User:Adamashraf

School of Arithmetics
All students must know their times table.

$$\frac{e+1}{e-1}-{\frac{2}{e-1}}\left(\sum_{j=1}^{2n-1}(-1)^{j+1}\sqrt[2n]{e^j}\right)=\frac{1}{4n+\frac{1}{12n+\frac{1}{20n+\frac{1}{28n+\cdots}}}}$$

$$\frac{(a-1)\sqrt[2n]{a}}{\sqrt[2n]{a^2}-1}=\sqrt[2n]{a}+\sqrt[2n]{a^3}+\sqrt[2n]{a^5}+\cdots+\sqrt[2n]{a^{2n-1}}$$

Inspired from the work of Ramanujan

Addition with carry
Exercise

Add the following questions

Subtraction with borrowing
Exercise

Subtract the following questions

Partition method
Example

634 - 149

Answer = 485

Show working out

1. 10 - 9 + 4 = 5

2. 10 - 4 + 2 = 8

3. 5 - 1     = 4

Exercise

Subtract the following questions using partition method

Multiplication using 5 and 10
Example

Work out

8 × 6 = 48

Exercise

Work out the following using this method of 5 and 10

Long multiplication
Example 1

Work out 34 × 23 = 782 Show working out

carry → ¹3 4 × 2 3         --           1 0 2         + 6 8           7 8 2

Exercise

Work out

Example 2

Work out 123 × 345 = 42435 1 2 3          × 3 4 5          --             6 1 5           4 9 2        + 3 6 9      --         4 2 4 3 5

Exercise

Work out

Long multiplication with zeros at the end
Example

Work out 2300 × 340 = 782000

Show working out

1) Ignore the zeros. Multiply normally

23 × 34 = 782 2) Count the numbers of zeros, there are only three zeros.

Now just add them zeros to the end of the answer

2300 × 340 = 782 000

Exercise

Work out

Use the fact table to work out the following

Quick multiplication (sum of two last digits must be 10)
Example 1

Work out 23 × 27 = 621 ( 3 + 7 = 10 )

Show working out

1) Increase the first digit by 1 and then multiply by the second number first digit

(2+1) × 2 = 6

2) Now multiply the last two digits together

3 × 7 = 21

3) Therefore 23 × 27 = 621

Exercise

Work out

Example 2

Work out 53 × 27 = 1431 ( 3 + 7 = 10 )

Show working out

1) Increase the first digit by 1 and then multiply by the second number first digit

(5+1) × 2 = 12

2) Now multiply the last two digits together

3 × 7 = 21

3) 1221 This is not the answer! 4) Last Calculating (5-2)     ↑       7×3 → → → →  21     5 3 × 2 7 = + 1221             ---                  1431

Exercise

Work out

Quick method of multiplication
 The cross method 

a b            × c d   - Rules:

1) d × b

2) d × a + c × b

3) c × a

Example

Work out 13 × 21 = 273

Show working out

1 3    × 2 1    ---     2 7 3

1) 1 × 3 = 3

2) 1 × 1 + 2 × 3 = 1 + 6 = 7

3) 2 × 1 = 2

Multiplication using odd sequence
Rules are:

1. First digit, we write down the odd sequence

2. Second digit, we double it

3. Second digit, we multiply by itself

Example 1

Work out 23 × 23 First digit ↑ 2 3    ↓    Second digit

Answer : 529

Show working out

Example 2

Work out 42 × 42

Answer : 1764

Show working out

Example 3

Work out 47 × 47

Answer : 2209

Show working out

Exercise

Work out thr following using the above method

Division for children(Subtraction method)
Example

Work out

6 ÷ 2 = 3 r 0         r stands for remainder Show working out

1. We are going to use subtraction method

6 - 2 = 4                  (1)

4 - 2 = 2                  (2)

2 - 2 = 0 → Remainder      (3)

How many times 2 is taken away from 6 to give us a remainder of zero of less than 2?

answer = 3 times with a remainder of zero.

So 6 ÷ 2 is equal to 3 with remainder 0

Exercise

Work out

Division for children(Diagram method)
Example

7 ÷ 2 = 3 r 1

(x x) 1 pair (x x) 2 pairs (x x) 3 pairs x  → Reaminder of 1

How many pairs of 2 are there in 7?

Answer = 3 pairs and remainder the 1

Exercise

Work out

Long Division
 Quick division with no remainder 

Example

6 ÷ 3 = 2

Exercise

Fill in the table

Quick division with remainder

Example

Work out

6 ÷ 4 = 1 r 2

Exercise

Work out the following

Long division with no remainder

Example Work out

124 ÷ 2 = 62

Show working out 62    2|124       12 ÷ 2 = 6 r 0 04        4 ÷ 2 = 2 r 0 0

Exercise

Work out the following

Long division with remainder

Eaxample

Work out

125 ÷ 2 = 62 r 1

Show working out

62r1 2|125      12 ÷ 2 = 6 r 0 04        5 ÷ 2 = 2 r 1 1

Exercise

Work out the following

Long division with remainder as a decimal (non recuring)

Example

Work out

54 ÷ 4 = 13.5

Show working out

13.5      4|54        5 ÷ 4 = 1 r 1 14      14 ÷ 4 = 3 r 2 2      20 ÷ 4 = 5 r 0 Borrow a zero

Exercise

Work out the following

Long divison with remainder as a decimal (recurring)

Example

Work out

47 ÷ 3 = 15.66...

Show working out

15.66 ...    3|47       4 ÷ 3 = 1 r 1 17     17 ÷ 3 = 5 r 2 20    20 ÷ 3 = 6 r 2 20   20 ÷ 3 = 6 r 2 This part will repeat itself ...

Exercise

Work out the following

Trick division of 9 and 8

Example 1

Work out

123 ÷ 9 = 13 r 6

Show working out

13r6 9|123 1) 1            2) 1+2=3            3) 3+3=6  → This last one is the reaminder

Exercise

Work out the following

Example 2

Work out 3446 ÷ 9 = 382 r 8

11       371 r 8 = 382 r 8 Add them from Right to Left! 9|3446    1) 3                2) 3 + 4 = 7                3) 7 + 4 = 11                4) 11 + 6 = 17 (17 ÷ 9 = 1 r 8)

Exercise

Work out the following

Number facts

Exercise

Work out

Removing zeros

Example 1

Work out

150 ÷ 30 = 5

Show working out

1) Remove the zeros and divide normally.

15 0 ÷ 3 0 = 15 ÷ 3 = 5

Example 2

Work out

45000 ÷ 900 = 50

Show working out

1) Remove the zeros and divide normally.

450 00 ÷ 9 00 = 450 ÷ 9 = 50

Exercise

Work out the following

Multiplication with signs
combination of signs (negative and positive)

Exercise

Work out

Exercise

Work out

Division with signs
combination of signs (negative and positive)

Exercise

Work out

Addition with signs
Rule of addition

If the signs (it could either be a plus or a minus) are the same, just add the numbers directly

Example

+ 2 + 2 = + 4

- 2 - 2 = - 4

Exercise

Add the following

Subtraction with signs
Rule of subtraction

If the signs are different, just subtract, but the biggest number takes the sign!

Example

-2 + 5 = +(5-2) = + 3

-5 + 3 = -(5-3) = - 2

Exercise

Subtract the following

Exercise

Work out the following

Addition and subtraction with signs
grouping method

Example

- 2 - 4 + 4 - 1 + 9 - 1 + 1 = + 6

Show working out

Exercise

Work out the following.

Money
1. Diagrams to numbers

2. Numbers to digrams

Currency of the U.K.

Conversion for money

£1 = 100 pences

Exercise

Express the following using only the above currency

Maths in words
Key words

The use of these key words in maths

Example

1. Find the total of £5, £1 and £3.

Total = £5 + £1 + £3 = £9

2. Find the sum of £5, £1 and £3.

Sum = £5 + £1 + £3 = £9

3. Find the product of 5 and 4

Product = 5 × 4 = 20

4. Find the quotient of 15 and 3

Quotient = 15 ÷ 3 = 5

5. Find the difference of 5 and 7

Difference = 7 - 5 = 2

Exercise

Work out the following

Example

John thinks of a number, then he multiplied it by 4 and added 2 to it and gives an answer of 14.

What number is John thinking of?

Answer:

John number is 3

Show working out

1. We work backward, meaning we start with the given answer

14

2. add 2 → subtract 2 ; here we do the reverse operation on the given answer. 14 - 2 = 12

3. multiply by 4 → divide by 4

12 ÷ 4 = 3

4. Answer is 3, so John is thinking of the number 3!

We always can check the answer to see whether it is correct or not.

John thinks of a number, then he multiply it by 4 and add 2 to it and gives an answer of 14.

3 × 4 + 2 = 14 ; multiplication go before addition 12 + 2 = 14       14 = 14 ;  it is correct! Exercise

Exercise

Exercise

What is the time ?
In telling the time we often use the following phases.

Analogue's clock

The long handle represents the hours ___________    |     12    |     |     ↑     |        |     |     |     |9    . →  3| The small handle represents the minutes |          |     |           |         |_____6_____|           Demonstration of the following phases below

___________    |     12    |     |     ↑     |        |     |     |     |9    . →  3|  Quarter past 12 means 15 minutes past 12 O'clock |          |     |           |         |_____6_____|              Quarter past 12

___________    |     12    |     |     ↑     |        |     |     |     |9    .    3|  Half past 12 means 30 minutes past 12 O'clock |    ↓     |     |           |         |_____6_____|               Half past 12

___________    |     12    |     |     ↑     |        |     |     |     |9 ←  .    3|  Quarter to 1 means 15 minutes to 1 O'clock |          |     |           |         |_____6_____|              Quarter to 1

Digital's clock

Hours ↑     4:14:20 → Seconds ↓        Minutes

Exercise

Show the following digital's time on the analogue's clock

Exercise

Show the following analogue'clock on the digital's clock

Q1. ___________    |     12    |     |           |        |           |     |9    .    3|     |           |     |           |         |_____6_____|

Q2. ___________    |     12    |     |           |        |           |     |9    .    3|     |           |     |           |         |_____6_____|           Q3. ___________    |     12    |     |           |        |           |     |9    .    3|     |           |     |           |         |_____6_____|

Order of operation (OOO)
$$\times{\div}{+}{-}$$

In maths, we perform calculation in the above order, starting from and end with -.

 Two operators 

example

Work out

$$6-3\div3=5$$

Show working out

6 - 3 ÷ 3    1) Division first → 3 ÷ 3 = 1

6 - 1      2) Subtraction → 6 - 1 = 5

5        3) Result

exercise

Fill in the table

Exercise

Three operators
Example

Work out

$$(15-2){\div{6}}+5=7$$

Show working out

(15-2) ÷ 6 + 5   1) Bracket first → 15-2=12   12 ÷ 6 + 5      2) Divsion second → 12 ÷ 6 = 2

2 + 5         3) Addition third

7           4) Result

Exercise

Fill in the table

Exercise

Four operators
Example

Work out

$$24-45{\div}9+3\times2=27$$

Show working out

24 - 45 ÷ 9 + 3 × 2     1) Division first → 45 ÷ 9 = 5

24 - 5 + 3 × 2        2) Multiplication → 3 × 2 = 6

25 - 4 + 6          3) Add or sub → 25 - 4 = 21        21 + 6             4) Add → 21 + 6 = 27 27              5) Result

Exercise

Exercise

Five operators
Example

Work out

$$60\div{10}-5\times(15\div{5})+1=-8$$

Show working out

60 ÷ 10 - 5 × (15 ÷ 5) + 1  1) Bracket first → 15 ÷ 5 = 3

60 ÷ 10 - 5 × 3 + 1     2) Divsion second → 60 ÷ 10 = 6

6 - 5 × 3 + 1        3) Multiplication third → 5 × 3 = 15

6 - 15 + 1         4) Add or sub → 6 - 15 = - 9             - 9 + 1            5) Subtract → -9 + 1 = - 8

- 8             6) Result

Exercise

Exercise

(OOO) with signs
Reminder of division and multiplication with signs

Example

Work out

$$-3\times(-3)-2\div2=8$$

Show working out

- 3 × (-3) - 2 ÷ 2  1) Division first → 2 ÷ 2 = 1

- 3 × (-3) - 1    2) Multiplication → - 3 × (-3) = + 9

9 - 1         3) Subtraction 9 - 1 = 8

8           4) Result

Exercise

Estimating
Why we use estimating?

In mathematics, we do need to be able to estimate a problem's situation and do calculation later to varify it.

Example

Find 58 + 9

1) First we roughly estimates it to numbers that we can easily calculated in own head

60 + 10 = 70

so we say the answer is roughly nearest to 70

The exact answer = 67

The exact answer is closes to the estimate answer.

Rounding off

Rounding numbers nearest to 10, 100, 1000, ... etc is a technique for calculating or an indication to expect what the real answer will be expect to be.

Why 10, 100, or 1000? Because they are easily can be calculated without any difficulty!

Table of comparing

Example

Round 67 to the nearest 10

Round 675 to the nearest 100

Round 5434 to the nearest 1000

Answer: 1) 70

2) 700

3) 5000

Show working out

1) First we look at the table, round number nearest to 10 we only need 1 digit

In number 67, we take 1 digit from it, starting from RHS to LHS, which is number 7

2) Now we compare number 7 by asking a question. Is number 7 is equal to number 5 or more?

3) If yes, then you add 1 to number 6 and make number 7 a zero

Question: Is number 7 is equal to 5 or more? Yes!

Number 7 at this position becomes zero! ↑  (6+1)0      ↓      Add 1 to number 6

so we have 70

Here is table to show how is work

Exercise

Round the following numbers

Further in estimating
See topic on division of decimal

Rounding decimal numbers to the nearest whole numbers

Example

Round the following to the nearest whole number.

1) 0.98

2) 3.91

3) 0.49

4) 301.1

Answers:

1) 0.98 ≈ 1

2) 3.91 ≈ 4

3) 0.49 ≈ 0.5

4) 301.1 ≈ 300

Exercise

Round the following numbers to the nearest whole numbers

Estimating with order of operation (OOO)

''Note to students. When estimating numbers always try and find a nice whole numbers that you can work with''

Example

Estimate the following

1) 0.9 + 1.02

2) $$\frac{1.94}{0.51}$$

3) 2.01 × 1.01 + 2.01

Answers:

1) 0.9 + 1.02 ≈ 2

2) $$\frac{1.94}{0.51}\approx4$$

3) 2.01 × 1.01 + 2.01 ≈ 4 Show working out

1) Estimate to the nearest whole number before doing any calculation

0.9 + 1.02 ≈ 1 + 1 = 2

$$\frac{1.94}{0.51}\approx{\frac{2}{0.5}}\approx4$$

3) 2.01 × 1.01 + 2.01 ≈ 2 × 1 + 2 ≈ 2 + 2 ≈ 4

Exercise

Estimate the following

Decimal place (d.p)

Another nice way to estimate a number is to round it to a certain decimal place.

Decimal point

This point is called the decimal point (d.p)        ↑ 3.14     ↓      After this decimal point there are two numbers

So 3.14 has two numbers after the decimal point and we says that this number 3.14 it has two decimal places and it is written as 2 d.ps

Example

Write down the numbers of decimal places after the decimal point of the following numbers.

1) pi = 3.14159

2) e = 2.718

3) Log2 = 0.3010299

Answers

1) pi = 3.14159 (5 d.ps)

2) e = 2.718 (3 d.ps)

3) Log2 = 0.3010299 (7 d.ps)

Exercise

Write down the numbers of decimal places after the decimal point of the following numbers.

Significant figures (s.f)

Example 1

Significant point start here ↑     1.34      ↓  ↓ Start    End

From the starting point to the end there are three numbers.

So number 1.34 it said to have 3 significant figures, it is written as 3 s.f.

Example 2

Note that zero does not count as a significant figure ↑  Significant point start here ↑  ↑      0 . 3  4          ↓  ↓     Start    End

From the starting point to the end there are two numbers.

So number 0.34 it said to have 2 significant figures, it is written as 2 s.f.

Example 3

Note that the starting zero does not count as a significant figure ↑  Significant point start here ↑  ↑      0 . 3  4  0 → This last number zero does not count as a significant figure ↓ ↓     Start    End

From the starting point to the end there are two numbers.

So number 0.340 it said to have 2 significant figures, it is written as 2 s.f.

Example

Write down the significant figures of the following numbers

1) 12.345

2) 0.00023

3) 1.0002

4) 1234

5) 0.003400

Answers:

1) 12.345 (5 s.fs)

2) 0.00023 (2 s.fs)

3) 1.0002 (5 s.fs)

4) 1234 (4 s.fs)

5) 0.003400 (2 s.fs)

Exercise

Write down the significant figures of the following numbers

Boundaries
An example is best to explain what is boundary of a number

$$\frac{7}{6}=1.1666...$$

We are going to round this number to 4 s.fs

$$\frac{7}{6}=1.167$$

This number 1.167 is only an approximate of the real value of $$\frac{7}{6}=1.1666...$$

The real value 1.1666... is lies between these two boundaries 1.1665 and 1.1675. 1.1665 is called the lower boundary

1.1675 is called the upper boundary

The real value of 1.1666... is lies between these two boundaries, so mathematically it is written as

1.1665 < 1.1666... < 1.1675

This real value will not go below the lower boundary and it will not go above the upper boundary.

Summary

Lower boundary < real value < Upper boundary

How to find the upper and lower boundaries of a number?

Upper boundary = approximate vale + 0.0005

= 1.167 + 0.0005

= 1.1675

Lower boundary = approximate vale - 0.0005

= 1.167 - 0.0005

= 1.1665

It is good to be able to work out the boundaries of a number so we can know rougly where the real value is lies at.

Example

Find the upper and lower boundaries of the following numbers

1 → 1.23

answer:  1.225 < Real value < 1.235

2 → 2.653

answer:  2.6525 < Real value < 2.6535

Show working out

1 → 1.23

Upper boundary = 1.23 + 0.005

= 1.235

Lower boundary = 1.23 - 0.005

= 1.225

1.225 < Real value < 1.235

2 → 2.653

Upper boundary = 2.653 + 0.0005

= 2.6535

Lower boundary = 2.653 - 0.0005

= 2.6525

2.6525 < Real value < 2.6535

Exercise

Find the upper and lower boundaries of the following numbers

Power
a) What is power?

Example of what is power.

Let us look closely at multiplication, especially the repetitve (curring often) one

two lots of 2's

$$2\times2$$

Five lots of 2's

$$2\times2\times2\times2\times2$$

As you can see if you want to write 100 lots of 2's

$$2\times2\times2\times2\times2\times2\times\cdots$$

this is very tedious!

So a shorter way of writing repetitive multiplication is called power notation

so five lots of 2's can be written as

$$2\times2\times2\times2\times2=2^5$$

$$2^5$$

This is read as two to the power of five!

exercise

Write the following repetitive numbers as power

exercise

Write the following as power as repetitive numbers

Example

Write this as power

$$2\times2\times2\times3\times3$$

$$2^3\times{3^2}$$

exercise

Write the following repetitive numbers as power

exercise

Write the following as power as repetitive numbers

Evaluation of power
We want to find the value of

$$2^4$$

we have to write this power back into its repetitve multiplication and work out from there!

$$2^4=2\times{2\times2}\times2=16$$

Exercise

Work out the value of the following power

Exercise

Work out the value of the following power

Power with (OOO)
POWER ÷ × + -

Example

Evaluate this

$$2^3\times3-4=16$$

Show working out

Exercise

Work out

More on power
Negative power

Example

Work out

$$2^{-3}=\frac{1}{8}$$

Show working out

1) Ignore the negative sign and work out the power normally

= $$2^3$$

= $$2\times2\times2$$

= 8

2) Now put back the negative sign. The negative signs telling you to inverse the answer

$$2^{-3}=\frac{1}{8}$$

Exercise

Work out

Square numbers
What is a square number?

When a number is raised to the power of 2! This is called a square number.

Here are table of square numbers

Here we see that power of

$$2^2=4$$

and

$$(-2)^2=4$$

they are the same !!!

So we conclude that the squaring of any number, the answer is always are positive!

Exercise

Work out

Quick methods of working out square numbers
1) 1st method the normal long multiplication

2 3         × 2 3            6 9        + ¹4 6      -          5 2 9

Exercise

Work out the following square numbers

2) 2nd method, called the square times method, will be demonstrate here!

Example 1 Work out

432 = 1849

Procedure

1. 42 = 1 6

2. 6×4 = 2 4

3. 32 = 0 9

Example 2

Work out

2342 = 54,756

Procedure

1. 22 = 4

2. 6 × 2 = 1 2

3. 32 = 0 9

4. 8 × 23 = 18 4

5. 42 = 1 6 Exercise

Work out the following square numbers by using the square times method

3) 3rd method will be demonstrate here!

232    5 2 9  1) 32 = 9

2) 2×2×3 = 12 (carry 1)

3) 22 = 4     4+1 = 5 (add with the carry)

Exercise

Work out the following square numbers

Square numbers involving zeros
Examples

a)

Work out

102 = 100

Show working out

1) 12 = 1

2) Double 1 zero = 2 zeros

100

b)

Work out

5002 = 250000

1) 52 = 25

2) Double 2 zeros = 4 zeros

250000 Exercise

Work out the following square numbers

The difference of two squares
What is the meaning of the The difference of two squares

Answer:

Two square numbers taken away from each other

Exercise

Work out the following square numbers

Another way of working out two squares taken away from each other,without the need of squaring them

$$4^2-2^2=(4+2)(4-2)=6\times2=12$$

Exercise

Work out the following square numbers

The confusing of square numbers!
(-2)2 and -22 !!!

Are they the same?

Let work them out individually

1)

(-2)2 = (-2)×(-2) = 4 ; Inclusive ; -2 is squared

2)

-22 = -2×2 = -4 ; Exclusive ; 2 is squared, minus sign is not included in the squaring

So they are not the same!

Exercise

Work out the following square numbers

Puzzle involving square numbers
Exercise

Work out

Exercise

True or False

More of square numbers relate to odd numbers
This is an odd numbers sequence.

Exercise

Work out

Well, well, well, what a surprise the sum of odd numbers is equal to square numbers!!!

Square roots
What is the square root of a number?

Example

Remember, when a number is multiplied by itself it is called a square number!

Perfect square numbers : 1, 4, 9, 16, ...

1)              1 × 1 = 1

2)              2 × 2 = 4

Explanation of what is a root of a number This 2 is one of the root of this perfect square number 4 ↓                   2 × 2 = 4                   ↑                   This 2 is another root of this perfect square number 4

So every square numbers has exactly two same roots.

When given a square number, we also would like to know what is the root of the square number.

This process of finding the root of a number is called finding the square root of a number. Symbol of a square root

We know that 4 is a perfect square number. To find the square root of its, this is how it is written:

$$\sqrt[2]{4}=2$$

Square root symbol ↑        $$\sqrt[2]{4}=2$$→Root of the square number ↓         Square number

Exercise

Find the square root of the following numbers

Example

Work out

$$\sqrt[2]{4}+\sqrt[2]{16}=6$$

Show working out

= $$\sqrt[2]{4}+\sqrt[2]{16}$$

= $$2+4$$

= 6

Exercise

Find the square root of the following numbers

Example

Work out

$$\sqrt[2]{9}-\sqrt[2]{1}=2$$

Show working out

= $$\sqrt[2]{9}-\sqrt[2]{1}$$

= $$3-1$$

= 2

Exercise

Find the square root of the following numbers

Example

Work out

1) $$\sqrt[2]{4}\times{\sqrt[2]{16}}=8$$

Show working out

= $$\sqrt[2]{4}\times{\sqrt[2]{16}}$$

= $$2\times4$$ = 8

2) $$2\sqrt[2]{9}=6$$

= $$2\sqrt[2]{9}$$ = $$2\times3$$ = 6

Exercise

Work out the following

Example

Work out

$$\sqrt[2]{16}\div2=2$$

Show working out

= $$\sqrt[2]{16}\div2$$

= $$4\div2$$ = 2

Exercise

Work out

More on square roots
The square root of number 4 can be written into two ways:

First way:

$$\sqrt[2]{4}$$

and

Second way:

$$4^{\frac{1}{2}}$$

In this section we will use the second way!

Example 1

Work out

$$4^{\frac{1}{2}}=2$$

Exercise

Work out

Further explanation of fractional power

$$4^{\frac{3}{2}}$$ In this fractional power. The number 3 (Numberator) represents the power and the number 2 (denumerator) represents the root.

Example 2

Work out

$$4^{\frac{3}{2}}=8$$

Show working out

1) First we find the square root of number 4

$$4^{\frac{1}{2}}=2$$

2) Second we power the answer

$$4^{\frac{3}{2}}=2^3$$

= 8

Exercise

Work out

Negative fractional power

Example 1

Work out

$$4^{-\frac{1}{2}}=\frac{1}{2}$$

Show working out

1) Ignore the negative sign and find the square root normally

$$4^{\frac{1}{2}}=2$$

2) The negative sign tell you that you have to inverse the answer!

$$4^{-\frac{1}{2}}=\frac{1}{2}$$ Exercise

Work out

Example 2

Work out

$$4^{-\frac{3}{2}}=\frac{1}{8}$$

Show working out

1) Ignore the negative sign

$$4^{\frac{3}{2}}$$

2) Second we find the square root of number 4

$$4^{\frac{1}{2}}=2$$

3) Third we power the answer

$$4^{\frac{3}{2}}=2^3$$

= 8

4) Fourth we bring back the negative sign, which tell us to inverse this answer

$$4^{-\frac{3}{2}}=\frac{1}{8}$$

Exercise

Work out

Cube numbers
What is a cube number?

When a number is raised to the power of 3! This is called a cube number.

Here are table of cube numbers

Here we see that power of

$$2^3=8$$

and

$$(-2)^3=-8$$

they are not the same !!!

So we see that cube of a positive number, the result is always positive

So we see that cube of a negative number, the result is always negative

Exercise

Work out

Exercise

Cube numbers involving zeros
Examples

a)

103 = 1000

1) 13 = 1

2) Triple 1 zero = 3 zeros

b)

5003 = 125000000

1) 53 = 125

2) Triple 2 zeros = 6 zeros

Exercise

Work out the following cube numbers

True or False

Cube roots
What is the cube root of a number?

Example

Remember, when a number is multiplied by itself three times, it is called a cube number!

Perfect cube numbers : 1, 8, 27, 64, ...

1)              1 × 1 × 1 = 1

2)              2 × 2 × 2 = 8

Explanation of what is a root of a number This 2 is one of the root of this perfect cube number 8 ↓                   2 × 2 × 2 = 8                   ↑                   This 2 is another root of this perfect cube number 8

So every cube numbers has exactly three same roots.

When given a cube number, we also would like to know what is the root of the cube number.

This process of finding the root of a cube number is called finding the cube root of a number. Symbol of a cube root

We know that 8 is a perfect cube number. To find the cube root of its, this is how it is written:

$$\sqrt[3]{8}=2$$

Cube root symbol ↑        $$\sqrt[2]{8}=2$$→Root of the cube number ↓         Cube number

Exercise

Find the Cube root of the following numbers

Example

Work out

$$\sqrt[3]{1}+\sqrt[3]{64}=5$$

Show working out

= $$\sqrt[3]{1}+\sqrt[3]{64}$$

= $$1+4$$

= 5

Exercise

Find the cube root of the following numbers

Example

Work out

$$\sqrt[3]{27}-\sqrt[3]{1}=2$$

Show working out

= $$\sqrt[3]{27}-\sqrt[3]{1}$$

= $$3-1$$

= 2

Exercise

Find the Cube root of the following numbers

Example

Work out

1) $$\sqrt[3]{8}\times{\sqrt[3]{64}}=8$$

Show working out

= $$\sqrt[3]{8}\times{\sqrt[3]{64}}$$

= $$2\times4$$ = 8

2) $$2\sqrt[3]{27}=6$$

= $$2\sqrt[3]{27}$$ = $$2\times3$$ = 6

Exercise

Work out the following

Example

Work out

$$\sqrt[3]{64}\div2=2$$

Show working out

= $$\sqrt[3]{64}\div2$$

= $$4\div2$$ = 2

Exercise

Work out

More on cube roots
The cube root of number 8 can be written into two ways:

First way:

$$\sqrt[3]{8}$$

This way is the normal symbolic written style

and

Second way:

$$8^{\frac{1}{3}}$$

This way is written as power of a fractional style

In this section we will use the second way!

Example 1

Work out

$$8^{\frac{1}{3}}=2$$

Exercise

Work out

Further explanation of fractional power

$$8^{\frac{2}{3}}$$ In this fractional power. The number 2 (Numberator) represents the power and the number 3 (denumerator) represents the cube root.

Example 2

Work out

$$8^{\frac{2}{3}}=4$$

Show working out

1) First we find the cube root of number 8

$$8^{\frac{2}{3}}=2$$

2) Second we power the answer

$$8^{\frac{2}{3}}=2^2$$

= 4

Exercise

Work out

Negative fractional power

Example 1

Work out

$$8^{-\frac{1}{3}}=\frac{1}{2}$$

Show working out

1) Ignore the negative sign and find the cube root normally

$$8^{\frac{1}{3}}=2$$

2) The negative sign tell you that you have to inverse the answer!

$$8^{-\frac{1}{3}}=\frac{1}{2}$$ Exercise

Work out

Example 2

Work out

$$8^{-\frac{2}{3}}=\frac{1}{4}$$

Show working out

1) 1st ignore the negative sign

$$8^{\frac{2}{3}}$$

2) 2nd we find the cube root of number 8

$$8^{\frac{1}{3}}=2$$

3) 3rd we power the answer

$$8^{\frac{2}{3}}=2^2$$

= 4

4) 4th we bring back the negative sign, which tell us to inverse this answer

$$8^{-\frac{2}{3}}=\frac{1}{4}$$

Exercise

Work out

Square and Cube numbers
Exercise

Evaluate

Method of finding cube root
Example

Find the cube root of 12167.

(12167)1/3 = 27

Show working out

1. Eliminate two digits from the number 12167

12 16 7

2. Now split this number into 3 sections

Eliminate section ↑↑      |  |     12| 16 | 7 → Last section | |     ↓↓ First section

3. In the first section take the cube root of 12 to the nearest whole number

(12)1/3 ≈ 2

4. The last section is untouch

5. Now combine the first and the last sections together

27

6. There are two possible answers

27  23

7. Now we have to take the cube of the last digit of both numbers

27 → 73 = 343  23 → 33 = 2 7

8. So which one is the correction answer?

23 is the correct answer. Why? Because the cube of 3 is 27. The last digit of the cube of 3 is the same as last section digit of the number 12167

Exercise

Find the cube root of the following numbers

Square roots and cube roots
Example

Work out

$$27^{\frac{2}{3}}+4^{\frac{3}{2}}=17$$

Show working out

1) 1st work out each separately

a)

= $$27^{\frac{2}{3}}$$

= $$3^2$$

= 9

b)

= $$4^{\frac{3}{2}}$$

= $$2^3$$

= 8

2) 2nd add them

= $$27^{\frac{2}{3}}+4^{\frac{3}{2}}$$

= 9+8

= 17

Exercise

Evaluate

Indices
What is an index form?

$$2^5$$ This is an index form

The 2 is called the base and the 5 is called the index. The plural of index is indices. Index ↑               25 } → Index form ↓        Base

Exercise

Write down the following

Exercise

Write down the index form

Multiplication of indices

Example 1

Simplify the following

1 → $$2^4\times{2^5}=2^9$$

2 → $$3^4\times{3^5}\times{3^2}=3^{11}$$

Show working out

1 → $$2^4\times2^5$$ → Same bases

= $$2^{4+5}$$ → Add the indices together

= $$2^9$$

Show working out

2 → $$3^4\times{3^5}\times{3^2}$$ → Same bases

= $$3^{4+5+2}$$ → Add the indices

= $$3^{11}$$

Exercise

Simplify the following

Example 2

Simplify

$$5^3\times{5^4}\times{3^3}\times{3^7}=5^7\times{3^{10}}$$

Show working out

→ $$5^3\times{5^4}\times{3^3}\times{3^7}$$ → Same bases

= $$5^{3+4}\times{3^{3+7}}$$ → Add them

= $$5^{7}\times{3^{10}}$$ Exercise

Simplify the following

Division of indices
Example 1

Simplify the following

1 → $$2^6\div{2^4}=2^2$$

2 → $$3^9\div{3^5}=3^{4}$$

Show working out

1 → $$2^6\div2^4$$ → Same bases

= $$2^{6-4}$$ → Subtract the indices from each other

= $$2^2$$

Show working out

2 → $$3^9\div{3^5}$$ → Same bases

= $$3^{9-5}$$ → Subtact the indices

= $$3^{4}$$

Exercise

Simplify the following

Multiplication and division of indices

Example 1

Simplify the following

1 → $$3^5\times3^6\div3^2=3^9$$

2 → $$9^5\div9^5\times9^5=9^5$$

3 → $$8^5\div8^{-6}\times8^2=2^{13}$$

Show working out

1 → $$3^5\times3^6\div3^2$$

= $$3^{5+6-2}$$ = $$3^{9}$$

2 → $$9^5\div9^5\times9^5$$

= $$9^{5-5+5}$$

= $$9^{5}$$

3 → $$8^5\div8^{-6}\times8^2$$

= $$8^{5-(-6)+2}$$

= $$8^{5+6+2}$$

= $$8^{13}$$

Exercise

Simplify the following

Example 2

Simplify

$$\frac{3^4\times3^7}{3^2\times3^3}=3^6$$

Show working out

→ $$\frac{3^4\times3^7}{3^2\times3^3}$$

= $$\frac{3^{4+7}}{3^{2+3}}$$

= $$\frac{3^{11}}{3^{5}}$$

= $$3^{11-5}$$

= $$3^{6}$$

Exercise

Simplify the following

Power of indices
Example 1

Simplify

1 → $$\left(2^4\right)^5=2^{20}$$

2 → $$\left(2^4\right)^{-5}=2^{-20}$$

3 → $$\left(2^{-3}\right)^{-4}=2^{12}$$

Show working out

1 → $$\left(2^4\right)^5$$

= $$2^{4\times5}$$ = $$2^{20}$$

2 → $$\left(2^4\right)^{-5}=2^{-20}$$

= $$2^{4\times(-5)}$$ = $$2^{-20}$$

3 → $$\left(2^{-3}\right)^{-4}=2^{12}$$

= $$2^{-3\times(-4)}$$ = $$2^{12}$$

Exercise

Simplify

Example 2

Simplify

$$\left(2^3\times7^5\right)^9=2^{27}\times7^{54}$$

Show working out

→ $$\left(2^3\times7^5\right)^9$$

= $$2^{(3\times9)}\times7^{(5\times9)}$$

= $$2^{27}\times7^{45}$$

Exercise

Simplify

Reciprocal of indices
Reciprocal means an inverse of a number

Reciprocal of a whole number

Example 1

Find the reciprocal of

1 → $$2^{+3}$$ the answer is $$\frac{1}{2^{-3}}$$

Here the number is inverse and also the sign is also inverse as well.

Note that $$2^{3}$$ is the same as $$2^{+3}$$

2 → $$2^{-3}$$ the answer is $$\frac{1}{2^{+3}}$$

3 → $$\frac{1}{2^3}$$ the answer is $$2^{-3}$$

4 → $$\frac{1}{2^{-3}}$$ the answer is $$2^3$$

Exercise

Find the reciprocal of the following

Reciprocal of a fraction

Example 2

Find the reciprocal of the following fractions

1 → $$\left(\frac{2}{3}\right)^{-1}$$ the answer is $$\left(\frac{3}{2}\right)^{+1}$$

Note that $$\left(\frac{3}{2}\right)^{+1}$$ is the same as $$\left(\frac{3}{2}\right)^{1}$$ also same as $$\left(\frac{3}{2}\right)$$

Curiosity of maths

2+1 = 21 = 2

2 → $$\left(\frac{5}{2}\right)$$ and the answer is $$\left(\frac{2}{5}\right)^{-1}$$

Exercise

Find the reciprocal of the following

Reciprocal of power

Example 3

Find the reciprocal of

1 → $$\left(\frac{2}{3}\right)^2$$ the answer is $$\left(\frac{3}{2}\right)^{-2}$$

The power doesn't change at all!

Exercise

Find the reciprocal of the following

Evaluating using reciprocal
Note $$\frac{2}{3}$$ is the same as $$2\times{\frac{1}{3}}$$

Example 1

Find the value of:

1 → $$\frac{3}{4^{-1}}$$ the answer is $$3\times4$$ here the answer is 12

Show working out

1 → $$\frac{3}{4^{-1}}$$

= $$3\times{\frac{1}{4^{-1}}}$$

= $$3\times{4}$$

= $$12$$

Example 2

Find the value of:

1 → $$\frac{4^{-1}}{5}$$ the answer is $$\frac{1}{20}$$

Show working out

1 → $$\frac{4^{-1}}{5}$$

= $$\frac{1}{5}\times{4^{-1}}$$

= $$\frac{1}{5}\times{\frac{1}{4}}$$

= $$\frac{1}{4\times5}$$

= $$\frac{1}{20}$$

Exercise

Find the value of the following

Power of zero

Example

Work out

$$4^0=1$$

Any number to the power of zero is always equal to one

Exercise

Work out

Standard form
What is standard form?

Number written as a decimal with the power of 10.

Ordinary number     Standard form ↑         ↑               300 = {3.0} × 102                        ↓                     This number is always is between 1 to 9

First method of working out standard form

Positve power

Example will be nice to understand it.

1234 = 1234 × 100 → Note :- 100 = 1

123 4 = 123. 4 ×101

12 34 = 12. 34 ×102

1 234 = 1. 234 ×103 This last number is called standard form

Note:- Everytime the decimal point is moving towards the LHS the power of 10 is increases by 1 towards the RHS

Negative power

Example will be nice to understand it.

0.01234 = 0.01234×100

0. 0 1234 = 0. 1234×10-1

0. 01 234 = 1. 234×10-2 This last number is called standard form

0. 012 34 = 12. 34×10-3

Note:- Everytime the decimal point is moving towards the RHS the power of 10 is decreases by 1 towards the LHS Example 1

Write the following numbers as standard form

1 → 6437 = 6.437×103

2 → 0.075 = 7.5×10-2

3 → 4655.6 = 4.6556×103

Show working out

Just use the above rules of moving the decimal point.

6 437 = 6. 437 ×103           ←

0. 07 5 = 07 .5×10-2         →

4 655 .6 = 4. 655 6×103              ←    Exercise

Write the following ordinary numbers as in standard form

Second method of working out standard form

Here is a table of power

Positive power

Example 1

Change this ordinary number to standard form

$$4000=4.0\times{10^3}$$

Note 4 = 4.0

Show working out

1 → $$4000$$

= $$4.0\times{1000}$$ → Use the table

= $$4.0\times{10^3}$$ → This kind of representation of an answer is called the standard form

Example 2

Change this ordinary number to standard form

$$435=4.35\times{10^2}$$

Show working out

1 → $$435$$

= $$4.35\times{100}$$ → Use the table

= $$4.35\times{10^2}$$ → This kind of representation of an answer is called the standard form

Example 3

Change this ordinary number to standard form

$$6741.8=6.7418\times{10^3}$$

Show working out

1 → $$6741.8$$

= $$6.7418\times{1000}$$ → Use the table

= $$6.7418\times{10^3}$$ → This kind of representation of an answer is called the standard form

Exercise

Write the following ordinary numbers in standard form

Negative power

Example 1

Change the ordinary number to standard form

$$0.045=4.5\times10^{-2}$$

Show working out

1 → 0.045

= 0. 04 5

= $$\frac{4.5}{100}$$

= $$4.5\times{10^{-2}}$$

Example 2

Change the ordinary number to standard form

$$0.00457=4.57\times10^{-3}$$

Show working out

1 → 0.00457

= 0. 004 57

= $$\frac{4.57}{1000}$$

= $$4.57\times{10^{-3}}$$

Exercise

Change the following ordinary numbers to standard forms

Multiplication of standard form
Example 1

Work out and write the answer in standard form

$$\left(3\times10^3\right)\times\left(12\times10^5\right)=3.6\times10^9$$

Show working out

1 → $$\left(3\times10^3\right)\times{\left(12\times10^5\right)}$$

= $$\left(3\times12\right)\times{\left(10^3\times10^5\right)}$$ → Numbers together and power together

= $$36\times{10^{3+5}}$$

= $$36\times{10^8}$$ → This is not a standard form

= $$\left(3.6\times10^1\right)\times{10^8}$$

= $$3.6\times{10^{8+1}}$$

= $$3.6\times{10^9}$$

Example 2

Work out and write the answer in standard form

$$\left(0.3\times10^3\right)\times\left(0.12\times10^5\right)=3.6\times10^6$$

Show working out

1 → $$\left(0.3\times10^3\right)\times{\left(0.12\times10^5\right)}$$

= $$\left(0.3\times0.12\right)\times{\left(10^3\times10^5\right)}$$ → Numbers together and power together

= $$0.036\times{10^{3+5}}$$

= $$0.036\times{10^8}$$ → This is not a standard form

= $$\left(3.6\times10^{-2}\right)\times{10^8}$$

= $$3.6\times{10^{8-2}}$$

= $$3.6\times{10^6}$$

Exercise

Work out the following and write the answers as in standard form

Division of standard form
Example 1

Work out and write the answer in standard form

$$\left(24\times10^8\right)\div\left(2\times10^5\right)=1.2\times10^4$$

Show working out

1 → $$\left(24\times10^8\right)\div{\left(2\times10^5\right)}$$

= $$\left(24\div2\right)\times{\left(10^8\div10^5\right)}$$ → Numbers together and power together

= $$12\times{10^{8-5}}$$

= $$12\times{10^3}$$ → This is not a standard form

= $$\left(1.2\times10^1\right)\times{10^3}$$

= $$1.2\times{10^{3+1}}$$

= $$1.2\times{10^4}$$

Example 2

Work out and write the answer in standard form

$$\left(2\times10^7\right)\div\left(100\times10^2\right)=2.0\times10^3$$

Show working out

1 → $$\left(2\times10^7\right)\times{\left(100\times10^2\right)}$$

= $$\left(2\div100\right)\times{\left(10^7\div10^2\right)}$$ → Numbers together and power together

= $$0.02\times{10^{7-2}}$$

= $$0.02\times{10^5}$$ → This is not a standard form

= $$\left(2.0\times10^{-2}\right)\times{10^5}$$

= $$2.0\times{10^{5-2}}$$

= $$2.0\times{10^3}$$

Exercise

Work out the following and write the answers as in standard form

Addition and subtraction of standard form
Same power

Make sure the power are the same before you perform any calculation ↑   → ←    ↑            12.3 × 103 + 1.1 × 103

Example 1

Work out and write the answer in standard form

$$\left(12.3\times10^3\right)+\left(1.1\times10^3\right)=1.34\times10^4$$

Show working out

1 → $$12.3\times10^3+1.1\times10^3$$

= $$\left(12.3+1.1\right)\times10^3$$

= $$13.4\times10^3$$

= $$\left(1.34\times10^1\right)\times10^3$$

= $$1.34\times10^{3+1}$$

= $$1.34\times10^4$$

Example 2

Work out and write the answer in standard form

$$\left(12.3\times10^3\right)-\left(1.1\times10^3\right)=1.12\times10^4$$

Show working out

1 → $$12.3\times10^3-1.1\times10^3$$

= $$\left(12.3-1.1\right)\times10^3$$

= $$11.2\times10^3$$

= $$\left(1.12\times10^1\right)\times10^3$$

= $$1.12\times10^{3+1}$$

= $$1.12\times10^4$$

Exercise

Work out the following with the answers in standard form

Different power

Example

Work out and write the answer in standard form

$$\left(123\times10^2\right)+\left(11\times10^3\right)=2.43\times10^4$$

Show working out

1 → $$\left(123\times10^2\right)-\left(11\times10^3\right)$$

= (12. 3 × 103) + (11 × 103) → Change to the same power

= $$\left(12.3+11\right)\times10^3$$

= $$24.3\times10^3$$

= $$\left(2.43\times10^1\right)\times10^3$$

= $$2.43\times10^{3+1}$$

= $$2.43\times10^4$$

Power of standard form
Example

Write the ordinary number as standard form

$$\left(5\times10^4\right)^3=1.25\times10^{14}$$

Show working out

1 → $$\left(5\times10^4\right)^3$$ = $$\left(5^1\times10^4\right)^3$$ → Hidden 1 on the 5

= $$5^{1\times3}\times10^{4\times3}$$ → Multiply power with power

= $$5^3\times10^{12}$$ = $$125\times10^{12}$$

= 1 25 ×1012

= (1. 25 ×102)×1012

= 1.25×102+12

= 1.25×1014

Exercise

Write the following ordinary numbers as standard form

Application of standard form
Example

The mass of planet Earth is 5.9736×1024kg. Planet jupiter is 318 times larger in size than planet Earth.

Write down the mass of planet Jupiter and put the answer in standard form

Answer:

Jupiter's mass = 318 × Earth's mass = 318 × 5.9736×1024               = (318 × 5.9736)×1024                = 1899.6048×1024 → This is not yet in standard form = 1 899 .6048×1024 → From the decimal point moves three spaces to the left = (1.8996048×103)×1024               = 1.8996048×103+24                = 1.8996048×1027 → This is in standard form

Exercise

Fill in the table

Application with percentage

Example

A space aircraft travels at a speed of 1.22×104m/s at 11:00. After 30 minutes its speed changed to 1.78×104m/s.

1. Calculate the increase in speed of the aircraft, write the answer in standard form.

2. Calculate the percentage of the speed of the aircraft is increased by, write the answer to 2 d.p

Answers:

1. 6.0×102m/s

2. 4.92%

Show working out

1. Increase speed in aircraft = 1.78×104 - 1.22×104 = (1.78 - 1.22)×104                              = 0.06×104 → This is not yet in standard form = 0. 06 ×104                              = (6.0×10-2)×104                               = 6.0×104-2                               = 6.0×102m/s 2. Increase in percentage of speed = $$\frac{IncreaseSpeed}{InitialSpeed}\times100$$ = $$\frac{6\times10^2}{1.22\times10^4}\times100$$ = 4.92%

Exercise

Surd
What is surd?

An example is best to explain what is surd

Work out the following

$$\sqrt{9}=3$$

$$\sqrt{4}=2$$

$$\sqrt{2}=1.414...$$

The first two give whole numbers.

The last one give a decimal number.

so square root of 2 can not be expressed as whole number, so we keep it as it is $$\sqrt{2}$$ keeping the answer in this form it is called surd.

Addition of surds
Example 1

Work out

$$\sqrt{9}+\sqrt{4}=5$$

Show working out

$$\sqrt{9}+\sqrt{4}=3+2=5$$

Exercise

Example 2

Work out

$$2\sqrt{9}+5\sqrt{4}=16$$

Show working out

$$2\sqrt{9}+5\sqrt{4}=2\times3+5\times2=6+10=16$$

Note the number outside the square root indicates multiplication!

Exercise

Example 3

Work out

$$\sqrt{9}+\sqrt{5}=3+\sqrt{5}$$

Show working out

$$\sqrt{9}+\sqrt{5}=3+\sqrt{5}$$

$$\sqrt{5}$$ can not be expressed as a whole number s we leave it there.

Exercise

Example 4

Work out

$$\sqrt{2}+\sqrt{5}=\sqrt{2}+\sqrt{5}$$

Show working out

$$\sqrt{2}+\sqrt{5}=\sqrt{2}+\sqrt{5}$$

$$\sqrt{2}$$ and $$\sqrt{5}$$ both can not be expressed as a whole number so we leave it there.

Exercise

Example 5

Work out

$$\sqrt{2}+\sqrt{2}=2\sqrt{2}$$

Show working out

$$\sqrt{2}+\sqrt{2}=(1+1)\sqrt{2}=2\sqrt{2}$$

Exercise

Example 6

Work out

$$3\sqrt{2}+2\sqrt{2}=5\sqrt{2}$$

Show working out

$$3\sqrt{2}+2\sqrt{2}=(3+2)\sqrt{2}=5\sqrt{2}$$

Exercise

Mixed section

Example 7

Work out

$$\sqrt{4}+2\sqrt{3}+5\sqrt{3}+4\sqrt{9}=14+7\sqrt{3}$$

Show working out

$$\sqrt{4}+2\sqrt{3}+5\sqrt{3}+4\sqrt{9}$$

$$2+(2+5)\sqrt{3}+4\times3$$

$$2+7\sqrt{3}+12$$

$$14+7\sqrt{3}$$

Exercise

Subtraction of surds
Example 1

Work out

$$\sqrt{9}-\sqrt{4}=5$$

Show working out

$$\sqrt{9}-\sqrt{4}=3-2=1$$

Exercise

Example 2

Work out

$$4\sqrt{9}-5\sqrt{4}=2$$

Show working out

$$4\sqrt{9}-5\sqrt{4}$$

$$4\times3-5\times2$$

$$12-10$$

$$2$$

Note the number outside the square root indicates multiplication!

Exercise

Example 3

Work out

$$\sqrt{9}-\sqrt{5}=3-\sqrt{5}$$

Show working out

$$\sqrt{9}-\sqrt{5}=3-\sqrt{5}$$

$$\sqrt{5}$$ can not be expressed as a whole number s we leave it there.

Exercise

Example 4

Work out

$$\sqrt{2}-\sqrt{5}=\sqrt{2}-\sqrt{5}$$

Show working out

$$\sqrt{2}-\sqrt{5}=\sqrt{2}-\sqrt{5}$$

$$\sqrt{2}$$ and $$\sqrt{5}$$ both can not be expressed as a whole number so we leave it there.

Exercise

Example 5

Work out

$$\sqrt{2}-\sqrt{2}=0$$

Show working out

$$\sqrt{2}-\sqrt{2}$$

$$(1-1)\sqrt{2}$$

$$0\sqrt{2}$$

$$0$$

Exercise

Example 6

Work out

$$3\sqrt{2}-2\sqrt{2}=\sqrt{2}$$

Show working out

$$3\sqrt{2}-2\sqrt{2}$$

$$(3-2)\sqrt{2}$$

$$\sqrt{2}$$

Exercise

Mixed section

Example 7

Work out

$$\sqrt{100}-10\sqrt{3}-5\sqrt{3}-4\sqrt{1}=6-5\sqrt{3}$$

Show working out

$$\sqrt{100}-10\sqrt{3}-5\sqrt{3}-4\sqrt{1}$$

$$10-(10-5)\sqrt{3}-4\times1$$

$$10-5\sqrt{3}-4$$

$$6-5\sqrt{3}$$

Exercise

Mixed section of addition and subtraction of surds
Example

Work out

$$3\sqrt{4}-2\sqrt{9}+\sqrt{5}+2\sqrt{5}=3\sqrt{5}$$

Show working out

$$3\sqrt{4}-2\sqrt{9}+\sqrt{5}+2\sqrt{5}$$

$$3\times2-2\times3+(1+2)\sqrt{5}$$

$$6-6+3\sqrt{5}$$

$$3\sqrt{5}$$

Exercise

Work out the following

Signs with Add and Subtract with surds
Example

Work out

$$\sqrt{3}-2\times{\left(-2\sqrt{3}\right)}$$ or

$$\sqrt{3}-2\left(-2\sqrt{3}\right)$$

Note: The bracket means multiplication, so we don't need to write the multiplication sign again

Answer:

$$5\sqrt{3}$$

Show working out

$$\sqrt{3}-2\times{\left(-2\sqrt{3}\right)}$$

$$\sqrt{3}+4\sqrt{3}$$   →   - 2 × ( - 2 ) = + 4 $$(1+4)\sqrt{3}$$ → Same roots, just add the numbers outside of the roots $$5\sqrt{3}$$

Exercise

Simplify the following surds

Multiplication of surds
Example 1

Work out

$$\sqrt{4}\times\sqrt{9}=6$$

Show working out

→ $$\sqrt{4}\times\sqrt{9}$$

= $$2\times3$$

= $$6$$

Exercise

Work out

Example 2

Work out

$$\sqrt{4}\times\sqrt{7}=2\sqrt{7}$$

Show working out

→ $$\sqrt{4}\times\sqrt{7}$$

= $$2\sqrt{7}$$

We can not work out the $$\sqrt{7}$$ so we leave it as it is.

Exercise

Work out

Example 3

Work out

$$\sqrt{2}\times\sqrt{7}=\sqrt{14}$$

Show working out

→ $$\sqrt{4}\times{\sqrt{7}}$$

= $$\sqrt{2\times7}$$

= $$\sqrt{14}$$

We can not work out the root of $$\sqrt{2}$$ and $$\sqrt{7}$$ individually so we leave it as it is and multiply them together.

Exercise

Work out

 Special case: same roots

Example 4

Work out

$$\sqrt{2}\times{\sqrt{2}}=2$$

Show working out

→ $$\sqrt{2}\times{\sqrt{2}}$$

= $$\sqrt{2\times2}$$

= $$\sqrt{4}$$

= $$2$$

Simple rule:

$$\sqrt{Number}\times{\sqrt{Number}}=Number$$

Exercise

Work out

Example 5

Work out

$$\sqrt{4}\times\sqrt{7}\times{\sqrt7}=14$$

Show working out

→ $$\sqrt{4}\times\sqrt{7}\times{\sqrt7}$$

= $$2\times\sqrt{7}\times{\sqrt7}$$ → Apply the simple rule

= $$2\times7$$

= $$14$$

Exercise

Work out

Example 6

Work out

$$2\sqrt{4}\times{3\sqrt{9}}=36$$

Show working out

→ $$2\sqrt{4}\times{3\sqrt{9}}$$

= $$2\times2\times3\times3$$

= $$4\times9$$

= $$36$$

Exercise

Work out

Example 7

Work out

$$2\sqrt{3}\times{4\sqrt{2}}=8\sqrt{6}$$

Show working out

→ $$2\sqrt{3}\times{4\sqrt{2}}$$

= $$2\times4{\times}{\sqrt{3\times2}}$$ → Multiply, number with number and root with root

= $$8\times{\sqrt{6}}$$

= $$8{\sqrt{6}}$$

Exercise

Work out

Division of surds
Example 1

Work out

$$\frac{\sqrt{9}}{3}=1$$

Show work out

→ $$\frac{\sqrt{9}}{3}$$

= $$\frac{3}{3}$$

= $$1$$

Exercise

Example 2

Work out

$$\frac{\sqrt{6}\times{\sqrt{6}}}{3}=2$$

Show work out

→ $$\frac{\sqrt{6}\times{\sqrt{6}}}{3}$$ → Apply the simple rule

= $$\frac{6}{3}$$

= $$2$$

Exercise

Example 3

Work out

$$\frac{\sqrt{27}}{\sqrt{3}}=3$$

Show work out

→ $$\frac{\sqrt{27}}{\sqrt{3}}$$

= $$\sqrt{\frac{27}{3}}$$

= $$\sqrt{9}$$ → With division of surd we can put the numerator and the denumerator under one root

= $$3$$

Exercise

Cancellation of surds
Example 1

Work out

$$\frac{\sqrt4\times{\sqrt5}}{\sqrt5}=2$$

Show working out

→ $$\frac{\sqrt4\times{\sqrt5}}{\sqrt5}$$

= $$\frac{\sqrt4\times{\sqrt5}}{\sqrt5}$$ → Cancel the numerator with the denumerator, because $$\sqrt5$$ are common to both of them

= $$\sqrt4$$

= $$2$$

Exercise

Example 2

Work out

$$\frac{\sqrt5\times{\sqrt6}}{\sqrt3}=\sqrt{10}$$

Show working out

→ $$\frac{\sqrt5\times{\sqrt6}}{\sqrt3}$$

= $$\sqrt5\times{\frac{\sqrt6}{\sqrt3}}$$

= $$\sqrt5\times{\sqrt{\frac{6}{3}}}$$ → With division we can put the numerator and denumerator under one root

= $$\sqrt5\times{\sqrt{2}}$$

= $$\sqrt{5\times2}$$

= $$\sqrt{10}$$

Exercise

Simplifying surds
Example 1

Simplify

$$\sqrt{8}=2\sqrt{2}$$

Show working out

→ $$\sqrt{8}$$

= $$\sqrt{4\times2}$$ → Product of 8

= $$\sqrt4\times{\sqrt{2}}$$ → split out the square root

= $$2\times{\sqrt{2}}$$ → Work out the square root where it is possible = $$2\sqrt{2}$$

Exercise

Simplify the following

Example 2

Simplify

$$3\sqrt{8}=6\sqrt{2}$$

Show working out

→ $$3\sqrt{8}$$

= $$3\sqrt{4\times2}$$ → Product of 8

= $$3\sqrt4\times{\sqrt{2}}$$ → split out the square root

= $$3\times2\times{\sqrt{2}}$$ → Work out the square root where it is possible = $$6\sqrt{2}$$

Exercise

Simplify the following

Example 3

Simplify

$$\frac{\sqrt{27}}{3}=\sqrt{3}$$

Show working out

→ $$\frac{\sqrt{27}}{3}$$

= $$\frac{\sqrt{9\times3}}{3}$$ → Product of 27

= $$\frac{\sqrt9\times{\sqrt{3}}}{3}$$ → split out the square root

= $$\frac{3\times{\sqrt{3}}}{3}$$ → Work out the square root where it is possible

3 are common to both numerator and denumerator, so it can be cancelled out. = $$\sqrt{3}$$

Exercise

Simplify the following

Rational and irrational numbers
Example of rational numbers are:

$$\frac{3}{2}=1.5$$ → Non recurring decimal

$$\frac{10}{3}=3.333...$$ → Recurring decimal

$$\sqrt4=2$$

Any decimal numbers whether they are recurring or non recurring that can be expressed into fractions are called rational numbers

Example of irrational numbers are:

$$\pi=3.14159\cdots$$

$$\sqrt2=1.414...$$

$$\ln2=o.69....$$

Any decimal numbers that can not be expresses as factions are called irrational numbers

Exercise

Write down a few numbers that are rational and irrational

Rationalise the denumerator (surds)
Changing an irrational number into a rational number.

Example 1

An irrational number $$\sqrt{2}$$

We are going to change this into a rational number. How?

Just multiply two same irrational numbers together and you will a rational number

$$\sqrt{2}\times{\sqrt{2}}=2$$

Show working out

→ $$\sqrt{2}\times{\sqrt{2}}$$

= $$\sqrt{2\times2}$$

= $$\sqrt{4}$$

= $$2$$

Exercise

Change the following irrational number into rational number

Example 2

Rationalise the denumerator. What is meant here is that you have to make the denumerator a rational number

$$\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$$

Show working out

→ $$\frac{2}{\sqrt{3}}$$

= $$\frac{2\times{\sqrt3}}{\sqrt{3}\times{\sqrt3}}$$ → Multiply top and bottom by

$$\sqrt3$$

=$$\frac{2\sqrt3}{3}$$

Exercise

Rationalise the denumerator

Example 3

Rationalise the denumerator and simplify

$$\frac{6}{\sqrt{2}}=3\sqrt{2}$$

Show working out

→ $$\frac{6}{\sqrt{2}}$$

= $$\frac{6\times{\sqrt2}}{\sqrt{2}\times{\sqrt2}}$$ → Multiply top and bottom by

$$\sqrt2$$

= $$\frac{6\sqrt2}{2}$$

= $$\frac{6}{2}\times{\sqrt2}$$

= $$3\sqrt2$$

Exercise

Rationalise the denumerator

Further rationalisation of the denominator
Quick revision

Example

Expand the following

1. $$\left(2-\sqrt3\right)\left(2+\sqrt3\right)$$

Show working out

$$\left(2-\sqrt3\right)\left(2+\sqrt3\right) = 2^2-\left(\sqrt3\right)^2$$

= 4 - 3                    = 1

2. $$\left(4-2\sqrt3\right)\left(4+2\sqrt3\right)$$

Show working out

$$\left(4-2\sqrt3\right)\left(4+2\sqrt3\right) = 4^2-\left(2\sqrt3\right)^2$$

= $$16-2^2\times{\left(\sqrt3\right)^2}$$

= $$16 -4\times3$$ = 16 - 12

= 4

Example 1

Rationalise the denominator

1 ... $$\frac{1}{2-\sqrt3}$$

Answer:

... $$2+\sqrt3$$

Show working out

1. We have to find the opposite of $$2-\sqrt3$$. Which is $$2+\sqrt3$$

2. We are going to multiply this $$2+\sqrt3$$ to the numerator and denumerator of this fraction $$\frac{1}{2-\sqrt3}$$

$$\frac{1}{2-\sqrt3}\times{\frac{2+\sqrt3}{2+\sqrt3}}$$

3. Now we start to work out ...

$$\frac{1}{2-\sqrt3}\times{\frac{2+\sqrt3}{2+\sqrt3}}=\frac{2+\sqrt3}{\left(2-\sqrt3\right)\left(2+\sqrt3\right)}$$

4. We work out the denumerator separately

$$\left(2-\sqrt3\right)\left(2+\sqrt3\right)=2^2-\left(\sqrt3\right)$$

= 4 - 3                    = 1

5. We put back the answer of the denumerator to the fraction

$$\frac{2+\sqrt3}{\left(2-\sqrt3\right)\left(2+\sqrt3\right)}=\frac{2+\sqrt3}{1}$$

= $$2+\sqrt3$$

6. Final answer

$$\frac{1}{2-\sqrt3}=2+\sqrt3$$

Summary of the working out

$$\frac{1}{2-\sqrt3}=\frac{1}{2-\sqrt3}\times{\frac{2+\sqrt3}{2+\sqrt3}}$$

= $$\frac{2+\sqrt3}{\left(2-\sqrt3\right)\left(2+\sqrt3\right)}$$

= $$\frac{2+\sqrt3}{2^2-\left(\sqrt3\right)^2}$$

= $$\frac{2+\sqrt3}{4-3}$$

= $$\frac{2+\sqrt3}{1}$$

= $$2+\sqrt3$$

Exercise

Rationalise the following questions

Example 2

Rationalise the denumerator

$$\frac{1}{\sqrt5+\sqrt3}=\frac{1}{\sqrt5+\sqrt3}\times{\frac{\sqrt5-\sqrt3}{\sqrt5-\sqrt3}}$$

= $$\frac{\sqrt5-\sqrt3}{\left(\sqrt5+\sqrt3\right)\left(\sqrt5-\sqrt3\right)}$$

= $$\frac{\sqrt5-\sqrt3}{\left(\sqrt5\right)^2-\left(\sqrt3\right)^2}$$

= $$\frac{\sqrt5+\sqrt3}{5-3}$$

= $$\frac{\sqrt5+\sqrt3}{2}$$

Expansion of bracket (surds)
Here is a table showing 3 types of multiplication signs use in mathematics

In this section we are going to use the symbol

Example

Expand the following brackets

1 → $$3(2+\sqrt5)=6+3\sqrt5$$

2 → $$3(4+2\sqrt5)=12+6\sqrt3$$

3 → $$\sqrt3(\sqrt5+\sqrt6)=\sqrt15+\sqrt18$$

4 → $$\sqrt2(\sqrt2+\sqrt3)=2+\sqrt6$$

Show working out

1 → $$3(2+\sqrt5)$$

= $$3\times2+3\times{\sqrt5)}$$ → The outside number is mutliply by every numbers inside the bracket

= $$6+3\sqrt5$$ → Simplify 2 → $$3(4+2\sqrt3)=12+6\sqrt3$$

= $$3\times4+3\times{2\sqrt3)}$$

= $$12+6\sqrt3)$$

3 → $$\sqrt3(\sqrt5+\sqrt6)$$

= $$\sqrt3\times{\sqrt5}+\sqrt3\times{\sqrt6)}$$

= $$\sqrt{3\times5}+\sqrt{3\times6}$$

= $$\sqrt{15}+\sqrt{18}$$

4 → $$\sqrt2(\sqrt2+\sqrt3)$$

= $$\sqrt2\times{\sqrt2}+\sqrt2\times{\sqrt3)}$$

= $$2 + \sqrt{2\times3}$$

= $$2 + \sqrt{6}$$

Exercise

Expand the following brackets

Further expansion of surds
Example 1

Expand

$$3(2-3\sqrt2)=6-9\sqrt2$$

Show working out

→ $$3(2-3\sqrt2)$$ = $$3\times2-3\times{3\sqrt2)}$$

= $$6-9\sqrt2$$

Example 2

Expand and simplify

$$5(2+2\sqrt2)+2(4-3\sqrt2)=18+4\sqrt2$$

Show working out

→ $$5(2+2\sqrt2)+2(4-3\sqrt2)$$ → Expand the brackets

= $$5\times2+5\times{2\sqrt2}+2\times4-2\times{3\sqrt2}$$ → Simplify

= $$10+10\sqrt2+8-6\sqrt2$$ → Work out the numbers and roots

= $$10+8+10\sqrt2-6\sqrt2$$

= $$18+4\sqrt2$$

Exercise

Expand and simplify

Expansion of surds with two brackets
Same roots

Example

Expand and simplify

$$\left(2+\sqrt3\right)\left(3+\sqrt3\right)$$

Answer:

$$9+5\sqrt3$$

Show working out

1. Take the first number in the first bracket and multiply it by the second bracket, as shown below → $$2\left(3+\sqrt3\right)$$ = $$2\times3+2\times\sqrt3$$ = $$6+2\sqrt3$$

2. Take the second number in the first bracket and multiply it by the second bracket, as shown below → $$\sqrt3\left(3+\sqrt3\right)$$ = $$\sqrt3\times3+\sqrt3\times\sqrt3$$

= $$3\sqrt3+3$$ ; notice that $$\sqrt3\times{\sqrt3}=3$$

3. Now we combine them together and we have

→ $$6+3\sqrt3+2\sqrt3+3$$ ; Collect like terms and simplify = $$\left(6+3\right)+\left(2\sqrt3+3\sqrt3\right)$$ ; notice that $$2\sqrt3+3\sqrt3=(2+3)\sqrt3$$

Collect the numbers and square roots separately and simplify them individually, as shown above

= $$9+5\sqrt3$$ ; This is the final answer!

4. Final summary

$$\left(2+\sqrt3\right)\left(3+\sqrt3\right)=2\left(3+\sqrt3\right)+\sqrt3\left(3+\sqrt3\right)$$ ; Expand the brackets = $$2\times3+2\times\sqrt3+\sqrt3\times3+\sqrt3\times\sqrt3$$ ;Expand the single brackets

= $$6+2\sqrt3+3\sqrt3+3$$ ; Work out

= $$\left(6+3\right)+\left(2\sqrt3+3\sqrt3\right)$$ ; Collect like terms and simplify

= $$9+5\sqrt3$$ ; answer

Exercise

Expand and simplify the following

Different roots

Example 2

Expand and simplify

$$\left(2+\sqrt3\right)\left(3+\sqrt5\right)$$

Answer:

$$6+2\sqrt5+3\sqrt3+\sqrt15$$

Show working out

1. Take the first number in the first bracket and multiply it by the second bracket, as shown below → $$2\left(3+\sqrt5\right)$$ = $$2\times3+2\times\sqrt5$$ = $$6+2\sqrt5$$

2. Take the second number in the first bracket and multiply it by the second bracket, as shown below → $$\sqrt3\left(3+\sqrt5\right)$$ = $$\sqrt3\times3+\sqrt3\times\sqrt5$$

= $$3\sqrt3+\sqrt15$$ ; notice that $$\sqrt3\times{\sqrt5}=\sqrt{3\times5}=\sqrt{15}$$

3. Now we combine them together and we have

→ $$6+2\sqrt5+3\sqrt3+\sqrt15$$ ; Collect like terms and simplify where possible

Notice that above, the roots are all different, so we can not simplify them further as it is. = $$6+2\sqrt5+3\sqrt3+\sqrt15$$ ; This is the final answer 4. Final summary

$$\left(2+\sqrt3\right)\left(3+\sqrt5\right)=2\left(3+\sqrt3\right)+\sqrt3\left(3+\sqrt5\right)$$ ; Expand the brackets = $$2\times3+2\times\sqrt5+\sqrt3\times3+\sqrt3\times\sqrt5$$ ;Expand the single brackets

= $$6+2\sqrt5+3\sqrt3+\sqrt15$$ ; answer

Expand and simplify the following

Special case ; same numbers and roots but different in signs

Example 3

Expand and simplify

$$\left(2-\sqrt3\right)\left(2+\sqrt3\right)$$

Answer:

$$1$$

Show working out

When this kind of question comes up, where both brackets are identical, except they have different signs

We ignore the long process of expanding the brackets method

Here is the procedure how we are going to do it.

1. We pick the bracket with the minus sign and we are going to square both terms inside the brackets, as shown below

→ $$\left(2-\sqrt3\right)$$ = $$2^2-\left(\sqrt3\right)^2$$ ; Notice that $$\left(\sqrt3\right)^2=3$$

= 4 - 3 ; Simplify = 1 ; Final answer!

If you notice carefully, the answer is in the form of a rational number

Example 3

Expand and simplify

$$\left(6-2\sqrt3\right)\left(6+2\sqrt3\right)$$

Answer:

$$24$$

Show working out

1. Pick the bracket witht the minus sign and square the two terms inside the bracket

→ $$\left(6-2\sqrt3\right)$$

→ $$6^2-\left(2\sqrt3\right)^2$$ = $$36-2^2\times{\left(\sqrt3\right)^2}$$ = $$36-4\times3$$

= $$36-12$$

= 24

Example 4

Expand and simplify

$$\left(3\sqrt5-2\sqrt3\right)\left(3\sqrt5+2\sqrt3\right)$$

Answer:

$$33$$

Show working out

1. Pick the bracket witht the minus sign and square the two terms inside the bracket

→ $$\left(3\sqrt5-2\sqrt3\right)$$

→ $$\left(3\sqrt5\right)^2-\left(2\sqrt3\right)^2$$ = $$3^2\times{\left(\sqrt5\right)^2}-2^2\times{\left(\sqrt3\right)^2}$$ = $$9\times5-4\times3$$

= $$45-12$$

= 33

Exercise

Expand and simplify the following

Square and cube of surds
Squaring of a square root number

Example

Work out

$$(\sqrt{2})^2=2$$

Show working out

→ $$(\sqrt{2})^2$$

= $$\sqrt{2}\times{\sqrt2}$$

= $$2$$ → So squaring any square root number always give a rational number.

Exercise

Sqaure the following square root numbers

Cubing of a square root number

Example

Work out

$$(\sqrt{2})^3=2\sqrt2$$

Show working out

→ $$(\sqrt{2})^3$$

= $$\sqrt{2}\times{\sqrt2}\times{\sqrt2}$$

Note → $$\sqrt{2}\times{\sqrt2}=2$$ = $$2\sqrt2$$ → So cubing any square root number always give an irrational number.

Exercise

Sqaure the following square root numbers

Sequences
What is a sequence?

A list of numbers that follow a certain rule and displayed an orderly pattern!

Example

Natural number sequence

1, 2, 3, 4, 5, ...

Here is a table of favourite number sequences.

Number line

This is called the number line

←...--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|---...→               -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Negative side                                               Positive side

Whole number ↑→ → → → → → → → → → ...  ←...--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|---...→                -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Number starts from zero and onward are called whole number

Natural number ↑→ → → → → → → → → .... ←...--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|---...→                -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Number start from one and onward are called natural number

Negative number ... ← ← ← ← ← ← ← ← ← ←← ← ← ← ←↑      ←...--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|---...→                -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Number starts from minus one and onward are called negative number

Rational number 0.25  2.25                                       ↑     ↑       ←...--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|---...→                -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Numbers (for example 2.25) that are lie between two numbers such as 2 and 3, that can be change into fraction are called rational number

Short summary of a rational numbers

Decimal numbers that can be change into fraction are called ration numbers

Exercise

Name the following sequences

Find the next terms in a sequence
Given a sequence, it is possible sometime, we can work out the next numbers in the sequence by just see an obvious pattern display in the sequence.

Example

Sequence:

What are the next two numbers in the sequence? 2, 4, 6, 8, ...

It is quite obivous (10 and 12). We can see that this sequence is gone up in two.

Exercise

Work out the next two terms of the following sequences

Exercise

Work out the next two terms of the following sequences

Exercise

Make your own sequences!

Find the rules for sequences
Given the rules; find the sequences according to the rules

Given a certain rule, we can see what a sequence will look like under this rule.

Linear sequences

What are linear sequences?

They are the sequences with a constant common difference.

An example will explain what is a common difference.

Sequence : 2, 5, 8, 13, ...

+3 +3 +3 →             ↑  ↑  ↑            2, 5, 8, 13, ...

This sequence is gone up in 3 everytime. This number 3 is called the common difference of the sequence.

Exercise

Work out the common difference of the following sequences

Terms in a sequence

What are terms in a sequence?

Every numbers in a sequence has its location in the sequence. These locations of numbers in the sequence are called terms.

Explanation:

The location of the sequence are fixed (Unchangeable)

Knowing these two information, now we can find the rules for any linear sequence.

Finding the rules of linear sequences

Example

Find the rule for this sequence

Sequence : 2, 5, 8, 11, ...

Terms    →  1  2  3  4  ... any term Sequence →  2, 5, 8, 11, ...

Show working out

1) First, find the common difference of the sequence

The common difference = 3

2) Second, the 1st term take away from the common difference

2 - 3 = - 1 ; This number is called the constant.

3) Let say you want to find the 5th and 6th terms in the sequence

Take the common difference multiply by the 5th term and take one away from it.

5th = 3 × 5 - 1 = 15 - 1 = 14

6th = 3 × 6 - 1 = 18 - 1 = 17

To find any term in the sequence: The rule is any term multiply by the common difference and take one away from it.

In general, to find the rule of the sequence.

 Any term in the sequence is then multiply by the common difference and the constant it is added to or subtracted from it.

Exercise

Find the rules for the following sequences and find the 5th and 6th terms of the sequence

Find the sequences given the rules
Given the rules find the sequences

Linear sequences

Example

Find the sequence of this rule.

Rule : Any term × 3 + 1

Sequence : 4, 7, 10, 13, 16, ...

Show working out

1) 1st term → 1 × 3 + 1 = 3 + 1 = 4

2) 2nd term → 2 × 3 + 1 = 6 + 1 = 7

3) 3rd term → 3 × 3 + 1 = 9 + 1 = 10

4) 4th term → 4 × 3 + 1 = 12 + 1 = 13

5) 5th term → 5 × 3 + 1 = 15 + 1 = 16

Exercise

Find the sequences of the following given rules

Square sequences

Example

Find the sequence of this rule.

Rule : (Any term)2 × 3 + 1

Sequence : 4, 13, 28, 49, 76, ...

Show working out

1) 1st term → 12 × 3 + 1 = 3 + 1 = 4

2) 2nd term → 22 × 3 + 1 = 12 + 1 = 13

3) 3rd term → 32 × 3 + 1 = 27 + 1 = 28

4) 4th term → 42 × 3 + 1 = 48 + 1 = 49

5) 5th term → 52 × 3 + 1 = 75 + 1 = 76

Exercise

Find the sequences of the following given rules

Other rules for sequences

Example

Rule: The next number is 3 multiply of the previous number and then add one to it

We start 1 as the previous number

Sequence: 1, 4, 13, 40, 121, ...

Show working out

1) 1×3 + 1 = 3 + 1 = 4

2) 4×3 + 1 = 12 + 1 = 13

3) 13×3 + 1 = 39 + 1 = 40

4) 40×3 + 1 = 120 + 1 = 121

5) and so on ...

Exercise

Work out

Here are a list of rules, find the sequence of its

Fractions
What is fraction?

Demonstration of what is fraction

.........    ....|....           |....           |....                 .       .     .   |   .           |   .           |   .   .       .     |       -       -       .       .     .   |   .       .   |   .       .   |   .    .........     ....|....       ....|....       ....|....    A whole      broken into     1 part is       3 parts are 4 eqaul parts    missing           remain

1                           ¼                  ¾

Missing part of a whole is = $$\frac{1}{4}$$

Remaining parts of a whole are = $$\frac{3}{4}$$

 Diagrams to fractions 

Example Q1.

Below is a rectangle, divided into three equal parts. Write down the following: shaded part, unshaded part, fraction of shaded part and fraction of unshaded part. _______   |/_/_/_/|    |_______|    |_______|

Answer:

Exercise

Fill in the table

Q1. _______ |/_/_/_/|  |_______|  |_______|  |/_/_/_/|

Q2. ___________  |/_/|___|___|   |/_/|___|___|   |/_/|___|___|

Q3. ________________ |/_/|__|__|__|/_/|

Fractions to diagrams

Example

A rectangle shaded fraction is $$\frac{3}{5}$$. Draw a diagram to represent this shaded fraction

Numerator represents the shaded parts ; 3 Denumerator represents the total parts of a whole ; 5

Answer:

Diagram showing shaded parts of a whole _______   |/_/_/_/|    |/_/_/_/|    |/_/_/_/|    |_______|    |_______|

Exercise

Draw diagrams to represent the following shaded fractions

Names of top and bottom numbers of a fraction

$$\frac{top}{bottom}=\frac{numerator}{denumerator}$$

The top number is called the numerator

The bottom number is called the denumerator

Multiplication of fractions
Multiply them directly:

Numerator with numerator

Denumerator with denumerator

$$\frac{3}{7}\times{\frac{2}{5}}=\frac{3\times2}{7\times5}=\frac{6}{35}$$

Exercise

Work out the following

Division of fraction
a)

First method; we cross multiply

1) $$\frac{12}{2}\div{\frac{6}{2}}$$

Numerator with denumerator

$$\frac{12}{.}\div{\frac{.}{2}}=\frac{12\times2}{}$$

Denumerator with numerator

$$\frac{.}{2}\div{\frac{6}{.}}=\frac{}{2\times6}$$

Result

$$\frac{12}{2}\div{\frac{6}{2}}=\frac{12\times2}{2\times6}=\frac{24}{12}$$

b)

Second method; we inverse and multiply

$$\frac{12}{2}\div{\frac{6}{2}}$$

Now inverse the last fraction and then multiply

$$\frac{12}{2}\times{\frac{2}{6}}=\frac{12\times2}{2\times6}=\frac{24}{12}$$

Exercise

Work out the following

Addition of fraction
a) same denumerator

$$\frac{2}{7}+\frac{1}{7}=\frac{2+1}{7}=\frac{3}{7}$$

Exercise

Add the following

b) different denumerator

1) The denumerator are not the same

$$\frac{1}{5}+\frac{2}{7}$$

2) × 7 by top and bottom and × 5 by top and bottom

$$\frac{1\times7}{5\times7}+\frac{2\times5}{7\times5}$$

3) Now the denumerators are the same

$$\frac{7}{35}+\frac{10}{35}$$

4) Add

$$\frac{7+10}{35}$$

5) Result

$$\frac{17}{35}$$

Exercise

Add the following

Subtraction of fraction
Same denumerator

$$\frac{2}{7}-\frac{1}{7}=\frac{2-1}{7}=\frac{1}{7}$$

Exercise

Sub the following

Different denumerator

1) The denumerator are not the same

$$\frac{1}{5}-\frac{2}{7}$$

2) × 7 by top and bottom and × 5 by top and bottom

$$\frac{1\times7}{5\times7}-\frac{2\times5}{7\times5}$$

3) Now the denumerators are the same

$$\frac{7}{35}-\frac{10}{35}$$

4) Subtract

$$\frac{7-10}{35}$$

5) Result

$$\frac{-3}{35}$$

Exercise

Sub the following

Type of fractions
. Proper fraction

. Improper fraction

. Mixed fraction

Example of proper fraction

Numerator is smaller than denumerator!

$$\frac{2}{3},\frac{5}{13},...$$

Example of Improper fraction

Denumerator is bigger than numerator!

$$\frac{5}{2},\frac{7}{3},...$$

Example of mixed fraction

a whole number with proper fraction!

$$1\frac{2}{3},3\frac{4}{7},...$$

Here are a list of fractions

Put the following fractions into its proper name

Changing Improper fraction into mixed fraction
Example

$$\frac{8}{3}=2\frac{2}{3}$$

Show working out!

$$\frac{8}{3}=\frac{6+2}{3}=\frac{6}{3}+\frac{2}{3}=2+\frac{2}{3}=2\frac{2}{3}$$

Note!

$$2+\frac{2}{3}$$

is written as

$$2\frac{2}{3}$$

Exercise

Change the following improper fraction into mixed fraction

Exercise

Write the following mixed fraction into short form

Changing mixed fraction into improper fraction
$$2\frac{3}{5}=\frac{13}{5}$$

Show working out!

$$2\frac{3}{5}=\frac{2\times5+3}{5}=\frac{10+3}{5}=\frac{13}{5}$$

Exercise

Change the following mixed fraction into improper fraction

Exercise

Change the following mixed fraction into improper fraction

Multiplication of mixed fraction
$$1\frac{1}{2}\times{2\frac{1}{4}}=\frac{27}{8}$$

Show working out!

First change the mixed fractions into its equivalent improper fraction

$$1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{2+1}{2}=\frac{3}{2}$$

$$2\frac{1}{4}=\frac{2\times4+1}{4}=\frac{8+1}{4}=\frac{9}{4}$$

Multiply them as the normal way!

$$\frac{3}{2}\times{\frac{9}{4}}=\frac{3\times9}{2\times4}=\frac{27}{8}$$

Exercise

Work out the following.

Exercise

Work out the following.

Division of mixed fraction
$$1\frac{1}{2}\div{2\frac{1}{4}}=\frac{12}{18}$$

Show working out!

First change the mixed fractions into its equivalent improper fraction

$$1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{2+1}{2}=\frac{3}{2}$$

$$2\frac{1}{4}=\frac{2\times4+1}{4}=\frac{8+1}{4}=\frac{9}{4}$$

$$\frac{3}{2}\div{\frac{9}{4}}$$

Now inverse the last fraction, and multiply them as the normal way!

$$\frac{3}{2}\times{\frac{4}{9}}$$

$$\frac{3}{2}\times{\frac{4}{9}}=\frac{3\times4}{2\times9}=\frac{12}{18}$$

Exercise

Work out the following.

Exercise

Work out the following.

Addition of mixed fractions
a) Same denumerator

$$1\frac{2}{5}+2\frac{1}{5}=3\frac{3}{5}$$

Show working out

Ignore the whole parts first, just add the fraction normally

$$\frac{2}{5}+\frac{1}{5}=\frac{2+1}{5}=\frac{3}{5}$$

Add the whole parts at the end

$$(1+2)\frac{3}{5}=3\frac{3}{5}$$

Exercise

Work out the following

Subtraction of mixed fractions
a) Same denumerator

$$4\frac{2}{5}-2\frac{1}{5}=2\frac{1}{5}$$

Show working out

Ignore the whole parts first, just subtract the fraction normally

$$\frac{2}{5}-\frac{1}{5}=\frac{2-1}{5}=\frac{1}{5}$$

Subtract the whole parts from each other

$$(4-2)\frac{1}{5}=2\frac{1}{5}$$

Exercise

Work out the following

Rearranging fraction in order of size
Same denumerator

Example

Rearrange the fractions into order, start with smallest to largest

First make sure the denumerators are the same, then rearrange the numerators in order of size from smallest to largest

Exercise

Rearrange the fractions from smallest to largest.

Different denumerator

Before we move into this topic, it is best to learn about factor first, it will make this topic must easy to grasp!

Factors

What is factor?

What are the factors of 6?

1 × 6 = 6

2 × 3 = 6

These numbers 1, 2, 3 and 6 itself are the factors of 6 or

1, 2, 3 and 6 itself are all divisible by 6. These number are the factors of 6.

We are going to use these factors table to help us work out rearranging fractions with the different denumerators.

Example

Rearrange the fraction into order of size, smallest to largest.

Exercise

Rearrange the fraction into order of size, smallest to largest.

Fraction of a value
What is $$\frac{3}{4}$$ of £20 = £15

Show working out

$$\frac{3}{4}{\times20}=\frac{20}{4}{\times3}=5\times3=15$$

Exercise

Work out the following

Fill in the table.

Application of fractions

1) John eats $$\frac{3}{5}$$ of 40 sweets.

How many sweets did he eat?

2) Steve said that $$\frac{4}{5}$$ of £100 is £60

Show your working that Steve is wrong and find the correct amount.

3) Steve eats 15 sweets out of 20 sweets.

What fraction of sweets did Steve eat?

Special case of Add and Subtract of fraction
1)

$$1+\frac{3}{7}= \frac{10}{7}$$

Show working out

$$1+\frac{3}{7}= \frac{7+3}{7}=\frac{10}{7}$$

2)

$$1-\frac{3}{7}= \frac{4}{7}$$

Show working out

$$1-\frac{3}{7}= \frac{7-3}{7}=\frac{4}{7}$$

Exercise

Work out

Factors
What are factors?

It is best to explain through an example.

What are the factors of 10

1) Write down all the priducts of 10

a) 1 × 10 = 10

b) 2 × 5 = 10

These numbers 1, 2, 5, and 10 itself are the factors of 10.

Or we can say 1, 2, 5, and 10 itself are all divisible by 10

Exercise

Find the factors of the following numbers

Example

Fill in the table

Here is a table showing the factors of numbers from 1 to 15

These numbers squence: 2, 3, 5, 7, 11, 13,... are called the prime numbers

or Numbers with only two factors are called prime numbers

2 is the only even prime number and the rest are odd!

Exercise

Find another 10 primes number greater than 13.

Exercise

True or false

Prime products
Example will explain what is prime product (p.p)

Find the prime products of 10

Answer: 10 = 2 × 5

Show working out

The product of 10 are

1) 10 = 1 × 10  (1 and 10 are both factors of 10, but they are not prime numbers)

2) 10 = 2 × 5   (2 and 5 are both factors of 10 and they are both the prime numbers)

so therefore the prime product of 10 are both 2 and 5

First method to work out the prime product of a number

Find the prime product of 12

1) 12 can be break down to 3 × 4 →

a) 12 = 3 × 4 (4 is not a prime)

b) 12 = 3 × 2 × 2  (3 and 2's are both prime numbers)

c) So the prime product of 12 are 3 × 2 × 2

Exercise

Find the prime product of the following

Second method of finding prime product is the tree method

HCF
HCF stands for Highest Common Factors

First method using factor

Example will explain what is HCF.

Find the HCF of 8 and 12

Answer: HCF of 8 and 12 is 4

Show working out

Exercise

Find the HCF of

Second method using prime product

Find the HCF of 8 and 12

HCF of 8 and 12 is 4

Show working out

1) Find the prime product of 8 and 12

2) Prime product of 8 →  8 = (2) × (2) × 2

3) Prime product of 12 →

12 = (2) × (2) × 3

4) Common factors of 8 and 12 →   2 × 2 = 4

Exercise

Find the HCF of

LCM
LCM stands for Lowest Common Multiple

First method of working out the LCM of numbers

Example will explain what is an LCM

Find the LCM of 4 and 6

Answer: The LCM of 4 and 6 is 12

Show working out

1) Write down the multiples of 4 and 6

2) The multiple of 4 are →

4, 8, 12, 16, 20, 24, ...

3) The multiple of 6 →

6, 12, 18, 24, 30, ...

4) Common multiple of 4 and 6 are →

12, 24, ...

5) Lowest Common Multiple of 4 and 6 is →

12

Here is a table of it

Exercise

Find the LCM of the following

Second method of finding the LCM by using prime product

Find the LCM of 4, 8 and 12

The LCM of 4, 8 and 12 is 24

Show working out

1) Write down the prime product of 4, 8 and 12 →

2) Prime product of 4 →

4 = 2 × 2 = 22

3) Prime product of 8 →

4 = 2 × 2 × 2 = 23

4) Prime product of 12 →

12 = 2 × 2 × 3 = 22 × 3

5) Write down the prime numbers that you see!

2 × 3

6) Choice the highest power of 2 and 3

23 × 3

7) Evaluate  23 × 3 = 24

8) The LCM of 4, 8 and 12 is 24 Here is a table of it

Decimals
What is a decimal point?

An example will explain what is a decimal point.

One pound and twenty five pences. How would you write this in number?

Answer = £1.25

From the LHS this section represent the £1  From the RHS this section represent the 25 pences LHS ↓ RHS £1.25                                             ↑                                              This dot(.) between the 1 and 25 is called the decimal point.

So we see that the purpose of this dot(.) called the decimal point is to separate the two different units from each other

Addition of decimal numbers

Example

Work out 11.3 + 0.05

Answer = 11.35

Show working out

a) First arrange these numbers into column and make sure the decmal point are in line.

11.3    + 0.05      11.35

Exercise

Work out the following

1) 123.67 + 1.54         2)   25.63 + 0.654        3)  0.098 + 0.913

Subtraction of decimal numbers

Example

Work out 11.3 - 0.05

Answer = 11.25

Show working out

a) First arrange these numbers into column and make sure the decmal point are in line.

11.3    - 0.05      11.25

Exercise

Work out the following

1) 123.67 - 1.54         2)   25.63 - 0.654        3)  1.098 - 0.913

Multiplication of decimal numbers

Example 1

If 12 × 13 = 156

Work out : 1.2 × 1.3

Answer: 1.2 × 1.3 = 1.56

Show work out

1) Multiply 12 × 13 normally without the decimal points on the numbers

12 × 13 = 156

2) Put back the decimal points on these numbers. Now count the decimal place on each number then add them     1 d.p + 1 d.p = 2 d.ps

1 d.p    1 d.p      Move two spaces from RHS to the LHS and put a decimal point here. ↓        ↓        ↓ 2 1 ← spaces 1 . 2 ×  1 . 3  = 1 . 5 6                        ↓                        2 d.ps

Example 2

If 12 × 13 = 156

Work out : 0.012 × 1.3

Answer: 0.012 × 1.3 = 0.156

Show work out

1) Multiply 12 × 13 normally without the decimal points on the numbers

12 × 13 = 156

2) Put back the decimal points on these numbers. Now count the decimal place on each number then add them     3 d.p + 1 d.p = 4 d.ps

3 d.ps         1 d.p      Move fours spaces from RHS to the LHS and put a decimal point here. ↓             ↓        ↓ 4 3 2 1 ← spaces 0 . 0 1 2  ×  1 . 3  = 0 . 0 1 5 6                             ↓                             4 d.ps

Example 3

If 12 × 13 = 156

Work out : 1200 × 0.13

Answer: 1200 × 0.13 = 156

Show work out

1) Multiply 12 × 13 normally without the decimal points on the numbers

1200 × 13 = 15600

2) Put back the decimal points on these numbers. Now count the decimal place on each number then add them     0 d.p + 2 d.p = 2 d.ps

0 d.p      2 d.p             Move two spaces from RHS to the LHS and put a decimal point here. ↓              ↓ 2 1 ← spaces 1 2 0 0 ×  0 . 1  3  = 1 5 6 . 0 0                                 ↓                                 2 d.ps

Exercise

If 12 × 12 = 144

Work out the following

Division of decimal numbers
To clearify matter

$$6\div3$$ it is the same as $$\frac{6}{3}$$

$$\frac{6}{3}$$ → $$\frac{numerator}{denumerator}$$

Rules for dividing decimal numbers

Numerator:

1) When the numerator decreases in value, the answer is decrease in value

2) When the numerator increases in value, the answer is increase in value

Denumerator:

1) When the denumerator decreases in value, the answer is increase in value

2) When the denumerator increases in value, the answer is decrease in value

Need teacher to explain this:

Sumamarise of rules

Decrease → Decrease Numerator (same): Increase → Increase

Decrease → Increase Denumerator (Opposite): Increase → Decrease

Example 1

If $$6\div3=2$$

Work out

$$\frac{60}{0.03}$$

Answer :

$$\frac{60}{0.03}=2000$$

Show working out

1) Divide 6 by 3 normally, to get an answer.

6 ÷ 3 = 2

2) Put the number back normally. Now apply the rules of the numerator and denumerator

Numerator ↓ increase by 1 position _   60 ÷ 0.03 = 2 0 0 0                     ↑ increase by 2 position Denumerator

Example 2

If $$6\div3=2$$

Work out

$$\frac{0.6}{300}$$

Answer :

$$\frac{0.6}{300}=0.002$$

Show working out

1) Divide 6 by 3 normally, to get an answer.

6 ÷ 3 = 2

2) Put the number back normally. Now apply the rules of the numerator and denumerator

Numerator ↓ decrease by 1 position _          0.6 ÷ 300 = 0.0 02                    ↑ decrease by 2 position Denumerator

Exercise

If 8 ÷ 4 = 2 then,

Work out the following

Exercise

Work out the following

Changing decimal into fraction
Example 1

Change 0.2 into fraction

Answer :

$$0.2=\frac{2}{10}$$

Show working out

1) Put the 2 onto the numerator

2) Because it is 1 decimal place, the denumerator is a 10

Example 2

Change 0.02 into fraction

Answer :

$$0.02=\frac{2}{100}$$

Show working out

1) Put the 2 onto the numerator

2) Because it is 2 decimal place, the denumerator is a 100

Exercise

Change the following decimal into fraction

Change fraction into decimal

Example 1

Change $$\frac{4}{100}$$ into decimal

Answer :

$$\frac{4}{100}=0.04$$

Show working out

1) Write down the numerator number   4

2) Denumerator is 100, it means you have to move two

decimal places of the numerator number from RHS to LHS

LHS ← RHS 2 1 ← position 4 ÷ 100 = 0 . 0 4                 ↑                2 decimal places Exercise

Change the following fraction into decimal

Partition of decimal numbers
An example be nice to explain this matter!

Before we go straight to decimal numbers, we first deal with pure numbers.

Example 1

Partition the number 23 into tens and units

$$23=20+3$$

Example 2

Partition the number 123 into hundred, tens and units

$$123=100+20+3$$

Exercise

Partition the following list of numbers

Now we deal with partition of decimal number.

Example 1

Parition the number 0.23 into tenths and hundredths

Answer :

$$0.23 =0.2+0.03$$

Show working out

1st decimal place ↓ 2nd decimal place ↓ ↓     1 2  → two decimal places 0. 2 3

1) Write down the 1st decimal place

0.2

2) Write the 2nd decimal place, but you have remove the number 2 and place it with a zero   0.03

3) So we have ...  0.23 = 0.2 + 0.03

Example 2

Parition the number 0.123 into tenths, hundredths and thousandths

$$0.123 =0.1+0.02+0.003$$

Exercise

Parition the following list of numbers

Exercise

Partition the following list of numbers

Partition of decimal numbers into fraction

Example 3

Partition number 0.23 into fraction

$$0.23=0.2+0.03=\frac{2}{10}+\frac{3}{100}$$

Exercise

Partition the following decimal numbers into fractions

Example 4

Partition number 1.423 into fraction

$$1.423=1+0.4+0.02+0.003=1+\frac{4}{10}+\frac{2}{100}+\frac{3}{1000}$$

Exercise

Partition the following decimal numbers into fractions

Change partitoned numbers back into decimals numbers
Example

Change partitoned numbers back into decimals numbers

$$0.2+0.06$$

Answer :

$$0.2+0.06 = 0.26$$

Show working out

1) Add the numbers by rearranging it in column, but make sure you have to place the decimal point in line with each other

0.2     + 0.06     ---        0.26

Exercise

Change partitoned numbers back into decimals numbers

Exercise

Change partitoned numbers back into decimals numbers

Change the partitioned fractions into decimal numbers

Example 1

Change the partitioned fraction into decimal number

$$\frac{3}{10}+\frac{5}{100}$$

Answer :

$$\frac{3}{10}+\frac{5}{100}=0.35$$

Show working out

1) First change the partitioned fraction into decimal numbers and then add them up as normal!

$$\frac{3}{10}+\frac{5}{100}=0.3+0.05$$ 0.3  + 0.05   --     0.35

2) And this is the answer!

$$\frac{3}{10}+\frac{5}{100}=0.3+0.05=0.35$$

Exercise

Change the partitioned fractions into decimal numbers

Changing recurring decimals to fractions
$$\frac{1}{3}=0.333...$$ → This fraction gives a recurring decimal

This recurring decimal it is written in a short form for neat purpose .                0.333... = 0.3

The dot on top of the number 3, showing that the number 3 is recurring.

Example

Write the following recurring decimals into it equivalent short form notation

1 → 0.3444...

2 → 0.343434...

3 → 0.012555...

Answers: . 1 → 0.3444... = 0.34                      ..  2 → 0.343434... = 0.34                         .  3 → 0.012555... = 0.0125

Exercise

Write the following recurring decimals into it equivalent short form notation

Recurring decimal

Example

Change the following recurring decimals into fractions . 1. 0.4 = $$\frac{4}{9}$$ .. 2. 0.45 = $$\frac{45}{99}$$

Show working out

1 represents no non-recurring digit between decimal pont and recurring number 4 ↑. 0 . 4        ↓      10 represents one recurring number after the decimal pont

. 0.4 = $$\frac{9}{10-1}=\frac{4}{9}$$

1 represents no non-recurring digit between decimal pont and recurring number 4 ↑. . 0 . 4 5        ↓       100 represents two recurring numbers after the decimal point .. 0.45 = $$\frac{45}{100-1}=\frac{45}{99}$$

Exercise

Change the following recurring decimal numbers to fractions

Recurring and non recurring decimals

Example

Change the following recurring decimal numbers to fractions . 1 → 0.23 = $$\frac{21}{90}$$

. 2 → 0.243 = $$\frac{219}{900}$$

Show working out

10 represents one non-recurring digit after the decimal point ↑ .  0 . 2 3       ↓      100 represents two numbers (Recurring and non-recurring) after the decimal point

.  0 . 2 3 = $$\frac{23-2}{100-10}=\frac{21}{90}$$

100 represents two non-recurring digits after the decimal point ↑ .   0 . 2 4 3        ↓      1000 represents three numbers (Recurring and non-recurring) after the decimal point

The number from the decimal point(243) take away the non-recurring number(24) ↑   ↑          .   0 . 2 4 3 = $$\frac{243-24}{1000-100}=\frac{219}{900}$$

Exercise

Change the following recurring decimal numbers into fractions

Imperial measurements (Mass)
Here is a table of conversion for mass.

Conversion from large to small units

Rule for conversion from large to small units, we multiply

Example

Convert the following:

1) 10kg to pounds

2) 10 Stones kilograms

3) 5 stones to pounds

Answers :

1) 10kg to pounds = 22 pounds

2) 10 stones to kilograms = 63kg

3) 5 stones to pounds = 70 pounds

Show working out 1) We use the table

Conversion from small to large units

Rule for conversion from small to large units, we divide

Example

Convert the following:

1) 1000 pound to kg

2) 20 kg to stones

3) 100 pounds to stones

Answers :

1) 1000 pounds to kg = 454.5kg

2) 20kg to stones = 3.2st

3) 100 pounds = 7.2st

Show working out 1) We use the table

Imperial measurements (Lenghts)
Here is a table of conversion for lenghts.

Exercise

Correct the following units and its names

Conversion from large to small units

Conversion from large to small units we multiply

Use the above table to help you to convert the following units

Example

Convert :

1) 12 metres to feet

2) 100 metres to yards

3) 15 miles to yards

4) 200 yards to cm

Answers :

1) 12 metres = 39.36 feet

2) 100 metres = 1093.6 yards

3) 15 miles = 26400 yards

4) 200 yards = 18288 centimetres

Show working out

Exercise

Convert the following units.

Conversion from small to large units

Conversion from small to large units we divide

Example

Convert the following units

1) 40 yards to metres

2) 2500 metres to miles

3) 400,000 centimetre to kilometres

Answers:

1) 40 yards = 56.8 metres

2) 2500 metres = 1.55 miles

3) 400,000 centimetres = 4 kilometres

Show working out

Use the table above to help you to convert from units to units

Exercise

Convert the following units to units

1) 500000 millimetres to miles

2) 6000 metres to miles

3) 12500 yards to kilometres

Imperial measurements (Currency)
Conversion table for currency

Conversion from large to small we multiply

Example

Convert the following units

1) 125 pounds to Canadian dollars

2) 12 Euros to Australian dollars

3) 35 pounds to Australian dollars

Answers:

1) 125 pounds = 203.63 canadian dollars

2) 12 Euros = 17.09 australian dollars

3) 35 pound = 60.9 australian dollars

Use the the table of conversion to work out

Conversion from small to large we divide

Imperial measurements (Time)
Here is a conversion table for time

Conversion from large to small units

Conversion from small to large units

Imperial measurements (Speed)
Speed of an object are measured in the following units of measurement

Changing from units to units

Here we are going to show you how to change from one unit to another unit of measurements

Example 1

Change 36km/h to m/s

Answer:

10m/s

Show working out

1. Change km to m

36km = 36×1000 = 36000m

2. Change h to s

1h = 1×3600 = 3600s

3. 36km/h = $$\frac{36000}{3600}$$ = 10m/s

Exercise

Change the following km/h to m/s

Example 2

change 225mph to km/s

1. Change mile to km

2250 miles = 225 × 1.6 = 3600km

2. Change h to s

1h = 1 × 3600 = 3600s

3. 2250mph = $$\frac{3600}{3600}$$

= 1km/s

Exercise

Change the following mph to km/s

Percentage
Example will explain what are percentages

The following fractions has special names

Any fractions that are out of 100 are called percentages

$$\frac{1}{100}$$___: has an equivalent symbol of 1%

This is read as one percent.

Exercise

Fill in the table

Note: The denumerators are sometimes called a base

Changing a fraction of none base of 100 into the equivalent base of 100

$$\frac{3}{10}=\frac{30}{100}$$

The fraction $$\frac{30}{100}$$ is the equivalent of $$\frac{3}{10}$$

Show working out

$$\frac{3}{10}=\frac{3\times10}{10\times10}=\frac{30}{100}$$

Exercise

Change the following none base of 100 into the its equivalent base of 100

Change fraction into percentage

Change $$\frac{3}{10}$$ into percentage

answer: $$\frac{3}{10}$$ = $$30\%$$

Show working out

1) Change $$\frac{3}{10}$$ into its equivalent base of 100

$$\frac{3}{10}=\frac{3\times10}{10\times10}=\frac{30}{100}$$

2) $$\frac{30}{100}$$ = $$30\%$$

Exercise

Change the following none base of 100 into the its equivalent base of 100

Changing decimal into percentage

Example

Change 0.4 into percentage

Answer = $$40\%$$

Show working out

1) Multiply the decimal by 100

$$0.4\times100=40\%$$

Exericse

Change the following decimals into percentages

Changing fraction into decimal

Example 1

Change $$\frac{30}{100}$$ into decimal

answer = $$0.30$$

Show working out (Need teacher to explain this on the board)

1) Move two spaces from RHS of the number 30 to the LHS and place there a decimal point.  Moving two spaces, because of 2's zeros                    LHS ← RHS                   2 1 ← spaces    Decimal point → 0.3 0

Example 2

Change $$\frac{3}{5}$$ into decimal

answer = $$0.6$$

Show working out (Need teacher to explain this on the board)

1) First change $$\frac{3}{5}$$ into its equivalent fraction with the base of either 10, 100, ...

2) $$\frac{3}{5}=\frac{3\times2}{5\times2}=\frac{6}{10}$$

3) Move one spaces from RHS of the number 6 to the LHS and place there a decimal point.  Moving two spaces, because of 1 zero                  LHS ← RHS                   1 ← spaces    Decimal point → 0.6

Exercise

Change the following fractions into its decmal values

Exercise

Change the following fractions into deciamls

Percentage, Fraction and Decimal

Exercise

Fill in the table

Percentage of a Value
Example

What is 20% of £200?

Answer = £40

Show working out

1) Change 20% into fraction

$$20\%=\frac{20}{100}$$

2) Now multiply this fraction by the value

$$\frac{20}{100}\times200=\frac{200}{100}\times20=2\times20=40$$

or use this method. Just multiply the percentage with the value and move the answer back two decimal places

LHS ← RHS 2 1 ← Space 20 × 200 = 4 0 .0 0                 ↑                  Decimal point

Exercise

Work out the percentages of the following values

Value increase by certain percentage

Example

Sally wages is £400.00 and it is increase by 20%. What is Sally new wage?

Answer = £480

Show working out

1) First calculate what is 20% of £400

$$\frac{20}{100}\times400=80$$

2) Now add this value of £80 to £400, which gives £480.

Exercise

Work out the percentages increase of the following values

Value decrease by certain percentage

Example

Sally wages is £400.00 and it is decrease by 20%. What is Sally new wage?

Answer = £320

Show working out

1) First calculate what is 20% of £400

$$\frac{20}{100}\times400=80$$

2) Now subtract this value of £80 from £400, which gives £320.

Exercise

Work out the percentages decrease of the following values

General formula for calculating percentage
100%                        100±%    _______           ±%          _______   |       |       _______       |       |   |       |      |       |      |       |   |       |      |_______|      |       |   |_______|                     |_______|                     Now Original        value          After value                         value

This symbol ± represents here to increase or decrease in a value

Example 1

A T.V costs £200 in Dixon. Dixon is offering a 30% off discount.

Calculate:

What is the cost of the T.V after the sale price?

What is the value of the discount price?

1. First it is best to translate these information into a box diagram as shown below.

100%-30%                                    ↑         100%           30%            70%    _______        Discount       _______ |      |       _______       |       |   |  T.V  |      |       |      |       | |      |      |_______|      |       |   |_______|                     |_______|                  Discount £200         price         Sale price Show working out

2. Second, find the value for 1%.

1% = $$\frac{OriginalValue}{Percentage}=\frac{NowValue}{Percentage}=\frac{AfterValue}{Percenatge}$$

1% = $$\frac{OriginalValue}{Percentage}$$ → We are given only this information

1% = $$\frac{200}{100}$$

= £2.00

3. To find the sale price

Sale price = Value of 1% × 70% = 2 × 70             = £140.00

4. To find the discount price

Discount price = Value of 1% × 30% = 2 × 30                 = £60.00

Example 2

Dixon is offering a 30% off discount of a T.V, which means the customer is saving £60.00 worth of money.

Calculate:

What is the original price of the T.V?

What is the value of the price his is paying for?

1. It is best try to put these words into a diagram where it will help you to understand the question, like the one below.

100%-30%                                    ↑         100%           30%            70%    _______        Discount       _______ |      |       _______       |       |   |  T.V  |      |       |      |       | |      |      |_______|      |       |   |_______|                     |_______|                     £60     Original                      Sale price price

Show working out

2. Second, find the value for 1%.

1% = $$\frac{OriginalValue}{Percentage}=\frac{NowValue}{Percentage}=\frac{AfterValue}{Percenatge}$$

1% = $$\frac{NowValue}{Percentage}$$

1% = $$\frac{60}{30}$$ → This is what we are been given

= £2.00

3. To find the original price

Original price = Value of 1% × 100% = 2 × 100             = £200.00

4. To find the sale price

Sale price = Value of 1% × 70% = 2 × 70

= £140.00

5. Or, we can calculate in this way

Sale price = Original price - Discount price = 200 - 60

= £140

Exercise

Fill in the table

Ratios
Symbol uses in ratio is

Example

1) 2 : 4

2) 1 : 2 : 5

What is ratio?

Examples will be very nice to explain what it is ...

Ratio are used in comparing measurement and dividing quantities

Ratio is another way of writing fraction

An example to illustrate this situation...

This family have five children. two of which are boys and other three are girls.

Fraction of boys → $$\frac{2}{5}$$

Fraction of girls → $$\frac{3}{5}$$

but we can write this in another way such as boys to girls 2 to 3 or 2:3. This is called ratio

Writing situation as in ratios

Example 1

15 pupils in total play sports.

3 pupils play tennis

5 pupils play football and

7 pupils play base ball

Write this as ratio.

Answer :

tennis : football : base ball

3  :     5    :    7

Exercise

Write the following situation as ratio

Example 2

12 students play the following sports, tennis, football and chess in the ratios of 1:3:8 respectively.

How many students play,chess tennis and football?

Answer :

play chess = 8

play tennis = 1

play footabll = 3

Exercise

Work out how many pupils play in each sport

Exercise

Fill in the table

Simplifying ratios
Example 1

Simplify this ratio 4 : 6 : 10

Answer → 2:3:5

Show working out

1) Write the ratio as prime product

2×2 : 2×3 : 2×5

2) Remove the common factors from the ratio by dividing by itself

( 2×2 : 2×3 : 2×5 ) ÷ 2

3) And we have the simplified ratio!   2 : 3 : 5

Exercise

Simplify the following ratios

Changing ratios to fractions
Example

Change the ratio into fraction

The ratios of football to cricket are respectively 2 : 3

What fractions are football and cricket

Answers :

1) fraction of football → $$\frac{2}{5}$$

2) fraction of cricket → $$\frac{3}{5}$$

Show working out

1) Add up the ratios and we get the total of the ratios as

2 + 3 = 5

2) Write the ratios as fractions by respectively put the numerator over the total ratios

1) fraction of football → $$\frac{football}{total}=\frac{2}{5}$$

2) fraction of cricket → $$\frac{cricket}{total}=\frac{3}{5}$$

Exercise

Change the following ratios into fractions

Exercise

Fill in the table

Work out the value of ratios
Example

5 children shared 30 sweets in the ratios boys to girls respectively 2 : 3

How many sweets does the boys and girls get?

Answer :

1) The boys will get 12 sweets

2) The girls will get 18 sweets

Show working out

1) First change the ratios into fractions

Ratios boys to girls → 2 : 3 fraction of boys → $$\frac{2}{5}$$

fraction of girls → $$\frac{3}{5}$$

2) Now multiply the fractions of boys and girls by the value 30

The boys will get → $$\frac{2}{5}\times30=12$$ sweets

The girls will get → $$\frac{3}{5}\times30=18$$ sweets

Exercise

Work out how many sweets does the boys and girls in their sharing of sweets

Exercise

Fill in the table

Directly proportion
What is directly proportion means?

An increase in a value leads to the increase in another value

A decrease in a value leads to the decrease in another value

Increase → Increase Rules of directly proportion: Decrease → Decrease

Example 1

An increase situation

A book costs £2.00

Two books cost £4.00 (Increase in book quantities increase in prices )

Example 2

A decrease situation

Two books cost £4.00

A book costs £2.00 (Decrease in book quantities decrease in prices)

Calculating directly proportion situation

Example 1

A pen costs 5 pences. Find the cost of ten pens

Answer : The cost of 10 pens = 50 pences

Show working out

1) The cost of 1 pen multiply that by the quantities

cost of 1 pen = 5 pences

cost of 10 pens = 5 × 10

Therefore the cost of 10 pens = 50 pences

Rule for calculating :

Cost of quantities of items = Cost of 1 item × quantities

Exercise

Find the cost of the following items

Example 2

3 books cost £15. Find the cost of 7 books.

Answer :

Cost of 7 books = £35

Show working out

1) First find the cost one book.

cost of one book → $$\frac{15}{3}=5$$

2) Now multiply the cost of one book by the qauanties

£$$5\times7$$

Therefore the cost of 7 books = £35

Exercise

Find the cost of the following books

Further ratios
This table gives the complete answer to a question

Let create a question from this table.

An amount(£) is shared between two children, Bob and Steve in the ratios 2:3 respectively.

Bob gets £20.00 for his portions.

Calculate:

1. The total amount that is shared between these two children.

2. How much did Steve gets for his portions?

Show working out

1 → First we construct a table and fill in the given information

2 → Second we begin our calculations.

We known that Bob gets £20.00 for his 2 portions of share.

2 portions → £20.00 so

[1 portion → £10.00]

We known that Steve gets 3 portions of share, so if one portion is £10.

[3 portions → £30.00], so Steve gets £30.00 of his shared

We known that Bob gets 2 portions and Steve gets 3 portions, so in total there are 5 portions.

[Again 1 portion is £10.00 worth, so 5 portions are worth £50.]

The total amount shared between them is £50.00

3. Thirdly we put these results into the table and we get ...

Exercise

Fill in the table

Inversely proportion
What is inversetly proportion means?

An increase in a value leads to the decrease in another value

A decrease in a value leads to the increase in another value

Increase → Decrease Rules of directly proportion: Decrease → Increase

Example 1

6 workers take 18 days to do a job.

9 workers would take shorter to do a job than 6 workers, because the numbers

of workers has increase (Increase in one quantity leads to the decrease in another quantity)

Example 2

9 workers take 12 days to do a job.

6 workers would take longer to do a job than 9 workers, because the numbers

of workers has decrease (Decrease in one quantity leads to the inecrease in another quantity) Calculating inversely proportion of a quantity

Example 1

6 workers take 18 days to do a job. How many days would it takes 9 workers.

Answer : 9 workers take = 12 days

Show working out

1) First find how long one worker will take to do the job.

Simply multiply the number of workers by the days

One worker takes → 6 × 18 = 108 days

2) Now divide this 108 days by 9 workers. This will give you the how long   9 workers will take to do the job.

Nine workers take → 108 ÷ 9 = 12 days

Exercise

Find how long it would take workers to do a certain job.

Use the table to help you answers the following questions

Probability
What is probabilty?

Let give an example. Just say you have two marbles red and green on the table. Close you eyes and try and pick up a red marble. So what is the chance of you picking up the red marble at a random situation?

Of course, the answer could be either I pick up a red or a green. So we say the chane are not certainty but it is a 50/50 of the chance of either colours.

Probabilty it is about estimating the possible of coming close to an answer in a randomness situation with know certain fact that you know.

In probabilty we often used fractions, percentages and decimals to represent the outcome of an event

Examples of these units used in probability

Fraction

The chance of me coming to school is $$\frac{3}{7}$$, so the chance of me not coming to school is $$\frac{4}{7}$$

Percentage

The chance of me coming to school is 40%, so the chance of me not coming to school is 60%

Decimal

The chance of me coming to school is 0.3, so the chance of me not coming to school is 0.7

Notice that the probabilities (chances) of an event happening and not happening is always sum up to one

Probability of an event occuring + Probability of an event not occuring = 1

$$\frac{3}{7}+\frac{4}{7}=1$$

40% + 60% = 100%

0.3 + 0.7 = 1.0

The above formula it is written as

P(Event Occurs) + P(Event Not Occurs) = 1

We will use these primary formulae to represent the chane of events occuring and not occuring

P(Event occuring)$$=\frac{chosen}{choice}$$

P(Event not occuring) = 1 - P(Event occuring)

Exercise

Find the probaility of an event is not occuring

Example

On a table, four cards(Card 1, Card 2, Card 3 and Card 4) are placed facing upward, each with a number written on it, as shown below

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___1___|  |___1___|   |___1___|  |___1___| First situation ;

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___1___|  |___1___|   |___1___|  |___1___|

Now face all the cards the other way to hide the numbers

What is the probability at random picking a card with a number one on it?

Answer : Obviously it is a certainy of 100% a number one

Althought we know it for certain the outcome, now let work it out mathematically using the definition of calculating the probability of an event, such as this.

Show working out

1. Write the formula for calculating the probability of an event P(Event occuring)$$=\frac{chosen}{choice}$$

2. Choice ; on the table we have four choice of cards

3. Chosen ; we are only going to choose a card with a number one out of the four cards at random

Question: How many number one card are there out of the four cards that are laid on the table?

Answer : Four cards are with number one

4. Apply the formula

P(Picking a number one)$$=\frac{4}{4}$$

= 1 or 100%

Second situation ;

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___1___|  |___1___|   |___1___|  |___4___|

Now face all the cards the other way to hide the numbers

What is the probability at random picking a card with a number one on it?

Answer : The answer is no longer a certainty, roughly 75% chance will be a number one card because a one card has a different value

Show working out

1. Write the formula for calculating the probability of an event P(Event occuring)$$=\frac{chosen}{choice}$$

2. Choice ; on the table we have four choice of cards

3. Chosen ; we are only going to choose a card with a number one out of the four cards at random

Question: How many number one card are there out of the four cards that are laid on the table?

Answer : Three cards are with number one

4. Apply the formula

P(Picking a number one)$$=\frac{3}{4}$$

= 0.75 or 75%

Third situation ;

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___1___|  |___1___|   |___2___|  |___4___|

Now face all the cards the other way to hide the numbers

What is the probability at random picking a card with a number one on it?

Answer : The answer is no longer a certainty, roughly 50% chance will be a number one card because two cards has a different value

Show working out

1. Write the formula for calculating the probability of an event P(Event occuring)$$=\frac{chosen}{choice}$$

2. Choice ; on the table we have four choice of cards

3. Chosen ; we are only going to choose a card with a number one out of the four cards at random

Question: How many number one card are there out of the four cards that are laid on the table?

Answer : Two cards are with number one

4. Apply the formula

P(Picking a number one)$$=\frac{2}{4}$$

= 0.5 or 50%

Fourth situation ;

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___1___|  |___2___|   |___3___|  |___4___|

Now face all the cards the other way to hide the numbers

What is the probability at random picking a card with a number one on it?

Answer : The answer is no longer a certainty, roughly 20% chance will be a number one card because two cards has a different value

Show working out

1. Write the formula for calculating the probability of an event P(Event occuring)$$=\frac{chosen}{choice}$$

2. Choice ; on the table we have four choice of cards

3. Chosen ; we are only going to choose a card with a number one out of the four cards at random

Question: How many number one card are there out of the four cards that are laid on the table?

Answer : One card are with number one

4. Apply the formula

P(Picking a number one)$$=\frac{1}{4}$$

= 0.25 or 25%

Fifth situation ;

Card 1   Card 2      Card 3     Card 4 _______   _______     _______    _______   |___0___|  |___2___|   |___3___|  |___4___|

Now face all the cards the other way to hide the numbers

What is the probability at random picking a card with a number one on it?

Answer : The answer is a certainty of zero chance, because four cards has a different value

Show working out

1. Write the formula for calculating the probability of an event P(Event occuring)$$=\frac{chosen}{choice}$$

2. Choice ; on the table we have four choice of cards

3. Chosen ; we are only going to choose a card with a number one out of the four cards at random

Question: How many number one card are there out of the four cards that are laid on the table?

Answer : Zero card are with number one

4. Apply the formula

P(Picking a number one)$$=\frac{0}{4}$$

= 0

Probability of a single event
Below is a table show a list of common events in probability

Example 1

Find the probability of getting a 5 on rolling a fair dice.

Answer: $$\frac{1}{6}$$

Show working out

1. Choice ; 6 faces

2. Chosen ; a face with 5 number written on it.

Question: How faces on a dice have a number 5 written on it?

Answer: only one face

$$Probability(5)=\frac{1}{6}$$

Example 2

In a bag there are 10 marbles. 3 Red, 5 Green and 2 Blue. Find the probability of randomly choosing a red colour in the bag.

Answer: $$\frac{3}{10}$$

Show working out

1. Choice ; 10 marbles

2. Chosen ; a red marble

Question: How many red marbles are there?

Answer: only three marbles are red

$$Probability(RedMarble)=\frac{3}{10}$$

Exercise

Probability of double Events
Examples of double events in probability

'''Tom tosses a fair dice first then a coin second. What is the probability that he gets a 5 in a dice and a head in a coin'''

Answer : P(5 and Head) = $$\frac{1}{12}$$

'''Tom tosses a fair dice first then a coin second. What is the probability that he gets a 5 in a dice or a head in a coin'''

Answer : P(5 or Head) = $$\frac{4}{6}$$

So the words and and or indicate that this is the probabilities of two or events

Example 1

Tom tosses a fair dice first then a coin second. What is the probability that he gets a 5 in a dice and a head in a coin

Answer : P(5 and Head) = $$\frac{1}{12}$$

Show working out

1. Calculate the probabilities of the events of the dice and coin independently

P(5 of a dice) = $$\frac{1}{6}$$

P(Head of a coin) = $$\frac{1}{2}$$

2. The word and used in probability, here means multiplication. Now multiply the two events together

P(5 and Head) = P(5 of a dice) × P(Head of a coin)

= $$\frac{1}{6}\times{\frac{1}{2}}$$

= $$\frac{1}{12}$$

Example 2

Tom tosses a fair dice first then a coin second. What is the probability that he gets a 5 in a dice or a head in a coin

Answer : P(5 or Head) = $$\frac{4}{6}$$

Show working out

1. Calculate the probabilities of the events of the dice and coin independently

P(5 of a dice) = $$\frac{1}{6}$$

P(Head of a coin) = $$\frac{1}{2}$$

2. The word or used in probability, here means addititon. Now add the two events together

P(5 or Head) = P(5 of a dice) + P(Head of a coin)

= $$\frac{1}{6}+{\frac{1}{2}}$$

= $$\frac{1}{6}+\frac{3}{6}$$ = $$\frac{4}{6}$$

Exercise

Calculate the probability of the following events

Probability of indepenable events
Example

Charlies has a bag contains 10 marbles, 2 red, 3 black and 5 yellow colours respectively.

He take a marble at random and record down its colour and replace the marble back into the bag.

The probability charlies takes a red is $$\frac{2}{10}$$ → First event

The probability charlies takes a black is $$\frac{3}{10}$$ → Second event

The probability of the second event is totally independent from the probability of the first event.

Why? Because, everytime he takes a colour at random and replaces it back, the probability of taken a marble is the same through out any number of event, because in the bag there is still 10 marbles.

The probabilities of any number of events that does not change the total quantity, this is known as an indedpendable event.

Probability of depenable events
Example

Charlies has a bag contains 10 marbles, 2 red, 3 black and 5 yellow colours respectively.

He take a marble at random and record down its colour and does not replace the marble back into the bag.

The probability charlies takes a red is $$\frac{2}{10}$$ → First event

The probability charlies takes a black is $$\frac{3}{9}$$ → Second event

The probability of the second event is totally dependent from the probability of the first event.

'''Why? Because, the total marbles in the bag has reduced by one marble. So the probability of getting a black marble is more of the chance than when the total quantity was 10 marbles in it.'''

The probabilities of any number of events that does change the total quantity, this is known as a dependable event.

Example Charlies has a bag contains 10 marbles, 2 red, 3 black and 5 yellow colours respectively.

He take a marble at random and record down its colour and does not replace the marble back into the bag.

The probability charlies takes a red is $$\frac{2}{10}$$ → First event

The probability charlies takes a red is $$\frac{1}{9}$$ → Second event

The probability charlies takes a black is $$\frac{3}{8}$$ → Third event

The probability charlies takes a black is $$\frac{2}{7}$$ → Fouth event

The probability charlies takes a yellow is $$\frac{5}{6}$$ → Fifth event

and so on ...

Exercise

Calculate the probabilities of the following depenable events (the marble is not replaced back into the bag)

Further probability of double depenable events
AND FUNCTION ; Both, Same

Example 1

The sweet container has 3 red, 5 blue and 7 orange sweets. Thomas takes one sweet and eats it and later he takes another one and eats it also.

Calculate the probability that he will eat two blue sweets.

Answer: P(Two blue sweets) = $$\frac{2}{21}$$ ≈ 10% → Roughly 10% chances that he will eat two blue sweets

Show working out

1. First work out the porbability that he eats the first blue sweet

P(First blue sweet) = $$\frac{5}{15}$$

2. Second work out the probability that he eats the second blue sweet, but one sweet has gone from the container

P(Second blue sweet) = $$\frac{4}{14}$$

3. The probability that he will eat the two blue sweets is an and function

means that we have two multiply these two probabilties together

P(Two blue sweets) = $$\frac{5}{15}\times{\frac{4}{14}}$$ = $$\frac{20}{210}$$ = $$\frac{2}{21}$$ = $$\frac{20}{210}\times100$$ ≈ 10%

Exercise

Calculate the probabilities of the following

OR FUNCTION ; Either, any

Example 2

A sweet container has 3 red, 5 blue and 7 orange sweets. Thomas takes one sweet and eats it and later he takes another one and eats it.

Calculate the probability that he will eat two sweets.

a. The first one will be red sweet and the second one could be of any colours

So the possible outcomes are as follow:

P(Red and red sweets) or  P(Red and blue sweets) or   P(Red and orange sweets)

P(Red and red sweets) = $$\frac{3}{15}\times{\frac{2}{14}}=\frac{1}{35}$$

P(Red and blue sweets) = $$\frac{3}{15}\times{\frac{5}{14}}=\frac{1}{14}$$

P(Red and orange sweets) = $$\frac{3}{15}\times{\frac{7}{14}}=\frac{1}{10}$$

P(Red or any colours) = P(Red and red sweets) + P(Red and blue sweets) + P(Red and orange sweets)

P(Red or any colours) = $$\frac{1}{34}+\frac{1}{14}+\frac{1}{10}=\frac{239}{1190}$$

Exercise

This one had been done for you

Calculate the probability of the following

Exercise

Calculate the probability of the following

R; Red sweet G; Green sweet B; Black sweet