CSEC Maths Sample Pages

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CSEC

CSEC® MATHEMATICS

• Full coverage of the CXC® syllabus for first examination from May–June 2018 • Real-world contexts throughout so students understand the practical application of maths • Clear guidance on the School Based Assessment • Written in clear, accessible language suitable for all students • Differentiated practice exercises help students understand the level they are working at, and help them reach the level required for a good grade • Worked examples • Practice examination papers for paper 1 and paper 2 • End-of-chapter summaries consolidate learning and provide additional practice questions

About the author Raphael Johnson holds a Masters degree in mechanical engineering (UCLV – Cuba) and a Diploma in Education specializing in the teaching of mathematics from the University of the West Indies, St. Augustine, Trinidad. He has many years’ experience teaching maths at CSEC®, CAPE® and teachers’ college levels in Grenada. He is currently teaching maths, physics and other mechanical engineering courses at T. A. Marryshow Community College, St Georges, Grenada.

www.collins.co.uk/caribbeanschools ISBN 978-0-00-830446-1

9 780008 304461

CSEC Mathematics 2.indd 1

MATHEMATICS

Raphael Johnson

• Extra guidance on the maths investigation question

CSEC® MATHEMATICS

Collins CSEC® Mathematics provides a new approach to the study of the CSEC® mathematics syllabus. Fully grounded in the practical application of maths in the world today, it includes real-world and everyday contexts for all topics to help Caribbean students understand why maths is so important.

®

Raphael Johnson Reviewers: Melissa Andrade, Joanne Baker, Marcus Caine, Lenore Dunnah

16/9/19 10:46 pm

Contents Introduction How to use this book

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30

vi 1.31 viii 1.32

Number theory and computation 2 History of numbers 4 Classification of numbers 7 Ordering a set of real numbers 10 Place value of whole numbers 11 Revision: Addition, subtraction, multiplication and division 13 Defined arithmetic operations 19 Associativity, commutativity and distributivity 19 Identity for addition and multiplication 21 Additive and multiplicative inverses 21 Multiplication and division involving zero 22 Order of arithmetic operations (BODMAS) 23 Powers of numbers 24 Reciprocal of a number 26 Factors and multiples 27 Fractions: revision 30 Operations with fractions 35 Decimal fractions: revision 46 Operations with decimals 50 Approximating numbers 52 Standard form (scientific notation) 54 Mixed operations with decimals 55 Percentages 57 Conversion between units of measure 60 Ratio 63 Direct and indirect proportion 65 Rates 66 Arithmetic mean (average) 67 Sequences 69 Number bases 71 Operations with binary numbers 72

CSECMaths_Chapter 00_FM.indd 3

Base 4 and base 8 The use of calculators in number theory and computation

74

End of chapter summary Examination-type questions for Chapter 1

80

78

85

2

Consumer arithmetic

86

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Discount, sales tax, profit and loss Simple and compound interest Hire purchase and mortgages Rates and taxes Utilities payments Currency and foreign exchange Salaries and wages Income tax Insurance

88 93 98 102 104 112 116 119 120

End of chapter summary Examination-type questions for Chapter 2

122 124

3

Algebra

3.1

Directed numbers and arithmetic operations on them 128 The meaning and importance of algebra 134 Algebraic notation 136 Laws of indices 137 Using the four basic operations to simplify algebraic expressions 139 Algebraic fractions 142 Substitution 144 Binary operations 145 Using the distributive law to expand algebraic expressions 146 Simple factorisation of an algebraic expression 147 Simple linear equations in one unknown 148 Inequalities 152 Transposition – changing the subject of a formula 154

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

126

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3.14 3.15 3.16 3.17 3.18 3.19 3.20

Simultaneous linear equations in two unknowns Factorising by grouping Factorising quadratic expressions Solving quadratic equations Simultaneous equations: one quadratic and one linear Identities and equations – what is the difference?

Direct and inverse variation End of chapter summary Examination-type questions for Chapter 3

166 167 168

6

Sets

6.1

Definition of a set and related concepts Intersection, union and complements of not more than 3 sets Venn diagrams Using set theory to solve problems in number theory, algebra and geometry

6.2 6.3 6.4

172 176

4

Geometry and 7 trigonometry 178

4.1 4.2

Geometrical concepts Drawing and measuring line segments and angles Constructing lines, angles and polygons Symmetry in plane figures Properties of angles, lines and polygons Three-dimensional shapes Circles Transformation geometry Trigonometry Spatial geometry: projections and scale drawing Trigonometry in three dimensions

4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

End of chapter summary Examination-type questions for Chapter 4

iv

155 158 159 162

180 182 187 195 199 212 217 224 233 243 246 250 254

5

Measurement

256

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14

Quantities and units Length Mass More units of length and mass Time Weight Speed Density Perimeter Area Margin of error in measurement Scale drawing Volume Surface areas of solids

258 258 262 263 265 269 269 271 272 276 281 283 286 294

End of chapter summary Examination-type questions for Chapter 5

299

7.1 7.2 7.3 7.4 7.5

304 306 313 316 322

End of chapter summary Examination-type questions for Chapter 6

328

Relations, functions and graphs

330

Relations and functions Coordinate geometry and graphs of linear functions Linear inequalities Quadratic functions Graphs of other functions End of chapter summary Examination-type questions for Chapter 7

326

332 340 350 356 366 371 374

8

Statistics

8.1

Sample statistics and population parameters 378 Classifying and representing data 379 Statistical diagrams 383 Measures of central tendency 394 Probability 406

8.2 8.3 8.4 8.5

376

End of chapter summary Examination-type questions for Chapter 8

416

9

Vectors, matrices and transformations

418

9.1 9.2 9.3 9.4

Concepts associated with vectors Position vectors Matrices Vectors, transformations and matrices End of chapter summary Examination-type questions for Chapter 9

413

420 427 432 443 452 457

301

Contents

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10

The School-Based Assessment in Mathematics Introduction Managing the project Devising a project topic Collecting data Problem-solving strategies Project report Project mark scheme Sample Mathematics SBA Examples of SBA topics

11

Examination Practice Examination 1 Examination 2 Examination 3

12

Using your calculator Using the calculator in the mathematics classroom

13 458

Teaching notes

513

Exercises 515 Problems: examples, strategies and solutions 516 Math investigations – examples 518 The use of calculators 522 In support of the use of calculators 522 Criticisms of the use of the calculator 523 Manipulatives and teaching aids 523

458 459 460 462 462 464 466 466 469

Answers 525

470

Index 567

470 482 494

Acknowledgements 577

506 506

Contents

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2 CHAPTER

Consumer arithmetic

Objectives By the end of this chapter, you should be able to: • calculate discount, sales tax, profit and loss • calculate percentage profit and percentage loss • express profit, loss, discount, mark up and purchase tax as a percentage of a basic value • solve problems involving marked price, selling price, cost price, profit, loss or discount • solve problems involving payments by instalments, such as hire purchase and mortgages • solve problems involving simple interest, compound interest (use of ­formulae), appreciation and depreciation, measures and money (currency conversion), rates and taxes, utility bills, invoices and shopping bills, salaries and wages, insurance and investments.

You should already know: • the basics of number theory and computation.

Introduction In this chapter you will learn how to perform calculations relating to normal, everyday business transactions, including calculations associated with personal budgets, personal shopping, payment of bills and salaries.

The chapter will also help you to develop an appreciation for speed and accuracy in everyday financial transactions; it addresses some advantages and disadvantages of particular investments and models and solutions to real-world problems.

Additionally, the chapter clarifies a number of financial calculations that many people find challenging, such as simple and compound interest, taxes, hire purchase, mortgages and utilities payments.

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Examples of consumer arithmetic in real life

Black Friday deals offer massive discounts on various items.

Some shops offer discount cards to regular customers.

Prices of groceries may be displayed electronically.

Profit sharing means that employees have a share in profits.

In duty-free shops the prices of goods are dramatically reduced due to the removal of duties.

The bar graph shows a steady percentage increase in workers’ salaries.

Money in the form of notes and coins: an important ingredient in the financial process.

Financial success – your mortgage loan is approved!

A mortgage is an important financial undertaking.

Cash may be deposited or withdrawn at a bank.

Money may be changed from one currency to another.

Money may be transferred electronically, in online purchasing.

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2.1 Discount, sales tax, profit and loss 2.1.1 Discount Shops usually display their goods showing the price they want you to pay. This is the marked price. However, they often offer a reduction in the price, or a discount, so you may pay less than the marked price.

Discount offered on a TV set

Calculation of discount You can calculate the discounted price of an item by subtracting the amount of the discount from the marked price: discounted price = marked price – discount The discount is often expressed as a percentage of the marked price. To calculate the percentage, divide the discount by the original (marked) price and multiply by 100: percentage discount =

Key term Discount is the amount, expressed as a quantity ($) or percentage (%), that the seller takes off the marked price so that the customer pays less for the item.

discount ($) × marked price ($)

When you know the percentage discount, you can determine the discounted price by first calculating the discount and then subtracting that amount from the marked price. discounted price = marked price – discount percentage of marked price Alternatively:

Example 1

discounted price =

(100 _ percentage discount) × marked price 100

A student was given a discount of $35.00 on a book priced at $80.00. How much did she pay for the book?

Discounted price = marked price – discount = $80 – $35

Example 2

= $45.00

88

A shirt priced at $60.00 was sold at $45.00. What was the percentage discount?

60 – 45 15 × 100 = × 100 60 60 = 25%

2 Consumer arithmetic

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Example 3

During a Christmas sale a television marked at $2000.00 was offered with a 20% discount. What was the final price of the television?

20% of 2000 =

20 × 2000 100

= $400.00 ⇒ final cost = 2000 – 400 = $1600.00 Alternatively, since a 20% discount is offered, the price of the television becomes 100% – 20% = 80% of its original value.

80% of $2000.00 =

80 × 2000 100

Example 4

= $1600.00

Tom bought his cell phone at a discounted price of $2000.00. If the discount was 10%, what was the original or marked price?

Using the formula: discounted price = marked price – discount percentage of marked price 2000 = marked price – 10% of marked price = (1 – 0.1) marked price = 0.9 marked price Marked price =

2000 0.9

= $2222.22

Exercise 2A 1

A radio marked for $200.00 is sold at a 5% discount. What is the discounted price?

2

A barber offers his customers a discount of 5¢ in the dollar. How much does a trim priced at $15.00 cost?

3

A guitar originally priced at $240.00 was sold for a discounted price of $200.00.

What is the percentage discount?

4

A shopkeeper sells envelopes at $2.00 each. He offers a discount of 5% on each envelope. Calculate the discounted cost of 45 envelopes.

2.1 Discount, sales tax, profit and los

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5

A boy buys a book at a discounted price of $80.00. If the discount given is 8%, what is the marked price?

6

A store owner gives a 5% discount on all his bicycles. For a bicycle priced at $240.00 (without discount) he decides to give an additional discount of 3%. What is the final price of the bicycle?

2.1.2 Sales tax

Example 5

On almost everything you buy, the government charges a sales tax to help them raise money to pay for the services they provide for the country.

Key term Sales tax is the amount of money that the government adds to the basic price of goods or services; it is normally calculated as a percentage of the basic price.

A boy buys a bicycle priced at $300.00 and sales tax of 10% is added. How much does he pay for the bicycle?

10% of 300 =

10 × $300 100

= $30.00 So the amount to be paid = $300 + $30 = $330

Exercise 2B

90

1

A girl buys a dress originally priced at $60.00, with an additional sales tax of 7%. What is the final price of the dress?

2

A refrigerator originally priced at $2000.00 is sold with an additional sales tax of 14%.

What is the final price of the refrigerator?

3

The cost of a KFC kid’s meal increases from $10.00 to $15.00 with sales tax. What is the percentage sales tax?

4

The cost of a shirt, after a 4% sales tax is added, is $50.00. What was the original price of the shirt?

2 Consumer arithmetic

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2.1.3 Profit and loss (per cent) Whenever goods and services are bought and sold, there is likely to be a difference between the cost price (CP) and the selling price (SP). Then the ­person offering the goods or services will make a profit or a loss.

Profit (P) is selling price – cost price (P = SP – CP)

Cost price is the amount paid to purchase the goods and services.

Selling price is the amount the goods and services are sold for.

When the selling price is more than the cost price the dealer makes a profit.

Example 6

When the selling price is less than the cost price the dealer makes a loss.

A man bought a used vehicle for $18 000.00 and sold it for $20 000.00. What was the profit?

Example 7

Loss (L) = cost price – selling price (L = CP – SP)

Key terms

A student pays $120.00 for a textbook and sells it to another student for $80.00. Did he make a profit or loss and what was the amount?

Profit = SP – CP = 20 000 – 18 000 = $2000

CP = $120.00; SP = $80.00 ⇒ CP > SP ⇒ loss The loss is 120 – 80 = $40.

2.1.4 Percentage profit and loss Example 8

Profit and loss may be expressed as percentages of the cost price.

A student bought a laptop for $1200.00 and sold it for $1400.00. What was his percentage profit?

1400 – 1200 Percentage profit = × 100% 1200 =

200 × 100% 1200

Example 9

= 16.7%

Key terms Percentage profit =

Percentage loss =

profit × 100% cost price

loss × 100% cost price

Allana bought a cell phone for $600.00 and sold it to Mary for $580.00. Determine her percentage loss.

Percentage loss =

cost price – selling price × 100% cost price

=

600 – 580 × 100% 600

=

20 × 100% 600

= 3.3% 2.1 Discount, sales tax, profit and los

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Exercise 2C 1

A boy sells his toy train to his friend for $5.60. Given that he paid $6.25 for the toy train how much was his profit or loss?

2

One bottle of soft drink costs the shopkeeper $2.00 to buy. The shopkeeper sold a case of 24 bottles of soft drink at $1.50 each. What was the loss or profit he incurred?

3

A shopkeeper buys 80 exercise books for $200.00 and sells them to schoolchildren at $2.00 each. Does he make a profit or loss and how much is it?

4

A boy bought a cricket bat for $80.00 and sold it for $60.00. Calculate the percentage loss.

5

James bought a bus for $80 000 and sold it for $90 000. Determine the percentage profit.

6

Andrea purchased a pen for $8.95 and then sold it to her friend for $7.90. Calculate the percentage loss.

7

A woman made a profit of 5% when she sold a household item for $25.50. What was the cost price of the item?

8

A householder bought a new washing machine for $4850.00 and sold it to a neighbour, to make a profit of 7%. What was her selling price?

2.1.5 Percentage change

Example 10

Generally, when any quantity increases or decreases, you can calculate the percentage change in the ­numerical value in a similar way as you find percentage profit and loss.

The number of boys in class 2J increased from 8 to 10. What is the percentage increase in the number of boys?

Percentage increase = =

10 – 8 × 100 8 2 × 100 8

Example 11

= 25% When a woman reduced the number of hours she worked, the amount she was paid was reduced from $2800.00 to $2400.00 per month. Calculate the percentage decrease in the amount she was paid.

Percentage decrease = =

2800 – 2400 × 100 2800 400 × 100 2800

= 14.3%

92

2 Consumer arithmetic

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Take cost price as 100%.

A price that leads to a loss will be less than 100% of the CP.

Tips

A price that leads to a profit will be more than 100% of the CP. When calculating percentage profit or loss, cost price is always the denominator.

Exercise 2D 1

John’s overall performance as a cricketer was enhanced when his tally of wickets moved from 16 to 20. What was his percentage increase in wickets?

2

The students’ performance in mathematics increased from 24 to 30 passes. What was the percentage increase in passes?

3

The number of accidents in a country made a big jump from 12 to 20 during the course of one year. What was the percentage increase?

4

According to one statistician, the number of people using social media over the last 9 years increased by 95% to 9.5 billion. What was the original number?

5

The amount a man weighed decreased by 0.5% after 3 days of exercise. He now weighs 220 lbs. What did he originally weigh?

6

After receiving a 7.2% increase in pay, an electrician was paid $2045.00 per month. What would his pay be if it had been increased by 11.3%? Calculate the difference between the two increases.

2.2 Simple and compound interest When someone borrows money from a bank, they are expected to pay the bank a charge, for borrowing the money. Likewise, when you put money into a savings ­account, the bank generally pays a fee because they can use the money you deposit to finance other transactions. This charge, or fee, is called interest and is paid at regular intervals, based on a percentage set by the bank.

Key term

Key terms

Interest is the profit earned from investing or saving a sum of money in a bank or company, or the amount paid for borrowing a sum of money from a financial institution or lender.

The amount borrowed or deposited is the ­principal, the percentage rate is the rate at which interest is calculated and the total of the principal and interest, at the end of the period of the loan or investment, is the amount.

The money invested or borrowed is the principal. The percentage earned on the investment, or paid on the loan, is the percentage rate and is usually calculated annually. The interest earned (or paid) is calculated as this percentage of the principal. The total obtained by adding the interest to the principal is called the amount.

2.2 Simple and compound interest

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