Topic 1: Integers€¦ · Topic 1: Integers Introduction An integer is a number that includes positive whole numbers, zero and negative whole numbers. They are numbers that have not

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TEP023 Foundation Mathematics

Topic 1: Integers

Topic 1

Integers

Duration

2 weeks

Content Outline

PART 1 Introduction

Place Value

Addition

Subtraction

Multiplication

Division

Long Division

Rounding

Estimating

Powers

Roots

Order of Operations

Application Problems

PART 2 Multiples and Factors

Negative Integers

Adding and Subtracting Integers

Adding and Subtracting Large Numbers

Multiplying and Dividing Integers

Problem Solving

Topic 2

Decimals

Topic 3

Fractions

Topic 4

Ratios

Topic 5

Percentages

Topic 6

Algebra

Topic 7

Equations and Formulae

Topic 8

Measurement

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Study Guide Topic 1 Integers

Topic 1: Integers Introduction

An integer is a number that includes positive whole numbers, zero and negative whole numbers. They are numbers that have not been broken up into fractions or parts. This section will review the ways in which we use integers and is an opportunity to remember some of the mathematics you learnt at school and learn a few new things as well.

Integers can be positive (1, 2, 3, 4, 5...), negative (-1, -2, -3, … ) and zero (0).

Place Value

Our numbering system uses ten as a base for grouping.

millions hundreds of

thousands

tens of

thousands

thousands hundreds tens units

The number 9 876 543 could be written in expanded form:

9 000 000 + 800 000 + 70 000 + 6000 + 500 + 40 + 3

In words it would be nine million, eight hundred and seventy six thousand, five hundred and forty three.

Example 1: What is the place value of the 4 in the following numbers? a) 9 459 4 has a place value of hundreds

b) 14 000 4 has a place value of thousands

9 876 543

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Addition

When adding numbers:

line them up so that all the units are under each other, all the tens, all the hundreds and so on.

begin by adding the units column on the right.

carry when you need to.

Example 1: Calculate 5691 + 934

Step 1: Line up the numbers with the units in a column, the tens in a column and so on.

5 6 9 1

+ 9 3 4

Step 2: Add up each column starting with the units on the right hand side.

4 + 1 = 5

Step 3: Put the right hand number (5) under the units.

5 6 9 1

+ 9 3 45

Step 4: Add up the numbers in the next column (tens).

9 + 3 = 12

Step 5: Put the right hand number (2) under the tens and carry the rest (the 1) into the hundreds column.

1

5 6 9 1+ 9 3 4

2 5

Step 6: Add up the numbers in the next column (hundreds) plus the carried number.

6 + 9 + 1 = 16

Put the right hand number (6) under the hundreds and carry the rest (the 1) into the thousands column.

1 1

5 6 9 1+ 9 3 4

6 2 5

Step 7: Add up the numbers in the next column, add on the carried number.

5 + 1 = 6.

Put the right hand number (6) under the thousands, there is no carry on.

1 1

5 6 9 1+ 9 3 4

6 6 2 5

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Subtraction

When subtracting numbers:

line them up so that all the units are under each other, all the tens, all the hundreds and so on.

begin by subtracting the unit on the right.

borrow when you are subtracting a larger number from a smaller number.

Example 1: Calculate 5069 934


5 0 6 9

9 3 4

Step 2: Subtract each column starting with the units on the right hand side.

9 4 = 5

Step 3: Put the number (5) under the units.

5 0 6 9 9 3 4

5

Step 4: Subtract the numbers in the next column (tens)

6 3 = 3

Step 5: Put the number (3) under the tens.

5 0 6 9 9 3 4

3 5

Step 6: Subtract the numbers in the next column (hundreds).

0 – 9 = ?? we can’t do this.

You need to borrow from the next column.

Step 7: Borrowing from the 5 in the thousands column, drops it down by 1 to 4 and the 0 in the hundreds column is increased from 0 to 10.

So, the 5 becomes 4 and the 0 becomes 10.

10 – 9 = 1, put the number (1) in the hundreds column.

4 5 10 6 9 9 3 4

1 3 5

Step 8: Subtract the numbers in the next column.

4 – 0 = 4

4 5 10 6 9 9 3 4 41 3 5

In subtraction, ‘borrow’ means to take one from the one higher

place value and add to the next lower.

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Want more practice? Go to: www.mathsisfun.com/numbers/subtraction-regrouping.html


3 0 0 0 6 2

Step 2: Subtract each column starting with the units on the right hand side.

0 2 = ??

Step 3: You need borrow from the tens BUT the tens column has nothing in it. So in this question you will need to borrow from the thousands column.

Step 4: The 3 thousand becomes 2 thousand and then you get 10 hundreds.

2

3 10 0 0 6 2

Step 5: Now you need to borrow from the hundreds column so the 10 hundreds becomes 9 hundreds and we get 10 tens… finally we can borrow from the tens column.

2 9

3 1 0 10 0 6 2

You can do steps 2 to 5 in one go

2 9 9

3 1 0 0 10 6 2

Step 6: Subtract the units

10 – 2 = 8

Subtract the tens

9 – 6 = 3

Subtract the hundreds

9 – 0 = 9

Subtract the thousands.

2 9 9

3 1 0 0 10 6 2 2 9 3 8

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Multiplication

Multiplication is a shortcut for adding the same number repeatedly.

For example, one book costs $10 so 6 books could be worked by saying 10 + 10 + 10 + 10 + 10 + 10. This is the same as 6 groups of 10 or 6 10.

Number 2 3 4 5 6 7 8 9 10

2 4

3 6 9

4 8 12 16

5 10 15 20 25

6 12 18 24 30 36

7 14 21 28 35 42 49

8 16 24 32 40 48 56 64

9 18 27 36 45 54 63 72 81

10 20 30 40 50 60 70 80 90 100

Multiplying by 10, 100, 1000…

When multiplying by 10, 100, 1000… we can just add zeros.

Example 1: Find the following:

a) 234 1000

= 234 000

b) 23 200

= 23 2 100

= 46 100

= 4600

To multiply well and to be able to use this ability in solving problems you must know your 10 times table.

Below are multiplication tables from 2 to 10 Memorise them!!

Three zeros in the answer since we are multiplying by 1000

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Multiplication can be written in a number of ways. If we wish to multiply 6 by 127, we can write:

6 127 or 6•127 or 6 (127)

Multiplication is indicated by these signs: and •

When you need to multiply by large numbers, for example, 127 movie tickets that cost $6 each, it is important to use the method shown below. You need to know how to do this without a calculator, so have a go. You can use a calculator to check your answers. (Later in this topic we will look at estimating your answer before you start).

Example 2: 127 6

Step 1: Write the question with the numbers as shown. 1 2 7 6

Step 2: Multiply the unit numbers from both lines.

6 7 = 42

Put down the 2 and carry the four.

4

1 2 7 6

2

Step 3: Now multiply the bottom unit by the next top number.

6 2 = 12

Add on the number you carried 12 + 4 = 16.

Put down the 6 and carry the 1.

1 4

1 2 7 6

6 2

Step 4: Multiply the bottom unit by the next top number.

6 1 = 6

Add on the number you carried 6 + 1 = 7

1 4

1 2 7 6

7 6 2

In your assignments, when you are asked to show your solutions, you do not have to rewrite or outline each step

as shown above. Your lecturer only needs to see the completed solution but make sure you include all your working

out (ie. borrowing, carrying).

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Example 3: 127 36

Step 1: Do steps 1 to 4 as shown in example 2. Initially, ignore the 3 (the tens) in the 36.

You have finished the first part of the problem.

1 2 7

x 3 6

7 6 2

Step 2: Now you need to multiply the top number by the tens in the bottom number so you ignore the 6 (units).

Since you are multiplying by the number in the tens column, first you need to put down a 0.

1 2 7

x 3 6

7 6 2

0

Step 3: Multiply the top number with the tens (the 3) in the bottom number starting with the units in the top number.

3 7 = 21

Put down the 1 and carry the 2.

2

1 2 7

X 3 6

7 6 2

1 0

Step 4: Multiply the number on the bottom line in the tens position (the 3) by the ten (the 2) top number.

3 x 2 = 6

Add on the number you carried.

6 + 2 = 8

Put down the 8.

Step 5: Multiply the bottom number (3) with the hundreds (1) top number.

3 x 1 = 3

2

1 2 7

X 3 6

7 6 2

+3 8 1 0

Step 6: Now everything on the top has been multiplied by everything on the bottom. The 762 shows the units multiply the top line and 1270 shows the tens multiply everything on the top line. To get the final answer we must add the two sets of answers together.

2

1 2 7

X 3 6

7 6 2

+ 3 8 1 0

4 5 7 2

Ignore the 3 and work out 1276 first

YOU MUST PUT IN A ZERO HERE

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Division

Division is the opposite of multiplication. Division is the sharing out of things equally. For example sharing out 24 blocks of chocolate between 8 people or working out how many 8 seater buses are needed to move 152 people. To work out division, knowing your tables is essential!

Division is the process of finding out how many times one number can be subtracted from another. If we wish to divide 60 by 4, we may write this in a number of ways:

60 ÷ 4 = 15,

60

4= 15, 60/4 = 15,

154 60

Division is indicated by these three signs: ÷,

, /,

Terminology:

60 ÷ 4 = 15

Example 1: Since 3 8 = 24 then 24 8 = 3

If you are not sure you could draw eight circles and count out 24 lines – one into each circle at a time and then count how many are in each circle.

To divide larger numbers, it is easier to take it one number at a time.

Not all numbers divide exactly into another number without a remainder. We will look at what to do with remainders in Topic 2: Decimals.

Dividing by 10, 100, 1000… This will be covered in Topic 2.

dividend

divisor

quotient

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Step 1: Set out the problem like this: 2 840

Step 2: Take it one number at a time. Start with the first number after the ' ' and see how many times 2 goes

into 8.

8 2 = 4 Write the 4 above the 8.

4

2 840

Step 3: Now the next number. See how many times 2 goes into 4.

4 2 = 2

Write the 2 above the 4.

42

2 840

Step 4: Finally use the last number, the '0'.

0 4 = 0. Write the 0 above the 0.

So 840 2 420

420

2 840


Step 1: Set out the problem.

Step 2: Start with the first number after the ' '. See how

many times 3 goes into 1. 3 does not go into 1.

Step 3: Now take the next number also (2), with the carry over 1 makes 12.

12 3 = 4

Write this above the 12.

4

3 1254

Step 4: The next number is ‘5’.

5 3 = 1 and 2 left over (remainder).


Carry the remainder 2 over.

2

4 1

3 125 4

Step 5: The last number is 24 (with the carry over).

24 3 = 8


1254 3 = 418

2

4 1 8

3 125 4

3 1254

Step 5 shows your complete working out

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Long Division

Long division is used when the divisor is a large number (larger than 10). Long division breaks the calculation up into pieces.

Example 1: 23828

Step 1: 23828 is written as: 23 828

Step 2: 23 does not go into 8 so we include the next number which gives 82

8223 goes 3 times with a remainder

The 3 is placed at the top.

382823

Step 3: Calculate the exact value of 233

233 = 69

Write this number below the 82

3

23 828

69

Step 4: Subtract the bottom number from the top number

82 – 69 =13

(this is your ‘remainder’)

3

23 828

69

13

Step 5: Bring down the next number from the dividend.

3

23 828

69

138

Step 6: 13823 goes 6 times with no remainder.

Answer : 23828 =36

36

23 828

69

138

138

0

If you want more examples of long division go to: www.mathsisfun.com/long_division.html

You may need to think a bit here to get the correct number

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Rounding

Rounding takes place for different reasons.

Numbers are rounded so that important information is not hidden behind unnecessary detail.

For example: 12 co-workers win Lotto and share $11 970 183 prize equally amongst themselves. How much does each person win?

Using your calculator it would show that they each win $997 515.25.

Do you think the winners would say, ‘I have just won $997 515.25!!’?

It would be more likely for them to say, ‘I have just won a million dollars!!’

The winners were rounding the size of their win in order to get across the important idea, which in this case is the magnitude of the prize.

Numbers are rounded in order to fit many requirements, for example currency.

For example: My supermarket bill comes to $94.92. How will I pay for this when our currency does not include 1 and 2c coins any more?

Prices are now rounded to the nearest 5 cents.

Numbers are rounded in order to quickly find estimates.

For example: I have $30. Will I have enough money to fill my 20 L jerry can with fuel if it costs 152.2 c/L?

In this case it is easy to think of the price of fuel as approximately $1.50 per litre. That means filling up the jerry can will cost a little more than:

1.5020 = $30

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Rules on Rounding

RULE 1

If the first digit dropped is less than 5, the last digit kept is left as it is

Example 1:

30

31

32

33

34

Rounded to the nearest 10 is 30

RULE 2

If the first digit dropped is 5 or greater than 5, the last digit kept is increased by 1 or rounded up.

Example 2:

35

36

37

38

39

Rounded to the nearest 10 is 40

In rounding off it is important to remember that the number keeps its place value. For example, 52 000 rounded off to the nearest ten thousand is 50 000 not 5 or 50 or 500.

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Example 3: Here are some more examples.

8 750 000 rounded to the nearest million is 9 000 000

28.467 rounded to the nearest unit is 28

153 rounded to the nearest 10 becomes 150

1509 rounded to the nearest 100 becomes 1500

362 rounded to the nearest 10 is 360




Estimating

Before performing a calculation with large numbers, a rough answer should be estimated. This practice can prevent many errors.

Estimating is the practice of making a ‘reasonable guess’ of an answer. Estimates are easiest to do when you use rounded numbers.

In estimation you want to round off the numbers in the question so that you can EASILY work out an approximate answer without having to do any long calculations. You answer may not be the same as the person next to you. Your level of accuracy will depend on your ability to work questions out in your head.

Also, when you are using a calculator, it is important to understand what information to give the calculator and to be able to trust the answer the calculator gives you. To be sure the answer is correct; you can estimate what the answer should be. If the answer is close you can probably trust it. If the answer is very different, you need to check your method and the buttons you pressed on the calculator.

means ‘approximately equal to’

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Example 1: Estimate the answer to:

5590 + 287 + 23 + 7398

Step 1: Round off each number, in this case we will round to the nearest hundred:

5590 6000 287 300 23 0 7398 7000

Step 2: Add the rounded numbers:

5590 + 287 + 23 + 7398

6000 + 300 + 7000 = 13 300

Example 2: Estimate 39 8

398

40 10

=400

Example 3: Estimate 1217 92

1217 92

1200 100

=12

Example 4: Estimate 37

4072

37

4072

40

4070

=40

110

3 (since 12040 = 3)

.

Estimation Practice:

You can go online and have a go at some estimation games: www.mathsisfun.com/numbers/estimation-game.php.

A horizontal line means division. The top of the line must be worked out, then the bottom and lastly the

division:

(72 + 40) 37

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Powers

Powers, also called indices or exponents make it easier to show repeated multiplication of the same number. For example: 24 is easier to write than 2 × 2 × 2 × 2.

The power of a number says how many times to use the number in a multiplication.

In this example:

82 = 8 × 8 = 64

Some more examples are below.

Expanded Index Notation Spoken as Result

4 4 42 4 to the power of 2

or 4 squared

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2 2 2 23 2 to the power of 3

or 2 cubed

8

3333 34 3 to the power of 4 81

In general:

...na a a a

What if the Exponent is 1, or 0?

If the exponent is 1, then you just have the number itself (example 91 = 9)

If the exponent is 0, then you get 1 (example 09 = 1)

an : multiply a by itself, so there are n of those a's

the 8 is called the base

the 2 is the index, power or exponent

n

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Roots

The opposite process of raising a number to a power is to find the root.

Square Root

The square root of a number is the value that when multiplied by itself (squared) gives the original number.

is the mathematical symbol for square root

For example: 3 squared is 9, so the square root of 9 is 3 because when 3 is multiplied by itself you get 9.

32 = 9 and 9 3

3 Square

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Square Root

The best way to understand it is to look at a few more examples of squares and square roots:

Example 1: What is the square root of 25?

We should know that 25 = 5 × 5, if you square 5 (5 × 5) you will get 25.

25 5 5 5

So the answer is 5.

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Cube Root

The cube root of a number is the value that when cubed gives the original number.

3 is the mathematical symbol for cube root

23 = 8 and 3 8 2

2 Cube

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Cube Root

Example 2: What is the cube root of 27?

We need to know that 27 = 3 × 3 × 3, if you cube 3 you will get 27.

3 327 3 3 3 3

Generalising - The nth Root

...n n na a a a

n of them

... the nth root is the number that is used n times in a multiplication to get the original value

This is the general way of talking about roots.

You would say "the cube root of 27 equals 3"

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Some more examples are below.

Using your calculator

You would not be expected to do the last two examples in your head. You would do them with a calculator. Look at your instructions book to work out how your calculator operates, then test out your idea with one you know the answer to.

Example 1: Use your calculator to check that 2 16 4 .

On most calculators you will press: 16 =

If your answer is not 4, try: 16 =

Example 2: Use your calculator to check that 3 8 2 .

Numbers under a square root sign or a power should be treated as if they have ‘hidden’ brackets around them. For example,

6 2 5 (6 2 5)

Equation Working Think Result

36 6 6 6 is the square root of 36

because 62 = 36 6

3 64 3 4 4 4 The cubed root of 64 is 3

because 43 = 64 4

4 16 4 2 2 2 2 The 4th root of 16 is 2

because 24 = 16 2

6 46656 6 666666 The 6th root of 46656 is 6

because 66 = 46656 6

5 100000 5 1010101010 The 5th root of 100000 is 10

because 105 = 100 000 10

On most calculators you will press: 3 a b 8 =

Remember: when there is no number in front of the root sign it means square

root.

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Order of Operations

When working with expressions that have more than one operation (+, , -, , powers, roots), it is important to do the calculations in the correct order.

Example 1: How much does a person get paid if they work for 8 hours at $10 per hour and 2 hours at $15 per hour?

This can be shown mathematically as:

810 + 215

Lets look at working this out two different ways:

If we work it out logically;

The person gets paid for 8 hours at $10 per hour: 810 = $80

and then $15 for 2 hours: 152 = $30.

the total earnings come to: $80 + $30 = $110.

Method 1: Working from left to right

810 + 215

= 80 + 2 15

= 8215

= 1230

The person is paid $1230. WOW!!

Method 2: Using logic

8 10 + 2 15

= 80 + 30

= 110

Here the person is paid $110… which is the correct amount.

Clearly we have to work out some rules for working out problems. This is very important when working with calculators as a calculator may have given you the first answer – the one that is wrong.

A set of rules has been worked out to make sure problems are worked out in the correct order and to make sure you get the right answer. It goes like this:

Step 1: Evaluate any expressions in brackets first

Step 2: Evaluate any exponents (powers or roots)

Step 3: Evaluate any division or multiplication working from left to right

Step 4: Evaluate any addition or subtraction working from left to right

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A way to remember this is: BEDMAS

B E D M A S

BRACKETS EXPONENTS DIVIDE MULTIPLY ADD SUBTRACT

The key is working one step at a time and carefully setting out your work. Don’t take shortcuts and don’t do too many steps in one go. Here are a few examples:

Example 2: Calculate (15 + 5) – 32 4 2

Steps to follow Working out

Step 1: Using BEDMAS

Brackets are first

15 + 5 = 20

Step 2: Exponents 32 is next

32 = 3 3 = 9

Step 3: Multiplication and Division …working from left to right

9 4 = 36

36 2 = 18

Step 4: Addition and Subtraction

20 – 18 = 2

(15 + 5) – 32 4 2

= (15 + 5) – 32 4 2

= 20 – 32 4 2

= 20 – 9 4 2

= 20 – 36 2

= 20 – 18

= 2

Remember:

When multiplication and division are being calculated, work from left to right

When addition and subtraction are being calculated, work from left to right

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Example 3: 3[4 + (32) – 5] – 6 + 15 5

this means the same thing as 3 (4 + ….

Steps to follow Working out

Step 1: Brackets, doing innermost brackets first.

[4 + (32) – 5]

6

Omit the inner brackets: [4 + 6 – 5]

Inside the bracket, follow the same rules. Addition and subtraction… working from left to right

[4 + 6 – 5] = 5

Step 2: Exponents :There are no exponents

Step 3: Multiplication and Division …working from left to right

3(5) = 3 5 = 15

15 5 = 3

Step 4: Addition and Subtraction is last … working from left to right

15 – 6 + 3

= 9 + 3

= 12

3[4 + (32) – 5] – 6 + 15 5

= 3[4 + 6 – 5] - 6 + 15 5

= 3(5) – 6 + 15 5

= 15 – 6 + 3

= 12

Nested Brackets show several groupings, one within the other.

First simplify the innermost brackets.

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Applications Problems

When applying your maths skills, you need to consider what you need to do to work out the answer. Try to write the problem in a mathematical sense.

Example 1: Four pieces of wood, 30 cm long are cut from a board 200 cm long. What is the length of the remaining piece?

There are a number of ways to approach this question. Here is one:

Step 1: We need to find out the amount of board left over.

Step 2: A diagram may be useful

200cm

30cm 30cm 30cm 30cm

Step 3: The board is 200 cm long take away 4 pieces, each 30 cm long.

200 cm – (4 x 30 cm)

Step 4: Calculate

200 – (4 x 30)

= 200 – 120

= 80

Step 5: Worded questions should have a written answer with units.

There is 80 cm of board left.

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Example 2: You are ordering Vitamin C tablets for the Health Clinic. You have 2 patients who need 2 tablets per day and a child who

needs 1 tablet per day. How long will a bottle of 260 tablets last?

Step 1: How many days will 260 tablets last?

Step 2: I need to share out 260 tablets. Sharing out is divide.

Every day I need 2 lots of 2 tablets and 1 tablet.

That’s 2 x 2 + 1

So my plan is 260 (2 x 2 + 1)

Step 3: You could estimate an answer here: 250 5 = 50.

Step 4: 260 (2 x 2 + 1) …brackets first

= 260 5 … 1

52

5 26 0

= 52

Step 5: The bottle will last 52 days.

Example 3: Dean’s total income for one year for his after school job was $3630 (gross). He paid tax of $562 which has to be deducted from his income. How much did Dean earn per week (net)?

Weekly income = (3630 – 562) 52

= 59

Dean’s weekly income would be $59.

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Multiples and Factors

Multiples

The multiples of a number are what you get when you multiply it by other numbers. It is like the multiplication table. Here are some examples: Multiples of 2 are 2, 4, 6, 8, 10… Multiples of 5 are 5, 10, 15, 20…

Common Multiples

If you have two or more numbers, and you list their multiples, the numbers that appear in both lists are common multiples.

Example 2: Find the first two common multiples of 3 and 4.

Step 1: List the multiples of these numbers.

Step 2: Look for the multiples found in both lists.

Multiples of 3 3, 6, 9, 12, 15, 18, 24 …..

Multiples of 4 4, 8, 12, 16, 20, 24…..

From this example we can see that 12 and 24 are common multiples of 3 and 4.

Least Common Multiples

Zero is a common multiple of every number. Excluding zero, the smallest common multiple is called the least common multiple or LCM. In our previous example the LCM is 12.

Example 3: Find the least common multiple for 4, 6, and 8.

Multiples of 4 8, 12, 16, 20, 24, 28, 32, 36 ...

Multiples of 6 12, 18, 24, 30, 36 ...

Multiples of 8 16, 24, 32, 40 ....

The least or lowest common multiple of 4, 6 and 8 is 24.

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Factors Factors are numbers you multiply together to get another number.

2 x 3 = 6 factor factor

Example 1: List all the factors of 36.

36 = 136 1 and 36 are factors of 36




36 = 66 6 is a factor of 36

Prime Number

A Prime Number is a whole number, greater than 1, that can be evenly divided only by 1 or itself.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17.

Prime Factors

‘Prime Factoring’ involves finding which prime numbers you need to multiply together to get the original number. A factor tree can be used. This method of finding the prime factors involves breaking down numbers into pairs of factors until all the factors are prime numbers.

Example 4: Write 24 as a product of its prime factors.

From this we get that 24 = 32 2 2 3 2 3

24

2 12

3 4

2 2

12 can be written as 34, the 3 is prime Breaking 24 into 212,

the 2 is a prime number.

Breaking 4 into 2 2

Using this, all the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

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Highest Common Factor

It is the largest of the common factors of two or more numbers.

Example 5: Find the Highest Common Factor (HCF) of 24 and 60.

METHOD 1:

find all factors of the given numbers

select the ones that are common to both, and

choose the greatest.

Factors of 24 1, 2, 3, 4, 6, 8, 12, 24

Factors of 60 1, 2, 3, 4 , 5, 6, 10, 12, 15, 20, 30, 60

The common factors of 24 and 60 are 1, 2, 3, 4, 6, and 12.

The HCF is 12.

METHOD 2:

find the prime factors of the given numbers using a factor tree (example 4 on previous page)

combine the common ones together.

24 = 2 2 2 3 and 60 = 2 2 3 5

The prime factors common to both numbers are 2, 2 and 3.

Therefore, the highest common factor is 2 2 3 12

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Negative Integers

So far, all the numbers used in this unit have been positive numbers.

Directed numbers include both positive and negative numbers (less than zero).

You use directed numbers all the time. Some examples are given below:

Example 1: Money

One way of showing negative numbers is by using a NUMBER LINE

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Let’s try this idea out with a few examples.

I get paid $4 for working this week. This means I have $4… so I am at 4 or +4.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

I spend my $4, so I have no money left.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Now it is the end of the week and I need to borrow money for bread and milk. I borrow $2 from my sister, this means I am down $2, so I am at –2.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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Example 2: Temperature

-10 -5 0 5 10 15 20 25 30 35 40 45 50

Freezing Towards boiling

Example 3: Height above sea level

+20m

Above sea level

Below sea level

-20m

The numbers we use all the time, 3, 4, 27 are all positive numbers, we just don’t write the + sign. You don’t need to write the + sign when you get confident. When using a calculator to evaluate these expression

there is a button which looks like this: / or ( ) .

Sea level 0 m

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Adding and Subtracting Integers

When adding or subtracting small directed numbers, you could use a number line:

When adding move to the right.

When subtracting move to the left.

Example 1: The temperature is 2oC and falls by 5oC. What is the temperature then?

Using a number line we cans see:

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

As a number sentence, the question can be written mathematically as:

2 – 5 = -3

Example 2: If the temperature at a ski field is -14o and then it increases by 20o, the result would be a temperature above zero.

-14 + 20 = 6

This answer can be shown on a number line

+20

–14 0 6

Example 3: Your bank account is overdrawn by $10 and you withdraw another $5. What is the balance of your account now?

Again moving around a number line is an easy way to solve this equation.

“Overdrawn by $10” means that you start at -10.

“Withdraw $5” means subtract or take away $5.

-19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7

-10 – 5 = -15

Starting at 2, we take 5 ‘hops' to the left. The answer is -3.

Starting at -10, we take 5 ‘hops' to the left. The answer is -15.

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Adding and Subtracting Large Numbers

As shown previously, when adding or subtracting small directed numbers, you can think of a number line but that becomes impractical with large numbers.

When you have to work with large numbers, you might need to use the rules given below:

Rule 1 for adding and subtracting:

When numbers have the same signs,

Give the result the same sign as the question.

Add the numbers, ignoring the sign.

Example 1: Calculate 21 + 57

Since the signs are the same, you follow rule 1.

Add 21 + 57 = 78

Example 2: Calculate -73 – 21


You will add the numbers ignoring the sign and give the answer the same sign in the question.

-73 21

= - (73 + 21)

= -94

Rule 2 for adding and subtracting:

When numbers have opposite signs,

Subtract the smaller from the larger, ignore the signs.

Give the result the sign of the number with the larger value.

Think of: I lost $73 and then I lost $21, so I lost $94 overall.

…starting with an easy question and using the rule

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Example 3: Calculate 73 – 34

Since the signs are different, you follow rule 2.

Step 1: Subtract the smaller number from the larger.

73 – 34 = 39

Step 2: The final answer will be positive since the larger number is positive.

Example 4: Calculate 85 – 107


Step 1:

Subtract the smaller number from the larger.

107 – 85 = 22

Step 2: The final answer will be negative since the larger number (107) is negative. 85 – 107

= - (107 – 85)

= -22

Example 5: Calculate -41 + 72


Step 1:


72 – 41 = 31

Step 2: The final answer will be positive since the larger number (72) is positive.

-41 + 72

= - (72 – 41)

= -31

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Rule 3: Simplify any double signs.

Subtracting a negative number (two minus or negative signs directly next to each other ‘ ‘ ) change the two minus signs to a plus

+

Subtracting a positive number or adding a negative gives a negative

+

+

Example 6: Find the following:

a) 6 – -2

‘ ’ becomes ‘+’ so,

6 – -2 = 6 + 2

= 8

b) -5 - 3

= -5 + 3

= -2

c) 6 + ( 14) + (+2) ( 10)

= 6 – 14 + 2 +10

= 4

Minus?.... Negative?.... You may be feeling a bit confused about these two words.

What’s the right place to use a ‘minus’?

What’s the right place to use a ‘negative’?

Don’t they mean the same thing?

Yes they do. The same question can be written different ways.

-3 – -4 = 3 – 4 = -3 – (-4)

They all say, ‘negative 3 minus negative 4’.

Want more examples? Go to: www.mathsisfun.com/positive-negative-integers.html

Want to more practice? Go to: www.maths.com/numbers.directed.CombineDoubleSignedNumbers.htm

www.maths.com/numbers.directed.Addition.htm

http://www.maths.com/numbers.directed.CombineDoubleSignedNumbers.htm

36


Putting it all Together

Now we are going to look at solving directed number questions with large numbers. Example 7: Calculate -401 + 722


Step 1:


7 2 2 4 0 1

3 2 1

Step 2: The final answer will be positive since the larger number (722) is positive. -401 + 722

= 722 – 401

= 321

Example 8: Calculate -169 + (-671)

First, using Rule 3, simplify the question:

169 + (671) = 169 – 671


Step 1:

Add the numbers together.

1 6 9 + 6 7 1

8 4 0

Step 2: The final answer will be negative since both numbers are negative.

169 – 671

= (169 + 671)

= 840

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Multiplying and Dividing Integers

Multiplying and dividing positive and negative numbers has special rules:

Want more explanation? Go to: www.mathsisfun.com/multiplying-negatives.html




+ + = + + – = –

–

– = + –

+ = –

Step 1: Choose the rule – + = –

Step 2: Work out the size of the answer 3 7 = 21

Step 3: Put them together in the answer 3 7 = 21

Step 1: Choose the rule – – = +

Step 2: Work out the size of the answer 63 3 = 21


Step 1: Choose the rule + – = –

Step 2: Work out the size of the answer 19 4 1

3 2 8 2 3


If the signs are the same, the answer is

positive

If the signs are different, the answer

is negative

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Problem Solving

Now is the time to use your knowledge of maths to solve some problems. The idea with word problems is to follow these steps to help you successfully solve the question.

1. Read the whole problem carefully and decide what it is asking.

2. Work out a plan:

Use simple numbers

Guess, check, refine

Look for patterns

Draw a diagram

3. Estimate a reasonable answer.

4. Solve the problem using the facts and your plan.

5. Check that your answer satisfies all information given.

6. Written questions need to have a worded answer. Remember to include units in the final answer: kg, $, cm, etc.

Example 1: Jayne swims 60 laps in 3 hours. Each hour she swam 4 less laps than the previous hour. How many laps did she swim in the first hour?

There are different approaches that can be taken to solve this question.

Here we will be using ‘Guess, Check and Refine’.

Step 1: GUESS

Work out an approximate number of laps to start with.

In 3 hours she swims 60 laps so,

60 3 = 20

She swims and average of 20 laps per hour.

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Step 2: CHECK

I am going to use the average as a starting point.

Each hour she swam 4 laps less than the hour before.

Show this on a table:

Hour First guess

1 20

2 16

3 12

TOTAL 48

Step 3: REFINE

The total number of laps swum in 3 hours is 48, but we need 60 so we are 12 laps short. These 12 laps need to be spread out over 3 hours: 12 3 4

We need to increase the number of laps each hour by 4.

Step 4: Answer the question.

In the first hour Jayne swam 24 laps.

Hour First guess Second guess

1 20 24

2 16 20

3 12 16

TOTAL 48 60

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Documents

Topic 1: Integers€¦ · Topic 1: Integers Introduction An integer is a number that includes positive whole numbers, zero and negative whole numbers. They are numbers that have not