User:General maths

user:hashim100

Fraction
Addition and subtracting of fraction

Question 1

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

The denumerator is not the same in both fractions, so can not be added

To add fraction the denumerator must be the same

Solution:

Step 1:

$$\cdots\frac{1}{5}\cdot{\frac{7}{7}}+\frac{1}{7}\cdot{\frac{5}{5}}$$

Step 2:

$$\cdots\frac{7}{35}+\frac{5}{35}$$

Same denumerator, just add the numerator

Step 3:

$$\cdots\frac{7+5}{35}$$

Step 4:

$$\cdots\frac{12}{35}$$

Step 5:

$$\cdots\frac{1}{5}+\frac{1}{7}=\frac{12}{35}$$

Question 2

$$\cdots\frac{1}{x}+\frac{1}{y}$$

Solution

Step 1:

$$\cdots\frac{1}{x}\cdot{\frac{y}{y}}+\frac{1}{y}\cdot{\frac{x}{x}}$$

Step 2:

$$\cdots\frac{y}{xy}+\frac{x}{xy}$$

Step 3:

$$\cdots\frac{y+x}{xy}$$

Step 4:

$$\cdots\frac{1}{x}+\frac{1}{y}=\frac{y+x}{xy}$$

Question 3

$$\cdots\frac{1}{x-1}+\frac{1}{x+2}$$

Solution

Step 1:

$$\cdots\frac{1}{x-1}\cdot{\frac{x+2}{x+2}}+\frac{1}{x+2}\cdot{\frac{x-1}{x-1}}$$

Step 2:

$$\cdots\frac{x+2}{(x-1)(x+2)}+\frac{x-1}{(x-1)(x+2)}$$

Step 3:

$$\cdots\frac{x+2+x-1}{(x-1)(x+2)}$$

Step 4:

$$\cdots\frac{2x+1}{(x-1)(x+2)}$$

Step 5:

$$\cdots\frac{1}{x-1}+\frac{1}{x+2}=\frac{2x+1}{(x-1)(x+2)}$$

Question 4

$$\cdots\frac{1}{2(x+3)}+\frac{1}{(x+2)^2}$$

Solution

Step 1:

$$\cdots\frac{1}{2(x+3)}\cdot{\frac{(x+2)^2}{(x+2)^2}}+ \frac{1}{(x+2)^2}\cdot{\frac{2(x+3)}{2(x+3)}}$$

Step 2:

$$\cdots\frac{(x+2)^2}{2(x+31)(x+2)^2}+\frac{2(x+3)}{(x+3)(x+2)^2}$$

Step 3:

$$\cdots\frac{(x+2)^2+2(x+3)}{(x+3)(x+2)^2}$$

Step 4:

We need to open bracket this one

$$\cdots(x+2)^2=(x+2)(x+2)=x(x+2)+2(x+2)=x^2+2x+2x+4=x^2+4x+4$$

Step 5:

$$\frac{x^2+4x+4+2(x+3)}{(x+3)(x+2)^2}$$

Step 6:

$$\cdots\frac{x^2+4x+4+2x+6}{(x+3)(x+2)^2}$$

Step 6:

$$\cdots\frac{x^2+6x+10}{(x+3)(x+2)^2}$$

Step 7:

$$\cdots\frac{1}{2(x+3)}+\frac{1}{(x+2)^2}=\frac{x^2+6x+10}{(x+3)(x+2)^2}$$

Law of indices
Multiplication of indices

$$1\cdots2\cdot2\cdot2=2^3$$

$$2\cdots2\cdot2\cdot2\cdot2=2^4$$

$$3\cdots2\cdot2\cdot2\cdot2\cdot=2^5$$

$$4\cdots2^3\cdot2^3=2^{2+3}=2^5$$

When the you are multiplying base with different indices, you add the indices raised to the common base

An example

$$1\cdots3^1\cdot3^3\cdot3^5=3^{1+3+5}=3^9$$

General formula of multiplication of indices

$$1\cdots{b^n}\cdot{b^m}=b^{n+m}$$

Home work

Add the following fraction

$$1\cdots\frac{1}{2x}+\frac{1}{x+2}$$

$$2\cdots\frac{1}{x+y}+\frac{1}{2(x+y)}$$

$$3\cdots\frac{x}{y}+\frac{1}{x+y}$$

$$4\cdots\frac{2}{x+2}-\frac{1}{x+1}$$

Solve the following indices

$$1\cdots{b^2}\cdot{b^x}$$

$$2\cdots{b^x}\cdot{c^y}$$

$$3\cdots3^a\cdot3^3\cdot3^4$$

$$4\cdots3^{\frac{1}{2}}\cdot3^{\frac{3}{4}}$$

Important arithmetic operation
$$1\cdots+3+3+3=+9$$

$$2\cdots-3-3-3=-9$$

$$3\cdots-6+9=+(9-6)=+3$$

Is plus because the 9 is greater than 6

$$4\cdots-9+6=-(9-6)=-3$$

Is minus because the 9 is greater than 6

Younger brother Lesson two : 3/2/8
1 Long addition

2 Addition of like and unlike terms

3 Addition of decimal

4 Addtion of fraction

5 Property of addition

6 Addtion of zero

7 Repeat addition relate to multiplication

Long Addition
Home work

$$1234+4351\cdots(1)$$

$$67464+5647\cdots(2)$$

$$45334+43351\cdots(3)$$

Addition of like and unlike terms
Examples

$$a+a+a=3\times{a}=3a\cdots(1)$$

$$a+a+b+b=2\times{a}+2\times{b}=2a+2b\cdots(2)$$

$$a+b=a+b\cdots(3)$$

$$x^2+3x^2=4x^2\cdots(4)$$

$$xy+xyz=xy+xyz\cdots(5)$$

Home work

$$x+y+x\cdots(1)$$

$$x+y+y+4\cdots(2)$$

$$x^2+y^2+x^2\cdots(3)$$

$$x^3+x^2\cdots(4)$$

$$(xy)^2+(yx)^2+xy\cdots(5)$$

Addition of fraction
Examples

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

$$\frac{1}{2}+\frac{3}{5}+\frac{2}{4}\cdots(2)$$

Definition of fraction:

Part over whole

$$\frac{part}{Whole}\cdots(1)$$

The part is called the numerator

The whole is called the denumerator

Addition of fraction:

When you are adding fractions you are adding the part of individual fraction together

Rule of addition of fraction states:

You can only add the part together if and only the whole of each fraction are the same

Example will clear the matter

Example 1:

$$\frac{2}{5}+\frac{1}{5}\cdots(1)$$

This fraction you can add 

Solution

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

Example 2:

$$\frac{2}{6}+\frac{1}{5}\cdots(1)$$

This fraction you can not add 

Solution

The soultion will be, you have to use basic rule of arithmetic to change both fractions to a common whole

To change 2 or more fractions to a common whole

Example 1:

$$\frac{2}{6}+\frac{1}{5}\cdots(1)$$

$$\frac{2}{6}\times{\frac{5}{5}}+\frac{1}{5}\times{\frac{6}{6}}\cdots(2)$$

$$\frac{12}{30}+\frac{6}{30}\cdots(3)$$

Now the whole are the same, just add the parts together

$$\frac{12}{30}+\frac{6}{30}=\frac{12+6}{30}=\frac{18}{30}\cdots(4)$$

Example 2:

$$\frac{1}{2}+\frac{2}{3}+\frac{4}{6}\cdots(1)$$

$$\frac{1}{2}\times{\frac{3}{3}}+\frac{2}{3} \times{\frac{2}{2}}+\frac{4}{6}\times{\frac{1}{1}}\cdots(2)$$

$$\frac{3}{6}+\frac{4}{6}+\frac{4}{6}\cdots(3)$$

Now the whole are the same, just add the parts together

$$\frac{3}{6}+\frac{4}{6}+\frac{4}{6}=\frac{3+4+4}{6}=\frac{11}{6}\cdots(4)$$

Home work
$$\frac{1}{3}+\frac{1}{6}\cdots(1)$$

$$\frac{2}{4}+\frac{5}{6}\cdots(2)$$

$$\frac{2}{4}+\frac{5}{12}+\frac{1}{3}\cdots(3)$$

$$\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\cdots(4)$$

Lesson two 3/1/8
In this lesson we will briefly cover the basic of these topics

1: Formulae, see user:hashim100 number:39

2: Trigonometry

3: The circle

4: The areas of triangle

5: Simultaneous equations, see user:hashim100

Trigonometry
We have a triangle ABC

The lenghts of the traiangle ABC will be represent by small case letter

The lenght of a is opposite angle A, b opposite angle B and c opposite angle C

The ratio a over b is defined by

$$SinA=\frac{a}{b}\cdots(1)$$

The ratio b over c is defined by

$$CosA=\frac{b}{c}\cdots(2)$$

The ratio a over b is defined by

$$TanA=\frac{a}{c}\cdots(3)$$

Application

If given angle A and lenght c, for example

We can find out all the other lenghts and angle, using the above defined ratio

If given lenght c and lenght a, for example

We can find out all the other lenghts and angle, using the above defined ratio

Sin(A) : 30 - 45 - 60 - 90
$$sin(30)=\frac{1}{2}\cdots(1)$$

$$sin(45)=\frac{1}{\sqrt{2}}\cdots(2)$$

$$sin(60)=\frac{\sqrt3}{2}\cdots(3)$$

$$sin(90)=1\cdots(4)$$

Cos(A) : 30 - 45 - 60 - 90
$$cos(30)=\frac{\sqrt{3}}{2}\cdots(1)$$

$$cos(45)=\frac{1}{\sqrt{2}}\cdots(2)$$

$$cos(60)=\frac{1}{2}\cdots(3)$$

$$cos(90)=0\cdots(4)$$

Tan(A) : 30 - 45 - 60 - 90
$$Tan(30)=\frac{1}{\sqrt{3}}\cdots(1)$$

$$Tan(45)=1\cdots(2)$$

$$Tan(60)=\sqrt{3}\cdots(3)$$

$$Tan(90)=Undefine!\cdots(4)$$

Derivation of the above angles are from the following special shapes

Equilateral triangle and a Square

Unity cirlce
Mean that the circle with the radius of 1 unit

This is very important and need to be expalin on the board

The solution and areas of traingles
The area of triangle can be determine in many, depend on the information is given

''If the base and height is given. The area of the triangle can be determine by this formula:''

$$A=\frac{b\times{h}}{2}\cdots(1)$$

''If all the sides are given. The area of the triangle can be determine by this formula:''

$$A=\sqrt{s(s-a)(s-b)(s-c)}\cdots(1)$$

Where s is called the semi-perimeter and is defined by:

$$s=\frac{a+b+c}{2}\cdots(2)$$

Angle C is between lenghts a and b. The area of the triangle can be determin by this formula:

$$A=\frac{a\times{b}\times{SinC}}{2}\cdots(1)$$

Example 1:

Base b = 4 and Height h = 5

$$A=\frac{b\times{h}}{2}$$

$$A=\frac{4\times5}{2}=\frac{20}{2}=10\cdots(1)$$

Example 2:

Lenghts a = 3, b = 4 and c = 5

$$s=\frac{a+b+c}{2}\cdots(1)$$

$$s=\frac{3+4+5}{2}=\frac{12}{2}=6\cdots(2)$$

$$A=\sqrt{s(s-a)(s-b)(s-c)}\cdots(3)$$

$$A=\sqrt{6(6-3)(6-4)(6-5)}=\sqrt{6(3)(2)(1)}=\sqrt{36}=6\cdots(4)$$

Example 3:

Angle C = 30 and lenghts a = 4 and b =5

$$A=\frac{a\times{b}\times{SinC}}{2}\cdots(1)$$

$$A=\frac{4\times{5}\times{Sin30}}{2}\cdots(2)$$

$$A=\frac{20\times{\frac{1}{2}}}{2}\cdots(3)$$

$$A=\frac{10}{2}=5\cdots(4)$$

The Circle
In the cirlce we have r, D , π , s , C , A , A(s) and Φ

r means radius

and it measured from the centre to the circumference

D means diameter and is equal to 2r

$$D=2\times{r}=2r\cdots(1)$$

π is a constant and is equal to roughly 3.14

$$\pi=3.14...$$

and is defined by

$$\pi=\frac{C}{D}\cdots(2)$$

s means the lenght of the sector

and is defined by

$$s=\theta\times{r}=\theta{r}\cdots(3)$$

C means the circumference

and is defined by

$$C=\pi\times{D}=\pi{D}\cdots(4)$$

A means the area

and is defined by

$$A=\pi\times{r^2}=\pi{r^2}\cdots(5)$$

A(s) means the area of the sector

and is defined by

$$A_s=\frac{\theta\times{r^2}}{2}=\frac{\theta{r^2}}{2}\cdots(6)$$

Φ means the angle subtended by the radii (plural of radius) or (the angle in between the radii)