determine the following sine ratios. Round to four decimal places.

a) sin 15° b) sin 30°

c) sin 60° d) sin 80°

What do you notice about these values?

SOL U T ION

a) sin 15° = 0.2588 b) sin 30° = 0.5000

c) sin 60° = 0.8660 d) sin 80° = 0.9848

The sine ratio determines that if you have a right triangle with an acute angle given,
regardless of the size of the triangle, the ratio of the side opposite that angle to the
hypotenuse will always be the same.

The value of the sine ratio increases as the angle gets bigger.

BUILD YOUR SKILLS

1. Calculate the value of sin A to two decimal places.

A
a) b)

5.2 in 8.1 in
6.9 m

A
9.6 in

4.3 m

2. Use your calculator to determine the value of each of the following sine ratios to four
decimal places.

a) sin 10° b) sin 48°

c) sin 62° d) sin 77°

 Chapter 7 Trigonometry of Right Triangles 299

3. Use your calculator to determine the value of sin 90°. Suggest a reason why this is so.

Example 2

The sine ratio can be used to help you find missing parts of a right triangle.

A ladder 8.5 metres long makes an angle of 72° with the ground. How far up the side of
a building will it reach?

SOL U T ION

Sketch a diagram.

ℓ = 8.5 m h

72°

The height, h, is opposite the 72° angle, and the ladder, ℓ, forms the hypotenuse of
the triangle. A right triangle is formed, with h as the side opposite the 72° angle, and
ℓ as the hypotenuse.

Use the sine ratio to calculate h.

opp
sin A =
hyp

sin 72° = h Substitute the known values.

8.5
8.5 × sin 72° = h × 8.5 Multiply both sides by 8.5.
8.5
8.5 × sin 72° = h
8.1 ≈ h
The ladder reaches approximately 8.1 metres up the side of the building.
300 MathWorks 10 Workbook

BUILD YOUR SKILLS

4. Calculate the length of the side opposite the indicated angle in the
following diagrams.

a)
A b)
58° 9.7 cm

23°
X
9.7 cm

5. A rafter makes an angle of 28° with the horizontal. If the rafter is 15 feet long, what
is the height at the rafter’s peak?
 Chapter 7 Trigonometry of Right Triangles 301

6. How high is a weather balloon tied to the ground if it is attached to a 15-metre

string and the angle between the string and the ground is 35°?

Example 3

 rad is building a ramp. The ramp must form an angle of 22° with the level ground and
B Wait until you
reach a point that is 1.5 metres above the ground. How long will the ramp be? have isolated the
unknown variable

SOL U T ION before doing the

calculation. This will
Sketch a diagram. minimize errors due
B to rounding.

c
1.5 m

22°
A C

Let c represent the length of the ramp. On the diagram, 1.5 metres is opposite the
22° angle.

Use the sine ratio to solve for c.

opp
sin A =
hyp

sin 22° = 1.5 Substitute the known values.

c
c × sin 22° = 1.5 × c Multiply both sides by c.
c
c × sin 22° = 1.5 Simplify.
c × sin 22° = 1.5 Divide both sides by sin 22° to isolate c.
sin 22° sin 22°
c = 1.5
sin 22°
c ≈ 4.004
The ramp is approximately 4 metres long.
302 MathWorks 10 Workbook

BUILD YOUR SKILLS

7. Find the length of the hypotenuse in the following diagrams.

a) b)
h 70°
7.8 mm
33°
A
h

12.1 cm

8. How long is a guy wire that is attached 4.2 metres up a pole if it makes an angle of
52° with the ground?
 Chapter 7 Trigonometry of Right Triangles 303

9. A boat is carried with the current at an angle of 43° to the shore. If the river is
approximately 15 metres wide, how far does the boat travel before reaching the
opposite shore?

NEW SKILLS: WORKING WITH ANGLE OF ELEVATION AND DEPRESSION

When you look up at an airplane flying overhead, the angle between the horizontal and angle of elevation: the
your line of sight is called an angle of elevation. When you look down from a cliff to a angle formed between
boat passing by, the angle between the horizontal and your line of sight is called an the horizontal and the
angle of depression. line of sight while looking
upward; sometimes
For more details, see pages 288–289 of MathWorks 10.
referred to as the angle of
inclination

angle of depression:
angle of elevation the angle formed between
horizontal
angle of depression the horizontal and the
line of sight when looking
downward
304 MathWorks 10 Workbook

Example 4

The angle of elevation of Sandra’s kite string is 70°. If she has let out 55 feet of string,
and is holding the string 6 feet above the ground, how high is the kite?

SOL U T ION

Sketch and label a diagram.

55 ft
h

70°

6 ft

Use the sine ratio to solve for the height of the kite.

opp
sin H =
hyp

sin 70° = h Substitute the known values.

55
55 × sin 70° = h × 55 Multiply both sides by 55.
55
55 × sin 70° = h
51.7 ≈ h
The kite is approximately 52 feet above where Sandra is holding it. Add 6 feet for
the distance between the ground and the start of the string. The kite is about 58 feet
above the ground.
 Chapter 7 Trigonometry of Right Triangles 305

BUILD YOUR SKILLS

10. George is in a hot air balloon that is 125 metres high. The angle of elevation from a
house below, to the balloon, is 18°. How far is George from the house?

11. The angle of elevation of a road is 4.5°. What is the length of the section of road if it
rises 16 metres?

12. The angle of elevation of a slide that is 3.6 metres long is 32°. How high above the
ground is the top of the slide?
306 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS

1. Calculate the sine of the indicated angle.

a) b)
8.9 cm
12.4 cm
7.9 cm

B
6.2 cm
A

2. Calculate the length of the indicated side.

a) b)

12.3 m

19.3 cm x
y

79°
68°

 Chapter 7 Trigonometry of Right Triangles 307

3. A ramp with a length of 21.2 metres has an angle of elevation of 15°. How high up
does it reach?

4. The angle of elevation from the bottom of a waterslide to the platform above is 20°.
If the waterslide is 25 metres long, how high is the platform?

5. A man walks at an angle of 68° north of east for 45 metres. How far north of his
starting point is he?
308 MathWorks 10 Workbook

7.3 The Cosine Ratio

NEW SKILLS: WORKING WITH THE COSINE RATIO TO SOLVE TRIANGLES

cosine ratio: in a right Another important trigonometric ratio of right triangles is the ratio of the side adjacent
triangle, the ratio of to the given acute angle to the hypotenuse. This is called the cosine ratio.
the length of the side
For a given angle A, the cosine ratio can be stated as follows.
adjacent a given angle
to the length of the length of side adjacent to ∠A
cosine ∠A =
hypotenuse (abbreviated
 length of hypotenuse
as cos) This ratio can be abbreviated as follows.

adj
cos A =
hyp
C

b
a

A
B c

For triangle ABC, the cosine of ∠A can be stated as the following.

cos A = c
b

For more details, see page 293 of MathWorks 10.

 Chapter 7 Trigonometry of Right Triangles 309

Example 1

Given the triangles below, find the indicated side. Q

L

a) b)
n = 9.6 in
r=?

33°
M N
ℓ=?
49°
P R
q = 7.8 cm

SOL U T ION

a) Use the cosine ratio to solve for ℓ.

adj
cos M =
hyp

cos 33° =  Substitute the known values.

9.6
9.6 × cos 33° =  × 9.6 Multiply both sides by 9.6 to isolate .
9.6
9.6 cos 33° =  Simplify.
 8.05 ≈  The notation
9.6 cos 33°, without
The missing side is approximately 8.1 inches long.
the multiplication
symbol, can be
A LT ERN AT I V E SOL U T ION used to mean

a) Since you know that the sum of the angles of a triangle is 180°, you can 9.6 × cos 33°.

calculate angle L and then use the sine ratio to solve for ℓ.

180° − 90° − 33° = 57°

opp
sin L =
hyp

sin 57° =  Substitute the known values.

9.6
9.6 sin 57° =  × 9.6 Multiply both sides by 9.6 to isolate .
9.6
9.6 sin 57° = 
8.05 ≈ 

Side ℓ is approximately 8.1 inches long.

310 MathWorks 10 Workbook

b) Use the cosine ratio to solve for r.

adj
cos P =
hyp

cos 49° = 7.8 Substitute the known values.

r
r cos 49° = 7.8 × r Multiply both sides by r.
r
r cos 49° = 7.8 Simplify.
r cos 49° = 7.8
cos 49° cos 49° Divide both sides by cos 49° to isolate r.
r ≈ 11.89

Side r is approximately 11.9 cm long.

BUILD YOUR SKILLS

1. Use your calculator to find the following pairs of ratios to four decimal places.

a) cos 23° = b) cos 83° =

sin 67° = sin 7° =

c) cos 45° = d) cos 37° =

sin 45° = sin 53° =

2. Find the measure of the indicated side in each triangle.

a) b)
8.4 cm
8.4 cm

12°
x
78°
x
 Chapter 7 Trigonometry of Right Triangles 311

c) d) x
x
60°

45° 6.1 cm
6.1 cm

Example 2

How far from the base of a house is a 40-foot ladder if the angle of elevation is 72°?

SOL U T ION

Sketch a diagram.

ℓ = 40″ h

72°

B
d

Use the cosine ratio to solve for d.

adj
cos B =
hyp

cos B = d

cos 72° = d Substitute the known values.
40
40 cos 72° = d × 40 Multiply both sides by 40.
40
40 cos 72° = d Simplify.
12.36 ≈ d
The ladder rests about 12.4 feet from the house.
312 MathWorks 10 Workbook

BUILD YOUR SKILLS

3. How far from the base of a flagpole must a guy wire be fixed if the wire is 12 metres
long and it makes an angle of 63° with the ground?

4. Reba walks 25 yards across the diagonal of a rectangular field. If the angle between
the width and the diagonal is 67°, how wide is the field?

5. A square pyramid has a slant height of 9 metres. The slant height makes an angle of
70° with the ground. What is the length of a side of the pyramid?
 Chapter 7 Trigonometry of Right Triangles 313

Example 3

The angle of a cable from a point 12.5 metres from its base is 52°. How long is the cable?

SOL U T ION

Sketch a diagram.

52°
A
12.5 m

Use the cosine ratio to solve for x, the length of the cable.

adj
cos A =
hyp

cos 52° = 12.5 Substitute the known values.

x
x cos 52° = 12.5 × x Multiply both sides by x.
x
x cos 52° = 12.5 Simplify.
x cos 52° = 12.5 Divide both sides by cos 52° to isolate x.
cos 52° cos 52°
x = 12.5
cos 52°
x ≈ 20.30

The cable is approximately 20.3 metres long.

314 MathWorks 10 Workbook

BUILD YOUR SKILLS

6. Arul needs to string a bridge line across the river from A to B. What must the length
of the bridge line be, given his measurements?

River
c

67°
B
5.5 m C

7. What is the length of a rafter that makes an angle of 35° with the floor of an attic
whose centre is 9.5 metres from the edge?

rafter

35°

9.5 m
 Chapter 7 Trigonometry of Right Triangles 315

8. An airplane starts descending at an angle of depression of 5°. If the horizontal

distance to its destination is 500 kilometres, what is the actual distance the airplane
will travel before it lands?

PRACTISE YOUR NEW SKILLS

1. Find the lengths of the indicated sides.

a) b)

12.3 cm

5.9 cm

52° 67°
x
a

c)
r d) 1.5 m

12° 61°
9.3 cm ℓ

316 MathWorks 10 Workbook

2. A screw conveyor is sometimes used to move grains and other materials up an

incline. How far from the base of a barn must a 20-metre screw conveyor be placed
if the angle of elevation to the loft is to be 30°?

3. What is the slant height of a cone if the diameter is 20 centimetres and the angle
made with it is 65°?

4. A hot air balloon travels 1.2 kilometres horizontally from its take-off point. The
angle of elevation from the take-off point to the balloon is 15°. How far did the
balloon travel?
 Chapter 7 Trigonometry of Right Triangles 317

5. What horizontal distance has a car travelled if the incline of the road averages 3.2°
and the car’s odometer reads 8.5 kilometres?

6. The horizontal distance between two clothesline poles is 3.4 metres. When wet
clothes are hung in the middle of the line, it sags at an angle of depression of 6°.
How long is the clothesline?
318 MathWorks 10 Workbook

7.4 The Tangent Ratio

NEW SKILLS: WORKING WITH THE TANGENT RATIO TO SOLVE TRIANGLES

tangent ratio: in a right You have studied two trigonometric ratios, the sine ratio and the cosine ratio. The third
triangle, the ratio of trigonometric ratio is the tangent ratio.
the length of the side
The tangent ratio is defined as the ratio of the side opposite an acute angle of a right
opposite a given angle
triangle to the side adjacent the angle. For angle A, the ratio can be stated as follows.
to the length of the side
adjacent to the angle length of side opposite ∠A
tangent ∠A =
(abbreviated as tan)
length of side adjacent to ∠A
This can be abbreviated as the following ratio.

opp
tan A =
adj
C

b
a

A
B c

For triangle ABC, the tangent of angle A can be stated as follows.

tan A = a
c

For more details, see page 301 of MathWorks 10.

 Chapter 7 Trigonometry of Right Triangles 319

Example 1

Find the indicated side of each triangle.

Y
a)
C b)
z
6.5 cm a

X 51°
38° 9.3 cm
A B

SOL U T ION

a) Use the tangent ratio to solve for a.

opp
tan A =
adj

tan 38° = a Substitute the known values.

6.5
6.5 tan 38° = a × 6.5 Multiply both sides by tan 6.5 to isolate a.
6.5
6.5 tan 38° = a
5.08 ≈ a

Side a is approximately 5.1 centimetres long.

b) Use the tangent ratio to solve for z.

opp
tan X =
adj

tan 51° = 9.3 Substitute the known values.

z
z tan 51° = 9.3 Multiply both sides by z.
z tan 51° = 9.3
tan 51° tan 51° Divide both sides by tan 51° to isolate z.

z = 9.3
tan 51°
z ≈ 7.53

Side z is approximately 7.5 centimetres long.

320 MathWorks 10 Workbook

BUILD YOUR SKILLS

1. Find the length of the indicated sides of the triangles.

a) b) 6 in
38° 75°
a
12.1 m

c) r 2m d) 9.4 ft
40°

p
50°
 Chapter 7 Trigonometry of Right Triangles 321

2. The angle of depression to a boat from the top of a 150-metre cliff is 20°. How far is
the boat from the base of the cliff?

3. When sand is piled onto a flat surface, it forms a cone. If the pile is 8 m wide, and
the angle between the ground and the slope of the pile is 28°, what is the height
of the pile?

h
28°

8m

PRACTISE YOUR NEW SKILLS

1. A 1.7-metre tall man stands 12 m from the base of a tree. He views the top of the
tree at an angle of elevation of 58°. How tall is the tree?
322 MathWorks 10 Workbook

2. Two buildings are 18.5 metres apart. The angle of elevation from the top of one
building to the top of the other is 18°. If the taller building is 15 metres tall, how tall
is the shorter building?

3. How far from the base of the house is the foot of a ladder if the angle of elevation is
70° and it reaches 15 feet up the side of the house?

4. About how tall is a tower if the angle of depression from its top to a point 75 metres
from the base is 62°?
 Chapter 7 Trigonometry of Right Triangles 323

5. A rafter’s angle of elevation with the horizontal is 25°. How far from the corner
could a 6-foot man stand up straight?

6. Determine the distance, AB, across the river, given the following measurements.

75°
C B
100 m
324 MathWorks 10 Workbook

7.5 Finding Angles and Solving Right Triangles

NEW SKILLS: WORKING WITH INVERSE TRIGONOMETRIC RATIOS

The trigonometric ratios discussed in this chapter are unaffected by the size of the
triangle, provided that the acute angle remains the same.

If you know the trigonometric ratio, you can calculate the size of the angle. This
requires an “inverse” operation. You can use your calculator to find the opposite of
the usual ratio calculation. You can think of the inverse in terms of subtraction and
addition: subtraction is the inverse, or opposite, of addition because it “undoes” the
operation.

For more details, see page 307 of MathWorks 10.

Example 1

Calculate each angle to the nearest degree.

a) sin A = 0.2546

b) cos B = 0.1598

c) tan C = 3.2785

SOL U T ION

Use the inverse function on your calculator.

a) sin A = 0.2546

A = sin −1(0.2546)
A ≈ 14.7

∠A is approximately 15°.

b) cos B = 0.1598

B = cos −1(0.1598)
B ≈ 80.8

∠B is approximately 81°.
 Chapter 7 Trigonometry of Right Triangles 325

c) tan C = 3.2785

C = tan −1(3.2785)
C ≈ 73.0

∠C is approximately 73°.

BUILD YOUR SKILLS

1. Calculate the angle to the nearest degree.

a) sin D = 0.5491 b) cos F = 0.8964

c) tan G = 2.3548 d) sin H = 0.9998

2. I n right triangle ∆XYZ, the ratio of the side opposite ∠X to the hypotenuse is 7 . When solving this
8
What is the approximate size of ∠X? problem on your
calculator, put
7
brackets around 8 .

3. What is the approximate size of an angle in a right triangle if the ratio of the side
opposite the angle to the side adjacent to the angle is 15 ?
8
326 MathWorks 10 Workbook

Example 2

Determine the angle of elevation to the top of a 5-metre tree at a point 3 metres from the
base of the tree.

SOL U T ION

Sketch a diagram.

5m

E
3m

When solving this  ou are given the height (h, 5 metres) and the length (ℓ, 3 metres) of the triangle,
Y
problem on your and you need to solve for the angle of elevation. Use the tangent ratio.
calculator, put
opp
5
brackets around 3 . tan E =
adj

tan E = 5 Substitute the known values.

3
E = tan −1 5
3 () Use the inverse funcction to solve for E.

E ≈ 59.0362

The angle of elevation is approximately 59°.

 Chapter 7 Trigonometry of Right Triangles 327

BUILD YOUR SKILLS

4. What is the angle of depression from the top of a 65-metre cliff to an object
48 metres from its base?

5. At what angle to the ground must you place a support if it is 6.8 metres long and
must reach 4.2 metres up the side of a tower?

6. At what angle to the ground is an 8-metre long conveyor belt if it is fastened

5 metres from the base of a loading ramp?

8m

5m
328 MathWorks 10 Workbook

NEW SKILLS: WORKING WITH RATIOS TO SOLVE TRIANGLES

Solving a triangle means finding the values of all the unknown sides and angles. In a
right triangle, you already know that one angle is 90°, so there are only five other parts
to consider: the three sides, and the two other angles. If you are given any two sides, or
any one side and one angle, you can use trigonometry to find the other values.

Example 3

Solve the right triangle. Give lengths to the nearest tenth.

B

c = 8.7 cm a

56°
A
b C

SOL U T ION

You are given two of the three angles, so you can solve for the third angle.

∠B = 180° − 90° − 56°

∠B = 34°

To solve for side a, you can use the sine ratio. Use ∠A, and the length of the
hypotenuse, c.

opp
sin A =
hyp

sin A = a
c
sin 56° = a Substitute the known values.
8.7
8.7 sin 56° = a × 8.7 Multiply both sides by 8.7 to isolate a.
8.7
7.2126 ≈ a
Side a is approximately 7.2 cm long.
 Chapter 7 Trigonometry of Right Triangles 329

To solve for b, use the cosine ratio.

You could have used
adj
cos A = the Pythagorean
hyp
theorem to find side
cos A = b
c b, but this would
have been less
cos 56° = b Substitute the known values.
8.7 accurate because
8.7 cos 56° = b × 8.7 Multiply both sides by 8.7 to isolate b. you would have used
8.7
an approximation for
8.7 cos 56° = b Simplify.
side a. It is always
4.8650 ≈ b
better to use the
numbers given, if
Side b is approximately 4.9 cm long.
possible, rather than
one you calculated.
BUILD YOUR SKILLS

7. Solve the given triangle without using the Pythagorean theorem.

Q

r 5.4 m

48°
P
q R
330 MathWorks 10 Workbook

8. The two equal angles of an isosceles triangle are each 70°. Determine the measures
of the rest of the triangle if it has a height of 16 cm.

9. The length of the rafter is 5.5 yards, and the side height of the building is 3.5 yards.
Determine the width of the building and its total height.

5.5 yd

15°

h
3.5 yd

w
 Chapter 7 Trigonometry of Right Triangles 331

Example 4

Solve the given triangle. S

t 16.3 miles

R
15.4 miles T
SOL U T ION

Calculate ∠R using the tangent ratio.

opp
tan R =
adj

tan R = ST
TR
tan R = 16.3 Substitute the known values.
15.4

( )
∠R = tan −1 16.3
15.4
Use the inverse funcction to solve for ∠R.

∠R ≈ 46.6263

∠R is approximately 47°.

Calculate ∠S using the measures of the angles in the triangle. You could have found
∠S first using the
∠S = 180° − 90° − 47°
tangent ratio.
∠S = 43°

Calculate the length of side t using the Pythagorean theorem.

You could have
2 2 2
r +s =t solved for side t

16.32 + 15.4 2 = t 2 using the sine or

cosine functions,
16.32 + 15.4 2 = t
but this would have
265.69 + 237.16 = t involved rounding
errors.
502.85 = t
22.42 ≈ t
Side t is approximately 22.4 miles long.
332 MathWorks 10 Workbook

BUILD YOUR SKILLS

10. Solve the following triangles.

L
a)
S b)

9.5 cm
1.5 m
6.8 cm

R
2.8 m T
M
N

11. What height is a pole, and how far away from it is a cable attached to the ground, if
the angle of elevation is 25° and the cable is 18 m long?

18 m
h

25°
d
 Chapter 7 Trigonometry of Right Triangles 333

12. Find the values of a, b, c, and d.

25 cm
a
15 cm

40°

b d

PRACTISE YOUR NEW SKILLS

1. Find the indicated angle in each of the following diagrams.

a) A

340 m
175 m
334 MathWorks 10 Workbook

b)

135 cm

200 cm

2. In a right triangle, one acute angle is 22° and the hypotenuse is 70 cm. Find the
lengths of the legs and the other angle measure.

3. What is the angle of elevation if a ramp with a height of 1 metre and a horizontal
length of 3 metres?
 Chapter 7 Trigonometry of Right Triangles 335

4. A grain auger is 25 feet long. The largest angle of elevation at which it can safely be
used is 75°. What is the maximum height to which it can reach and how far from
the base of the granary will it be, assuming that it dumps right at the edge?

5. Maura’s driveway has an angle of depression of 40° from the flat roadway. If it
levels off to the garage floor, which is 3 metres below the roadway, how long is the
driveway and how far into the lot is the garage entrance?

6. If a boat is 150 metres from the base of a cliff that is 90 metres high, what is the
angle of elevation from the boat to the cliff top?
336 MathWorks 10 Workbook

CHAPTER TEST

1. What is the length of a diagonal brace used to support a table that is 120 cm wide
by 50 cm tall?

2. The Pyramid of Khufu is approximately 140 metres tall. If the base is a square with
sides measuring 230 metres, what is the slant height from the centre of one of the
sides of the pyramid? (Hint the slant height is the hypotenuse of a right triangle.)

140 m

230 m

3. A plane travels 12 km along its flight path while climbing at a constant rate of 8°.
What is the vertical change in height during this time?
 Chapter 7 Trigonometry of Right Triangles 337

4. A ramp 12 metres long makes an angle of 15° with the ground. What is the height
of the ramp? If the ramp is doubled in length, what will the total height be?

5. A chute from an open window to the ground makes an angle of 52° with the side of
a building. If the window is 18 metres from the ground, how long is the chute?

6. A tree casts a shadow that is 10 metres long. If the angle of elevation to the top of
the tree from the ground at the end of the shadow is 60°, how high is the tree?
338 MathWorks 10 Workbook

7. The angle of elevation from the bottom of one building to the top of another
building is 78°. The angle of elevation from the bottom of the second building to the
top of the first is 62°. If the distance between them is 150 metres, how much taller is
the higher building than the shorter one?

8. In an A-frame building, the angle of elevation of the roof is 50° and the building is
12 metres wide.

a) How high is the building at

the centre?

50° h

2m
12 m

b) How high is it 2 metres in from an edge?

 Chapter 7 Trigonometry of Right Triangles 339

9. A box is 1.5 m long, 1.0 m deep, and 8.0 m tall. What is the length of the longest
object that can fit in the box?

8m

10. A lifeguard sits in a chair that is 2.5 metres high. He spots a child in trouble in the
water at an angle of depression of 23°. How far out from the chair is the child?
1.5 m 1m

11. What is the angle of elevation of a playground slide that is 1.2 m high and has a
horizontal length of 2.6 m?
340 MathWorks 10 Workbook

Glossary
alternate exterior angles: angles in opposite positions outside capacity: the maximum amount that a container can hold
two lines intersected by a transversal
circumference: the measure of the perimeter of a circle

1 2
complementary angles: two angles that have measures that
3 4 ℓ1 add up to 90°
corresponding angles: two angles that occupy the same
5 6 ℓ2 relative position at two different intersections
7 8

t 1 2
3 4
alternate interior angles: angles in opposite positions
between two lines intersected by a transversal and also on 5 6
alternate sides of the same transversal 7 8

1 2 corresponding sides: two sides that occupy the same relative

3 4 ℓ1 position in similar figures

5 6 ℓ2
7 8

t
angle bisector: a segment, ray, or line that separates two
halves of a bisected angle

cosine ratio: in a right triangle, the ratio of the length of

the side adjacent to a given angle to the length of the
hypotenuse (abbreviated as cos)
deduction: money taken off your paycheque to pay taxes, union
fees, and for other benefits and programs
divisor: in a division operation, the number by which another
number is divided; in a ÷ b = c, b is the divisor

angle of depression: the angle formed between the horizontal exchange rate: the price of one country’s currency in terms of
and the line of sight while looking downward another country’s currency
horizontal factor: one of two or more numbers that, when multiplied
line angle o together, form a product. For example, 1, 2, 3, and 6 are
of s f depr e
ssion factors of the product 6 because:
ig h t

1x6=6
2x3=6
angle of elevation: the angle formed between the horizontal
and the line of sight when looking upward; sometimes 3x2=6
referred to as the angle of inclination 6x1=6

ig h t gross pay: the total amount of money earned before deductions;

ne o
fs vation
li of ele also called gross earnings or gross income
angle
horizontal
 Glossary 341

hypotenuse: the longest side of a triangle, opposite the 90° sine ratio: in a right triangle, the ratio of the length of the side
angle opposite a given angle to the length of the hypotenuse
(abbreviated as sin)

hypotenuse supplementary angles: two angles that have measures that

add up to 180°
surface area: the total area of all the faces, or surfaces, of a
three-dimensional object; measured in square units

imperial system: the system most commonly used in the United Système International (SI): the modern version of the metric
States; the standard unit of measurement for length is the system; uses the metre as the basic unit of length
foot tangent ratio: in a right triangle, the ratio of the length of the
kilogram: the mass of one litre of water at 4°C side opposite a given angle to the length of the side adjacent
to the angle (abbreviated as tan)
markup: the difference between the amount a dealer sells a
product for (retail price) and the amount he or she paid for it taxable income: income after before-tax deductions have been
(wholesale price) applied, on which federal and provincial taxes are paid

mass: a measure of the quantity of matter in an object transversal: a line that intersects two or more lines
transversal
multiple: the product of a number and any other number. For
example, 2, 4, 6, and 8 are some multiples of 2 because:

2x1=2
2x2=4
2x3=6
2x4=8
true bearing: the angle measured clockwise between true north
net: a two-dimensional pattern used to construct three- and an intended path or direction, expressed in degrees
dimensional shapes
unit price: the cost of one unit; a rate expressed as a fraction in
net income: income after all taxes and other deductions have which the denominator is 1
been applied; also called take-home pay
vertically opposite angles: angles created by intersecting
percentage: a ratio with a denominator of 100; percent (%) lines that share only a vertex
means “out of 100”
perimeter: the sum of the lengths of all the sides of a polygon 1 2
promotion: an activity that increases awareness of a product or 3 4
attracts customers
proportion: a fractional statement of equality between two
ratios or rates volume: the amount of space an object occupies
Pythagorean theorem: in a right triangle, the sum of the weight: a measure of the force of gravity on an object
squares of the lengths of the legs is equal to the square of
the length of the hypotenuse
quotient: the result of a division; in a ÷ b = c, c is the quotient
rate: a comparison between two numbers measured with
different units
ratio: a comparison between two numbers measured in the same
units
342 MathWorks 10 Workbook

Answer Key

CHAPTER 1 12. $65.00:3 months, $65.00/3 months,

UNIT PRICING AND CURRENCY EXCHANGE or $65.00
3 months
1.1 PROPORTIONAL REASONING
13. $74.00:8-hour day, $74.00/8-hour day,
$74.00
BUILD YOUR SKILLS, P. 11 or 8-hour day

1. a) 1 b) 1 14. 1 cm:2500 km
4 4
c) 1 d) 5 15. $5.57/350 g
3 7
e) 4 f) 9 16. 75 g
9 20
g) 2 h) 1 17. approximately 6.9 kg
5 7
i) 1 PRACTISE YOUR NEW SKILLS, P. 21
8
2. a) x = 8 b) x = 24 1. a) x = 16 b) x ≈ 6.5

c) x = 7 d) x = 54 c) x = 42 d) x = 63

e) x = 1542 f) x = 60 e) x = 29 f) x ≈ 52.1

g) x = 276 h) x = 125 g) x ≈ 0.5 h) x = 60

3. a) 500 mL b) 1500 mL 2. a) 20 b) 1
3 5
4. 3:1 c) 2 d) 4.8
1 1
5. 4:9 3. 8 drops of yellow pigment

6. 1:1.13 4. 30 cm

7. 1:4 5. 5 cans of paint

8. 1:32 6. The smaller gear has 24 teeth.

9. 20 cups of flour and 10 cups of shortening 7. It will take Stephie approximately 11 minutes;
this is a rate problem.
10. 10.4 L of the first chemical and 34.6 L of the
second chemical 8. 2.25 cups or 2 1 cups of flour
4
11. Cheryl will use 15 L of paint and 9 L 9. 1020 parts
of thinner
 Answer Key 343

1.2 UNIT PRICE 8. A carton of 18 is a better buy, by

$0.14 per unit.
BUILD YOUR SKILLS, P. 27

1. $0.85 1.3 SETTING A PRICE

2. $0.095 or about 10 cents BUILD YOUR SKILLS, P. 37

3. a) approximately $1.70 1. a) 0.78 b) 0.93

b) $3.80 c) 1.25 d) 3.24

4. The first package is the better deal. e) 0.005 f) 0.0038

5. A dozen muffins for $14.99 is a better buy. g) 0.012 h) 1

6. Buying pieces of 6-foot lumber is a better deal. 2. a) 45 b) 675

7. a) The first roll is less expensive per foot. c) 98 d) 42

b) The difference in price is $0.01 per foot. e) 3.9 f) 0.525

8. $23.10 g) 112 h) 33.75

9. $52.32 3. a) 20% b) 33.3%

PRACTISE YOUR NEW SKILLS, P. 32 c) 162% d) 20%

1. a) approximately $0.71 e) 8.3% f) 1250%

b) $10.10 4. $562.50

2. a) at least 84 m2 (the area of the room) 5. $1.81

b) $684.60 6. $187.92

3. a) $160.00 7. $15.75

b) $5.00/sq ft 8. $104.95

4. a) $10.44 9. $1048.93

b) 3 cases cost $84.48; each brush costs $4.69 10. $44.64

5. 8 oz of Brie cheese for $4.95 is a better buy. 11. GST: $2.30; PST: $2.75

6. a) 54 loaves 12. $74.26

b) $270.00 13. $26.65

7. You would save $0.0014/mL.

344 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS, P. 46 PRACTISE YOUR NEW SKILLS, P. 56

1. a) 3.6 b) 22 1. $1481.85

c) 304 d) 47.25 2. a) $14.99 b) $34.99

2. a) $50.75 b) $304.50 3. $6.59

3. 135% 4. $59.95

4. $495.49 5. a) $669.75 b) $2679.02

5. $20.10 6. approximately 30%

6. $21.56 7. a) $87.45 b) $96.20

7. Harry will save $1.11 by buying the MP3 c) $34.98 d) $38.48

player in Alberta.
e) $57.72 f) 60%

1.4 ON SALE!
1.5 CURRENCY EXCHANGE RATES
BUILD YOUR SKILLS, P. 50
BUILD YOUR SKILLS, P. 60
1. $296.99
1. 90 038.95 Ft
2. $1.79
2. £118.22
3. $249.99
3. 535.41 kr
4. $17.46
4. €215.40
5. $37.85
5. 421.68 SFr
6. $255.00
6. 1236.10 Trinidad and Tobago dollars
7. about 14%
7. a) approximately 345.81 Brazilian reals
8. 20%
b) approximately 1742.27 Moroccan dirhams
9. 17%
c) approximately 1544.55 Ukrainian hryvnia
10. $109.52
d) approximately 639.78 Polish zloty
11. $206.21
8. $339.69 CAD
12. a) $112.97 b) 40%
9. $176.21 CAD
 Answer Key 345

10. $88.48 CAD 5. $47.74 CAD

11. You will get more units of the currency than of 6. $5.30 CAD
Canadian dollars.
7. $28.84
12. $384.48 CAD
CHAPTER TEST, P. 70
13. $32.57
1. 42 L
PRACTISE YOUR NEW SKILLS, P. 67
2. a) 5.5 L/100 km, 5.5 L:100 km, or 5.5 L
1. a) 47 746 Japanese yen 100 km
b) 22 L
b) 705 Turkish Lira
3. 296 km
c) 340 euros
4. $3.25
d) 3215 Chinese yuan
5. $14.43
e) 3659 Hong Kong dollars
6. $2.87/dozen
2. a) 6112 Mexican pesos
7. a) $1105.97
b) 5322 Estonian kroon
b) No. The original price is reduced by 35%,
c) 374 British pounds and then the sale price is reduced by 20%.
d) 54 213 South Korean won 8. 9%
e) 21 784 Indian rupees 9. 62%
f) 14 178 Russian rubles 10. $103.80
3. a) $91.63 CAD 11. $177.00 CAD
b) $638.09 CAD 12. a) €306.10
c) $367.52 CAD b) $183.76 CAD
d) $466.54 CAD

4. a) $552.21 CAD

b) $193.95 CAD

c) $344.29 CAD

d) $26.45 CAD
346 MathWorks 10 Workbook

CHAPTER 2 10. TIME CARD: HAE-RIN

EARNING AN INCOME Day Morning Afternoon Total
2.1 WAGES AND SALARIES Hours
IN OUT IN OUT
BUILD YOUR SKILLS, P. 76 Monday 7:45 9:00 5:00 7:45 1.25 h +
2.75 h
1. a) 4 1 b) 54 7 Tuesday 4:00 8:00 4h
7 9 Wednesday 9:00 11:00 2h
c) 34 19 d) 7 1 Thursday 9:00 11:00 3:00 5:00 2h+2h
29 2
Friday 3:00 6:00 3h
e) 24 7 f) 66 2 Saturday 9:00 12:00 3h
8 3
2. a) 61 b) 43 Hae-rin earned $255.20.
11 9
c) 263 d) 61 11. Pete will earn $971.68 for the week.
17 8
e) 64 f) 127 12. Ingrid will earn $537.81 per week.
5 12
3. $83.60 13. Nathalie will earn $5436.24 during
the summer.
4. $834.80

5. $1215.50 PRACTISE YOUR NEW SKILLS, P. 85

6. a) $3765.28 1. $107.50

b) $868.91 2. $92.70

7. $13.82/h 3. Juanita would earn more per week at the

second job, even though she would be working
8. $9.88/h fewer hours, so she should take the second job.
9. TIME CARD: MONTY 4. $129.96
Day Total
Hours 5. $300.16
IN OUT
Monday 3:30 6:45 3.25 h
6. $381.70
Tuesday
Wednesday 5:00 9:30 4.5 h 2.2 ALTERNATIVE WAYS TO EARN MONEY
Thursday 5:00 9:30 4.5 h
Friday 3:30 7:00 3.5 h BUILD YOUR SKILLS, P. 87

Monty earned $148.84. 1. $366.00

2. $51.75

3. $51.00
 Answer Key 347

4. $254.00/job 2. Since $275.00/month is more than 12% of his

regular pay, he should take that offer.
5. $3.75/quart
3. Darren: $198.40; Sean: $123.04
6. 550 words
4. $527.75
7. $94.31
5. $481.25
8. $15 018.00
6. The bonus of 28% of his regular pay is more
9. $701.05
than $1250.00, so Chen should take the
10. 7% 28% bonus.

11. 5% 7. $208.10/day

12. $2128.89 8. about 12%

13. $479.52 9. $121.30

14. $414.63 PRACTISE YOUR NEW SKILLS, P. 104

15. about 11% 1. $51 890.00

PRACTISE YOUR NEW SKILLS, P. 95 2. $795.63

1. a) $231.00 b) $189.00 3. a) Each waiter would receive $2028.40 in tips

for the month.
2. $160.00
b) Each waiter would make $1521.30.
3. $6250.00
4. $840.29
4. $12.63/h
5. $4887.62
5. a) The other contractor’s bid is $14 382.00.
Jeff could lower his bid to that amount; he
2.4 DEDUCTIONS AND NET PAY
would make a profit of $1683.90.
BUILD YOUR SKILLS, P. 107
b) $33.68/h
1. $112.50
2.3 ADDITIONAL EARNINGS
2. $1.50
BUILD YOUR SKILLS, P. 98
3. $78.11
1. $175.00
4. about 23%
348 MathWorks 10 Workbook

5. a) about 15% b) 5.9% CHAPTER 3

LENGTH, AREA, AND VOLUME
6. about 1.7%
3.1 SYSTEMS OF MEASUREMENT
7. $687.62
BUILD YOUR SKILLS, P. 120
8. $2301.43
1. a) 53.6 cm b) 43.6 cm
9. a) $575.00/week b) $470.27/week
c) 7.6 m
c) 8.3%
2. 780 cm
PRACTISE YOUR NEW SKILLS, P. 113
3. 240 m
1. a) $1098.84 b) $2211.96
4. 122 ft
2. about 23%
5. about 33.3 m
3. $790.21
6. 104 m

CHAPTER TEST, P. 114 7. 1099.5 cm

4. $12.54/h 8. Bernard needs 13 1 ft of 2 by 4 lumber and

2
52 1 ft of 2 by 2 lumber.
5. $19 500.00 2
9. 12 ft
6. $42 816.00
10. 740 boards
7. The second job would pay $40.00
more per year. 11. The cages will not fit along a 30′ wall. They are
2″ too wide.
8. 3%
12. The garden circumference is about 20 feet, so
9. $281.20
Gordon will need 20 geraniums.
10. $186.30
13. The height of the pipe is 6′ 7″. Craig is 6′ 6″
11. $873.75 tall, so he will be able to walk under the
finished pipe.
12. $75.87

13. $3274.16 PRACTISE YOUR NEW SKILLS, P. 129

14. $38.73 1. a) 3 1 feet b) 1 ft 4 in

2

15. a) $985.80 b) $24.65/h c) 2 2 yards d) 8800 yards

3
 Answer Key 349

2. 34′ 4″ 4. The height of the semi-trailer is 10 ft 8.4 in. It

will not fit through the tunnel.
3. 65′ 2″
5. $38.14
4. 64 packages
6. 10.5 lb
5. 11.4 blocks
7. $643.72
3.2 CONVERTING MEASUREMENTS
3.3 SURFACE AREA
BUILD YOUR SKILLS, P. 132
BUILD YOUR SKILLS, P. 145
1. 760 sq ft
1. 468 sq in
2. The diameter of the rug is 2.4 m; this is less
than the dimensions of the space Travis wants 2. 66 sq ft
to cover, so the rug will fit into the space.
3. 11 305 sq in
3. The label is 28.3 cm long.
4. a) 959 sq in b) 6.7 sq ft
4. 20.3 cm by 10.2 cm
5. 149.2 sq in
5. $2227.14
6. 1.4 sq ft
6. a) $28.56/linear yard
7. 374 sq ft
b) $31.73/linear metre
8. 214.2 sq in
c) $123.75
9. 112.7 sq ft
7. $554.82
10. $1234.80
8. a) 27 boxes b) $511.65
11. $1312.85
9. $1072.85
12. $179.70
PRACTISE YOUR NEW SKILLS, P. 139
PRACTISE YOUR NEW SKILLS, P. 155
1. a) 132 inches b) 11 feet
1. 35.3 sq ft
c) 3 yd 2 ft
2. 36 sq ft
2. 387.5 mi
3. 50 of the 39 cm by 39 cm tiles would be
3. 45 m by 25.5 m needed. 155 of the 18 cm by 27 cm tiles would
be needed.
350 MathWorks 10 Workbook

4. 47.3 sq in CHAPTER TEST, P. 169

5. 11.8 in by 7.5 in 1. a) 6 ft 6.4 in

6. 14.7 sq in b) Robert Pershing Wadlow: about 2.7 m;

He Pingping: about 0.7 m.
7. 922 320 sq ft
c) 2 m
3.4 VOLUME
2. Height of tunnel: 3.15 m. Franklin’s truck is higher
BUILD YOUR SKILLS, P. 159 than that, so it will not fit through the tunnel.

1. a) 12 960 in3 b) 7.5 ft3 3. 7 panels

2. 11.25 ft3 4. a) 129 tiles b) 47 geraniums

3. Volume of box: 72 in3. Volume of cube: 64 in3. 5. 2.3 gallons (likely rounded up to 3 gallons)
The contents of the box will not fit in the cube. 6. 354.8 sq ft
4. 3 bags 7. Soccer field dimensions: 118.8 yd by 79.6 yd.
5. 1875 cubic feet Yes, it does fit the league’s specifications.

6. 4446 in3 8. 158 400 cm3

7. a) 14.5 US gallons b) $43.36 9. 2.7 cubic yards

c) $120.29 10. 1 US gal = 3800 mL; 0.5 US gal = 1900 mL;

1 quart = 950 mL; 0.5 pint = 237.5 mL
8. 2675 ft3
11. a) 5 cup or 0.625 cup
9. 6.2 ft 8
b) 125 mL
PRACTISE YOUR NEW SKILLS, P. 166

1. a) 72 cubic feet b) 2.7 cubic yards

2. 687.5 mL

3. 150 mL

4. 10.4 US gallons

5. Fav will need about 4.7 L of fertilizer, so he

will need 2 bottles.

6. $366.12
 Answer Key 351

CHAPTER 4 5. The temperature rose from –2.2°F to 71.6°F.

MASS, TEMPERATURE, AND VOLUME
6. 5°C
4.1 TEMPERATURE CONVERSIONS
7. –77°C
BUILD YOUR SKILLS, P. 177
8. 1260°C
1. 176.7°C
4.2 MASS IN THE IMPERIAL SYSTEM
2. 114.4°C
BUILD YOUR SKILLS, P. 185
3. 1198.9°C
1. 12 lb 3 oz
4. Ashley’s dog has a temperature of 104°F. This
is outside (higher than) the normal range. 2. 17 lb 8 oz

5. The temperature is 48°F. The paint can be 3. 5.3 baskets of raspberries, rounded up to 6
safely applied. 4. The weight of the load is about 1.6 tons, so
6. a) 103°F it is unsafe and over the acceptable limit
of 1.5 tons.
b) Minimum temperature: –47.8°C;
maximum temperature: 9.4°C 5. 28 slabs

c) 57.2°C 6. 14.4 tons

7. The 2.5-lb jar is the best buy.

PRACTISE YOUR NEW SKILLS, P. 179
8. 5.3 skeins, rounded up to 6
1. a) 95°F b) 17.6°F
9. approximately 4693 chocolate bars
c) 329°F d) 69.8°F
10. $2.20/lb
e) –40°F f) 395.6°F
11. $0.08/lb
2. a) –28.9°C b) 26.7°C
12. $1.29/oz
c) 190.6°C d) –16.7°C

e) –17.8°C f) –18.9°C PRACTISE YOUR NEW SKILLS, P. 192

3. The blowtorch flame is 2372°F. The candle 1. a) 1.5 lb b) 3.945 tn

flame is 998.9°C. The blowtorch is hotter than c) 3 lb 6 oz d) 98 oz
the candle flame, by 542°F or 301.1°C.
e) 9080 lb f) 40 lb 14 oz
4. a) between 93.3°C and 126.7°C
2. 6 lb
b) between 18.3°C and 37.8°C
3. 6 2 oz
3
c) between 71.1°C and 98.9°C
352 MathWorks 10 Workbook

4. 3425 lb 6. $16.61/kg

5. The paint will weigh 13.5 lb, so Harinder can 7. 16 hamburgers

safely paint the structure.
4.4 MAKING CONVERSIONS
6. a) $1.67/lb b) $1.25/12 oz
BUILD YOUR SKILLS, P. 204
7. $1.73/lb
1. a) 67.9 kg b) 149 lb
4.3 MASS IN THE SYSTÈME
INTERNATIONAL 2. White beans weigh more per unit volume.

BUILD YOUR SKILLS, P. 196 3. $5.33/bu

4. 187 bushels
1. 5.75 tonnes
5. 22.9 tonnes
2. 414 g
6. Wheat is approximately twice as heavy as
3. a) $6.27 b) $17.90
sunflower seeds, per unit volume.
4. about 6.3 oz
7. about 2.8 kg
5. 3522.4 g
8. The sandstone weighs about 4.8 t, so the crane
6. 149.6 lb can be used.

7. $4.43 9. The total weight of the package is 1.7 kg, so

she will be able to send it at the cheaper rate.
8. 200 g at $3.85 is the better buy.

9. a) 4.2 mg b) $8.19 PRACTISE YOUR NEW SKILLS, P. 207

1. about 199 bushels

PRACTISE YOUR NEW SKILLS, P. 200
2. Total weight: 5216 kg. This is over the limit, so
1. a) 2500 kg b) 2800 g
the load cannot be safely carried.
c) 0.125 kg d) 0.0024 kg
3. approximately 909 kg
e) 2200 lb f) 3272.4 kg
4. about 8 adults
2. 1 tonne (t) ≈ 1.1 tons (tn)
5. 150 mL
3. 3450 g
6. 23.6 ha
4. 2045 kg

5. 6.6 lb
 Answer Key 353

CHAPTER TEST, P. 210 CHAPTER 5

ANGLES AND PARALLEL LINES
1. a) 77°F b) –3.9°C
5.1 MEASURING, DRAWING, AND
c) –40°C d) –31.7°C ESTIMATING ANGLES

e) 207.2°C f) 113°F BUILD YOUR SKILLS, P. 214

2. 932°F to 36 032°F 1. a) acute b) reflex

3. a) 196 oz b) 4600 lb c) obtuse d) acute
c) 2 tn 1284 lb d) 60 oz e) straight f) obtuse
e) 10.3 lb f) approximately 1 lb g) reflex h) reflex
4. 140 318 trees 2. ∠A measures about 40°.
5. 1 lb 7 oz ∠B measures about 75°.
6. 6.25 lb, or 6 lb 4 oz ∠C measures about 65°.
7. 14 eggs ∠D measures about 10°.
8. 1571 kg 3. The angle is about 20°.
9. approximately 1.2 tonnes 4. ∠A measures about 140° or 150°.

∠B measures about 230° or 240°.

∠C measures about 170°.

∠D measures about 330°.

5. The angle is about 100°.

6. ANGLE COMPLEMENTS AND SUPPLEMENTS

Angle Complement Supplement
45° 90° − 45° = 45° 180° − 45° = 135°
78° 90° − 78° = 12° 180° − 78° = 102°
112° Does not exist, 180° − 112° = 68°
because angle is
greater than 90°.
160° Does not exist, 180° − 160° = 20°
because angle is
greater than 90°.
220° Does not exist, Does not exist, because
because angle is angle is greater than
greater than 90°. 180°.
354 MathWorks 10 Workbook

7. a) 32° b) 148° b)

8. a) 90° b) 90°

9. 115°

10. 180°

11. 337.5°
c)
PRACTISE YOUR NEW SKILLS, P. 222

1. a) acute b) obtuse

c) reflex d) obtuse 3. 156°

e) acute f) reflex 4. 120°

2. a) The angle is slightly greater than 60°, 5. 3.3°

about 65°.
6. approximately 35°
b) The angle is slightly greater than 45°,
7. a = 45°; b = 50°; c = 55°; d = 30°
about 50°.

c) The angle is about 45°. PRACTISE YOUR NEW SKILLS, P. 229

d) Angle x is about 120°, and angle y is 1. a) Not perpendicular.

about 60°. b) Not perpendicular.
3. 25° c) Yes, the lines are perpendicular.
4. ∠A is more than 90°; it is about 110°. d) Yes, the lines are perpendicular.
5. a) 175° b) 220° 2. ANGLE CALCULATIONS
Angle Complement Supplement Resulting
5.2 ANGLE BISECTORS AND angle
measures
PERPENDICULAR LINES
after angle is
bisected
BUILD YOUR SKILLS, P. 226
73° 17° 107° 36.5°
78° 12° 102° 39°
1. 45°
15° 75° 165° 7.5°
2. a) 48° 42° 132° 24°
90° 0° 90° 45°
68° 22° 112° 34°
41° 49° 139° 20.5°
136° n/a 44° 68°
80° 10° 100° 40°
254° n/a n/a 127°
 Answer Key 355

3. x = 67.5°, y = 22.5° ∠7, using lines ℓ1 and ℓ2 with

4. a) y = 110°, x = 70° transversal ℓ4.

x and y are supplementary. 4. CB and BD intersect AB and AD.

b) x = 72° 5. Line t cannot be a transversal because it does

not pass through two distinct points. It is
c) x = 99° concurrent to ℓ1 and ℓ2 because they all pass
x and the 81° angle are supplementary. through the same point.

d) x = 245° 6. t and ℓ3 are intersected by ℓ1 and ℓ2.

5. 67.5° PRACTISE YOUR NEW SKILLS, P. 236

5.3 NON-PARALLEL LINES AND 1. a) ∠3 and ∠5; ∠2 and ∠8
TRANSVERSALS
b) ∠1 and ∠5; ∠2 and ∠6; ∠3 and
BUILD YOUR SKILLS, P. 233 ∠7; ∠4 and ∠8

1. a) alternate interior angles c) ∠2 and ∠5; ∠3 and ∠8

b) corresponding angles 2. ∠3 and ∠5; ∠4 and ∠6.

c) exterior angles on the same side of the 3. a) ℓ3 is the transversal that makes ∠1 and ∠2
transversal corresponding angles for ℓ1 and ℓ2.

d) interior angles on the same side of the b) ℓ4 is the transversal that makes ∠3 and ∠4
transversal alternate interior angles for ℓ1 and ℓ2.

2. a) ∠6 b) ∠3 4. ∠3 = 95°

c) ∠3 d) ∠3 ∠4 = 68°

3. a) ∠7, using lines ℓ3 and ℓ4 with transversal ℓ1. ∠5 = 112°

∠3, using lines ℓ1 and ℓ2 with ∠3 + ∠4 + ∠5 = 360°

transversal ℓ3.
5. ∠1 = 60°
b) ∠4, using lines ℓ3 and ℓ4 with transversal ℓ2.
∠2 = 120°
c) ∠10, using lines ℓ3 and ℓ4 with
∠3 = 60°
transversal ℓ2.
∠4 = 110°
d) ∠5, using lines ℓ3 and ℓ4 with
transversal ℓ2. ∠5 = 70°

∠6 = 110°
356 MathWorks 10 Workbook

5.4 PARALLEL LINES AND TRANSVERSALS 4. ℓ1 and ℓ3 are parallel.

BUILD YOUR SKILLS, P. 240 ℓ1 ℓ2
ℓ3

1. ∠1 and the 71° angle are interior angles on the

same side of the transversal.

∠1 = 109°
62° ℓ4
∠2 is supplementary to the 118° angle.
103° 103°
ℓ5
∠2 = 62° 77° 77°

ℓ6
∠3 corresponds to the 118° angle.
68°
∠3 = 118°

∠4 corresponds to ∠2, and is also

supplementary to ∠3.
5. ∠1 = 57°
∠4 = 62°
6. 4°
2. ∠1 = 112°
7. SOLVING ANGLE MEASURES
∠2 = 68° Angle measure Reason
∠1 = 54° It is vertically opposite the 54°
∠3 = 68° angle.
∠2 = 54° It is an alternate interior angle
∠4 = 112° to ∠1.
∠3 = 97° It is supplementary to the 83°
3. ∠A is 106°. It is supplementary to ∠B (74°) angle.
because they are interior angles on the same ∠4 = 83° It is an interior angle on the
side of the transversal, given parallel lines BC same side of the transversal
as ∠3, so is supplementary to
and AD and transversal AB. it. Also, it is vertically opposite
83°.
∠C is 106°. It is supplementary to ∠B (74°)
because they are interior angles on the same 8. ∠1 = 122°
side of the transversal, given parallel lines AB ∠2 = 90°
and CD and transversal BC.
9. ∠2 = 127°
∠D is 74°. It is supplementary to ∠A (106°)
because they are interior angles on the same PRACTISE YOUR NEW SKILLS, P. 246
side of the transversal, given parallel lines AB
1. ∠1 is supplementary to the 112° angle.
and CD and transversal AD.
∠1 = 68°
 Answer Key 357

∠2 is supplementary to the 112° angle, and 4. The top of stud A must be moved 1° to the
vertically opposite ∠1. right, to change the 89° angle to 90°.

∠2 = 68° The top of stud B must be moved 1° to the left,

to change the 91° angle to 90°.
∠3 is an interior angle on the same side of the
transversal as 112°, and is an alternate interior The top of stud D must be moved 1° to the left,
angle to ∠2. to change the 134° angle to 135°.

∠3 = 68°
CHAPTER TEST, P. 248
∠4 is supplementary to the 60° angle.
1. a) obtuse b) acute
∠4 = 120°
c) reflex d) straight
∠5 is vertically opposite ∠3.
e) right f) obtuse
∠5 = 68°
2. ANGLE CALCULATIONS
2. ∠1 = 57° Angle Complement Supplement Resulting
angle
∠2 = 33° measures
after angle
3. ℓ2
is bisected
ℓ1 58° 32° 122° 29°
94° Does not exist, 86° 47°
because angle
48° 48° 132°
ℓ3 is greater than
90°.
ℓ4 87° Does not exist, 93° 43.5°
46°
46°
because angle
134°
ℓ5 is greater than
46°
90°.
153° Does not exist, 27° 76.5°
132° 132°
ℓ6 because angle
is greater than
90°.
65° 25° 115° 32.5°
ℓ1 is parallel to ℓ2 because, with transversal ℓ3,
the corresponding angles are equal. 3. a) alternate interior angles

ℓ3 is parallel to ℓ6 because, with transversal ℓ2, b) interior angles on the same side of the
two corresponding angles are 132°. transversal

ℓ4 is parallel to ℓ5 because, with transversals ℓ1 c) vertically opposite angles

and ℓ2, the corresponding angles are equal.
d) corresponding angles
358 MathWorks 10 Workbook

4. ∠2 is supplementary to the 62° angle. 9.

1
∠2 = 118° ℓ1
2
∠4 is vertically opposite to the 62° angle or ℓ2
3
supplementary to ∠2. ℓ3

∠4 = 62°

∠3 is the alternate interior angle to the 62° Fred is correct. ∠1 is equal to ∠3 and ℓ1 is
angle, and is an interior angle on the same side parallel to ℓ3 since the corresponding angles
of the transversal as ∠2. are equal.
∠3 = 62° 10. ∠2 = ∠7
∠1 is an interior angle on the same side of the Using ℓ1 and ℓ2, and transversal t1, ∠2 and ∠7
transversal to the 67° angle. are alternate interior angles.
∠1 = 113° ∠5 = ∠7
5. ∠1 is an interior angle on the same side of the Using ℓ1 and ℓ2, and transversal t1, ∠5 and ∠7
transversal (line A) as ∠D (68°). are corresponding angles.
∠1 = 112° ∠4 = ∠7
∠2 is the corresponding angle to ∠C (75°), Using ℓ1 and ℓ3, and transversal t1, ∠4 and ∠7
given transversal AC. are alternate interior angles.
∠2 = 75°

6. ∠1 = 23°

∠2 = 23°

7. a) ∠1 = 72° b) ∠2 = 18°

8. a) 90° b) 185°
 Answer Key 359

CHAPTER 6 x = 45 cm
SIMILAR FIGURES
y = 70°
6.1 SIMILAR POLYGONS
z = 17.5 cm
BUILD YOUR SKILLS, P. 257
7. length: 16.25 cm; width: 12 cm; height: 7 cm
1. LP = 7 cm
8. length: 20 in; width: 17.5 in; height: 13.5 in
OP = 6 cm
9. Box 1: 9 cm by 6 cm by 3 cm;
ON = 4 cm Box 2: 6.75 cm by 4.5 cm by 2.25 cm;
MN = 6 cm Box 3: 5.1 cm by 3.4 cm by 1.7 cm

2. 22 feet by 12 feet PRACTISE YOUR NEW SKILLS, P. 262

3. ABCD∼QRST and EFGH∼LKJI 1. 855 cm or 8.55 m

4. Corresponding angles: 2. about 3.1 m and 3.75 m

∠A = ∠G 3. 108°, 204°, 63°, 120°, and 45°

∠B = ∠H 4. 48 mm by 20 mm; 1920 mm by 800 mm

∠C = ∠I 5. Funnel 1: height about 13.3 cm, spout length

about 8.3 cm, spout diameter about 1.7 cm
∠D = ∠J
Funnel 2: height about 10.7 cm, spout length
∠E = ∠K
about 6.7 cm, spout diameter about 1.3 cm
∠F = ∠L
Funnel 3: height 8 cm, spout length 5 cm,
Angle measures: spout diameter 1 cm

∠D = 73°
6.2 DETERMINING IF TWO POLYGONS ARE
∠H = 21° SIMILAR

∠E = 40° BUILD YOUR SKILLS, P. 266

5. ∠B = 100° 1. Yes, the two hexagons are similar. If a hexagon

∠X = 32° is regular, all the angles are congruent and
all sides are congruent. Therefore, the two
∠C = 48° hexagons Pierre drew will have congruent
6. w = 42° corresponding angles, and are similar.
360 MathWorks 10 Workbook

2. Yes, the picture and poster are similar. 6.3 DRAWING SIMILAR POLYGONS

3. Zora is correct if she is comparing rectangle BUILD YOUR SKILLS, P. 272

ABCD to rectangle PQRS. But if Zora compares
the longer sides of the two rectangles, and the 1.
shorter sides of the two rectangles, she will 4 cm
find that they are similar. She can compare
rectangle ABCD to rectangle QRSP. Since 100 8 cm
50
is equal to 60 , Zora can say that rectangle
30 2. See diagram on the next page
ABCD is similar to rectangle QRSP.
3. 1 in:5.5 ft
4. Yes, it is true that if you increase or decrease
the side lengths of a figure by the same factor, 4. scale factor: 1.25
the resulting figure will be similar to the
5. 26 cm by 15.6 cm by 6.5 cm
original. If you multiply or divide by the same
number, you are keeping the same proportion. 6. length: 70 cm; upper portion: 14 cm;
half-width: 35 cm
Examples will vary.

5. The framed picture is not similar to PRACTISE YOUR NEW SKILLS, P. 275
the original. 1. 75 mm
6. The two cylinders are not similar. 2.

PRACTISE YOUR NEW SKILLS, P. 269

3 cm
2 cm
1. a) 2500 m b) 7.5 cm

2. The prisms are not similar. 2 cm 3 cm

3. Colin is correct as there is enough information a) Yes, the squares are similar because the
given to determine that corresponding sides are proportional.
sides are proportional and all the angles are
b) No, the rectangles will not be similar.
right angles.
3.
4. Mats 1 and 2 are similar. Mat 3 is not similar
1.5 cm
to mat 1 or mat 2.
1 cm
4 cm
5. Consider two rectangles, one with sides ℓ and
w, and the other with sides double the length, 4. a) scale factor: approximately 0.0039
2ℓ and 2w. The area of the second rectangle is 4
times the area of the first; this is not the same b) 0.7 cm
proportion as the sides. 5. a) 1.25 feet b) 1:8
 Answer Key 361

6.3 Drawing Similar Polygons, Build Your Skills, Question 2.

Bench

Table

Pool
Planters

6.4 SIMILAR TRIANGLES PRACTISE YOUR NEW SKILLS, P. 282

BUILD YOUR SKILLS, P. 278 1. ED ≈ 4.7 m

1. a) x ≈ 3.9 cm 2. No, the triangular blocks of wood are not

similar. While the triangular faces are similar,
b) x ≈ 29.1 ft both blocks are 2 inches thick. In order for the
c) c ≈ 4.8 m blocks to be similar, their thicknesses would
have to be proportional.
2. ST ≈ 6.7 cm; you are not given enough
information to find the length of the 3. ∠F = ∠X
third side, TR. ∠G = ∠Y
3. No, not all isosceles triangles are similar. ∠H = ∠Z
4. You will be 10.5 m higher. FG ∼ XY
5. 17.6 cm and 36.8 cm GH ∼ YZ
6. ΔABC ∼ ΔACD ∼ ΔCBD FH ∼ XZ

4. about 17 ft
362 MathWorks 10 Workbook

CHAPTER TEST, P. 284 CHAPTER 7:

TRIGONOMETRY OF RIGHT TRIANGLES
1. 5 in, 15 in, 25 in, 35 in, and 60 in
7.1 THE PYTHAGOREAN THEOREM
2. x = 7.2 m
BUILD YOUR SKILLS, P. 290
y ≈ 3.5 m
1. a) y X
3. The river is approximately 42.7 m wide. Z

4. The man is approximately 2 m tall. z

x

5. When the side lengths of a rectangle are

Y
doubled, the area is quadrupled.
b) Q
6. When the ratios are rounded to 2 decimal
s r
places, the shapes are similar.
R S
7. a) True, all equilateral triangles are similar. q
In equilateral triangles, the angles are
f
c)
always 60º and the ratios of the sides will D E
always be equal.

b) False, all isosceles triangles are not similar.

e d
The angles do not need to be equal, nor do
the sides have to be proportional.

c) True, congruent triangles are similar. The F

corresponding angles are equal and the
d) r
ratios of sides will be 1:1. S T

8. The scale factor of the projection is about 8.3.

t s
9. 17′ is represented by 7.8″.

6′ is represented by 2.8″.
R
5′ 2″ is represented by 2.4″.

7′ 2″ is represented by 3.4″.

10. The painting will be 31.25 inches by

43.75 inches.

11. 95.7 cm by 126.5 cm

 Answer Key 363

2. a) A 4.

B b

a
ℓ h

b) Q
s
r
R d
q S
h2 + d2 = ℓ2
c) Y 5. x = z 2 − y2

y= z2 − x2
x w
6. x ≈ 8.0 cm

W X y ≈ 6.7 cm
y

7. approximately 12.5 feet

d) d E
F
8. Your route would be about 83.7 m shorter by
walking diagonally across the field.
e f

PRACTISE YOUR NEW SKILLS, P. 295

D
1. approximately 10 ft 7.2 in
3. In ΔABC:
2. approximately 14.4 m
a + c = (x + y)
2 2 2

3. approximately 11.1 m
In ΔABD:
4. approximately 41 in
x2 + z2 = c2
5. approximately 50.9 km
In ΔBDC:

z2 + y2 = a2
364 MathWorks 10 Workbook

7.2 THE SINE RATIO 7.3 THE COSINE RATIO

BUILD YOUR SKILLS, P. 298 BUILD YOUR SKILLS, P. 310

1. a) sin A ≈ 0.62 b) sin A ≈ 0.54 1. a) cos 23° = 0.9205

2. a) sin 10° = 0.1736 b) sin 48° = 0.7431 sin 67° = 0.9205

c) sin 62° = 0.8829 d) sin 77° = 0.9744 b) cos 83° = 0.1219

3. sin 90° = 1 sin 7° = 0.1219

Possible reasons will vary. c) cos 45° = 0.7071

4. a) a ≈ 8.2 cm sin 45° = 0.7071

b) x ≈ 3.8 cm d) cos 37° = 0.7986

5. about 7 ft sin 53° = 0.7986

6. about 8.6 m 2. a) x ≈ 8.2 cm b) x ≈ 1.7 cm

7. a) h ≈ 14.3 mm c) x ≈ 8.6 cm d) x ≈ 12.2 cm

b) h ≈ 12.9 cm 3. about 5.4 m

8. approximately 5.3 m long 4. about 9.8 yards

9. about 22 m 5. about 6.2 m

10. about 404.5 m 6. about 14.1 m

11. approximately 203.9 m 7. about 11.6 m

12. about 1.9 m 8. about 502 km

PRACTISE YOUR NEW SKILLS, P. 306 PRACTISE YOUR NEW SKILLS, P. 315

1. a) sin A ≈ 0.6371 b) sin B ≈ 0.7191 1. a) x ≈ 3.6 cm b) a ≈ 4.8 m

2. a) x ≈ 17.9 cm b) y ≈ 12.5 cm c) r ≈ 9.5 cm d) ℓ ≈ 3.1 m

3. about 5.5 m 2. approximately 17.3 m

4. about 8.6 m 3. approximately 23.7 m

5. about 41.7 m 4. about 1.24 km

5. about 8.49 km

6. about 3.42 m
 Answer Key 365

7.4 THE TANGENT RATIO 7. r ≈ 7.3 m

BUILD YOUR SKILLS, P. 320 q ≈ 4.9 m

1. a) x ≈ 9.5 m b) a ≈ 22.4 in ∠Q = 42°

c) r ≈ 2.4 m d) p ≈ 7.9 in 8.

2. about 412 m
20° 20°
3. about 2.1 m

PRACTISE YOUR NEW SKILLS, P. 321

17 cm

1. about 21 m
h = 16 cm
2. about 9 m

3. about 5.5 ft

4. about 141.1 m
70° 70°
5. about 12.9 ft 11.6 cm

6. about 373 m 9. width: about 10.6 yards; total height of

building: about 4.9 yards.
7.5 FINDING ANGLES AND SOLVING
RIGHT TRIANGLES 10. a) ∠R ≈ 28°

BUILD YOUR SKILLS, P. 325 ∠S ≈ 62°

t ≈ 3.2 m
1. a) D ≈ 33°
b) ℓ ≈ 6.6 cm
b) F ≈ 26°
∠M ≈ 45°
c) G ≈ 67°
∠L ≈ 45°
d) H ≈ 89°
11. pole height: about 8 m; the cable is attached
2. ∠X ≈ 61°
about 16 m away from the pole.
3. about 62°
12. a) a ≈ 23 cm b) b ≈ 18 cm
4. about 54°
c) c ≈ 16 cm d) d ≈ 12 cm
5. about 38°

6. about 51°
366 MathWorks 10 Workbook

PRACTISE YOUR NEW SKILLS, P. 333 CHAPTER TEST, P. 336

1. a) ∠A ≈ 31° b) ∠B ≈ 48° 1. 130 cm

2. B
2. about 181 m

3. about 1.67 km
70 cm
a 4. The ramp rose about 3.1 m.
22° Because of similarity of triangles, a 24-m ramp
A
b C
would rise double that of a 12-m ramp, or
∠B = 68° about 6.2 m.

a = 26 cm 5. about 29.2 m

b = 65 cm 6. about 17.3 m

3. about 18.4° 7. about 423.6 m

4. about 24.1 ft 8. a) about 7.2 m

5. The driveway is about 4.7 m long. The garage b) about 2.4 m

entrance is about 3.6 m into the lot. 9. about 8.2 m
6. about 31° 10. about 6 m

11. about 25°