Algebraic Fractions Examples (5.2, Syl MA5.2-6NA)

Algebraic Fractions Examples (5.2)


Firstly let's make the note that $a$ is the same thing as $1a$ and also $$\frac{a}{3}=\frac{1a}{3}$$

Example:

Add & Subtract algebraic fractions  =======

(1)    $${2a\over 7}+{3a\over 7}={2a+3a\over 7}$$

(2)   $${2a\over 5}+{a\over 3}={2a\times 3\over 7\times 3}+{a\times 5\over 3\times 5}={6a+5a\over 15}={11a\over 15}$$

(3)  $${3a\over 4}-{2a\over 9}={3a\times 9\over 4\times 9}-{2a\times 4\over 9\times 4}={27a-8a\over 36}={19a\over 36}$$

Improper to Mixed Numeral algebraic fractions =======

(4)  (5.3?)  $$\begin{array}{l}
{33a\over 7}\\
={(4\times 7 +5)a\over 7}\\
=4{5\over 7}a
\end{array}$$

Multiplying algebraic fractions =======

(5)  
$$ \begin{array}{l}
{2x\over 3}\times{3x\over 4}\\
={2x\times 3x\over 3\times 4}\\
={6x^2\over 12}\quad \div
\mbox{top & bottom by $6$}\\
={1x^2\over 2}\\
={x^2\over 2}
\end{array}$$

(6)   ${2a\over 7}+{3a\over 7}={2a+3a\over 7}$

Directed Numbers (Negative and positive numbers)

Multiplication of directed numbers

$$\begin{array}{rcr}
1\times 1&=&1\\
-1\times 1&=&-1\\
1\times -1&=&-1\\
-1\times -1&=&1\\
5\times -1&=&-5\\
-6\times 3&=&-18\\
-3\times -5&=&15
\end{array}$$

Addition and Subtraction of directed numbers

$$\begin{array}{rcr}
1 +1&=&2\\
-1+ 1&=&0\\
1+ -1&=&0\\
-1+ -1&=&-2\\
5+ -1&=&4\\
-5+ -1&=&-4\\
-5+1&=&-4\\
5-9&=&-4\\
-5-8&=&-13\\
-15-(-3)&=&-12
\end{array}$$
Notation wise, $a+(-b)$ is written as $a-b$.

Basic Algebra Rules : Addition, Subtraction and Multiplication

ADDITION IN ALGEBRA
$1a$ means $1$ of $a$, say, $1$ apple.

$1a$ is usually written without the $1$ as just $a$.

$2a$ means $2$ lots of $a$, sat $2$ apples.

$2\times 1a$=$2a$. This makes sense. Two times one apple gives two apples.

$2\times 3a=6a$.

$a+a=1a+1a=2a$
$2a+3a=5a$

Rule:$$ma+na=(m+n)a$$

Rule:$$ma-na=(m-n)a$$

MULTIPLICATION IN ALGEBRA

$2x\times 3y = 6xy$

just multiply everything together!

$a\times a=a^2$

$a^3\times a^4=a^{3+4}=a^7$

$mx\times n =mnx$ where $m,n$ are numbers,

FRACTIONS - Basic Operations

Addition/Subtraction

${2\over 7}+{3\over 7}={2+3\over 7}={5\over 7}$

 ${7\over 10}-{4\over 10}={7-4\over 10}={3\over 10}$

${1\over 4}+{2\over 5}$

$={1\times 5\over 4\times 5}+{2\times 4\over 5\times 4}$

$={5\over 20}+{8\over 20}$

$={13\over 20}$

Multiplication

 ${4\over 5}\times{5\over 6}={4\times 5\over 5\times 6}={20\over 30}={2\over 3}$

Division

 ${4\over 5}\div{5\over 6}=
{4\over 5}\times{6\over 5}={4\times 6\over 5\times 5}={24\over 25}$

'Over'
${{2\over 5}\over 6} ={2\over 5}\div 6={2\over 5} \times {1\over 6}={2\over 30}={1\over 15}$

${2\over {5\over 6}} =2\div {5\over 6}=2\times {6\over 5}={12\over 5}=2 {2\over 5}$

${{a\over b}\over c} ={a\over bc}$

${a\over {b\over c}}={ac\over b}$

COORDINATE GEOMETRY

Consider the two points $$A(x_1, y_1) \quad\mbox{and}\quad B(x_2, y_2)$$
The distance between the points $A$ and $B$ is $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\quad\quad\quad \mbox{DISTANCE}$$ 
The coordinates of the midpoint of the interval joining the points $A$ and $B$ are $$\left( {x_1+x_2 \over 2 }, {y_1+y_2 \over 2}\right)\quad\quad\quad \mbox{MIDPOINT}$$ 
The slope or gradient of the line through the points $A$ and $B$ is $$m={y_2-y_1 \over x_2-x_1 }={ \mbox{rise}\over \mbox{run}}\quad\quad\quad \mbox{GRADIENT}$$ 
The GRADIENT-INTERCEPT form of the equation of a line is $$y=mx+b$$ where $m$ is the gradient or slope of the line, and $b$ is the $y$-intercept. 

The POINT-GRADIENT form of a line is $$y-y_1=m(x-x_1)$$ where $m$ is the gradient of the line, and $(x_1,y_1)$ is ANY point on the line. 

The GENERAL FORM OF A LINE is $$ax+by+c=0$$ where $a,b,c$ are constants.The slopes or gradients of parallel lines are equal, that is, $m_1=m_2$.

The slopes of perpendicular lines are negative reciprocals of each other. This can be written as $$m_2=-{1\over m_1}\quad\quad\mbox{or}\quad\quad m_1\times m_2=-1$$ 
The $x$-intercept is found by putting $y=0$ and solving for $x$ in the equation of the line. The $y$-intercept is found by putting $x=0$ and solving for $y$ in the equation of the line.

VERTICAL LINES have the form, $$x=k$$ where $k$ is a constant (number).

HORIZONTAL LINES have the form, $$y=k$$ where $k$ is a constant (number). }

SURFACE AREA AND VOLUME

A PRISM is a solid whose cross sectional area is the same all along its length, and all its sides are flat.

In a RIGHT PRISM the edges of the sides are perpendicular to the base. If the sides are not perpendicular to the base it is an OBLIQUE PRISM.

A COMPOSITE SOLID is one which is made up of two or more solids (e.e. a cube and a hemisphere stuck together).
The APEX of a pyramid or cone is the very top point (vertex).

A RIGHT PYRAMID is one which has its apex aligned directly above the center of the base. The surface area of a pyramid comprises the area of the sides plus the base (unless stated otherwise).
The surface area of a closed (right) cylinder is $$SA=2\pi r^2+2\pi rh\quad\quad\quad \mbox{CLOSED CYLINDER}$$ where $h=$height, $r=$radius of the base.

A RIGHT CONE is one which has its apex aligned directly above the center of the base. The surface area of a right cone is $$SA=\pi r l+\pi r^2\quad\quad\quad \mbox{RIGHT CONE}$$ where $l=$slant height, $r=$radius of the base.

A SPHERE is a surface whose shape is the outside of a ball.\\ The surface area of a sphere of radius $r$ is $$SA=4\pi r^2\quad\quad\quad \mbox{SPHERE}$$
The surface area of composite solids involving a sphere and other solids excludes common areas (unless stated otherwise)\\ The volume of a right prism is $$V=Ah \quad\quad\quad \mbox{RIGHT PRISM}$$ where $h=$height of the prism, and $A=$cross-sectional area.

The volume of a cylinder is $$V= \pi r^2 h\quad\quad\quad \mbox{CYLINDER}$$ (really same as $V=Ah$ where $A=\pi r^2$ here.)

The volume of a sphere of radius $r$ is $$V={4\pi r^3 \over 3 }\quad\quad\quad \mbox{SPHERE}$$
The volume of a pyramid is $$V={1\over 3} Ah\quad\quad\quad \mbox{PYRAMID}$$ where $h=$perpendicular height, $A=$area of base.

The volume of a cone is $$V={1\over 3} \pi r^2 h\quad\quad\quad \mbox{CONE}$$ (really same as ${1\over 3}Ah$ where $A=\pi r^2$ here.)

Volume of composite solids is the sum of its volume parts. Conversion between volume and capacity: $$1 m^3=1 kL $$  $$ 1 cm^3=1 mL $$ 

AREAS OF SIMILAR FIGURES ARE RELATED:
If the matching sides of two similar figures are in the ratio $m:n$ then the areas are in the ratio $m^2:n^2$.
So if you double the sides of a figure its area becomes $4$ (since $2^2=4$) times bigger.

VOLUMES OF SIMILAR FIGURES ARE RELATED:
If the matching sides of two similar figures are in the ratio $m:n$ then (1) the ratio of their surface areas is $m^2:n^2$.
(2) the ratio of their volumes is $m^3:n^3$. So if you double the sides of a figure its volume becomes $8$ (since $2^3=8$) times bigger.

PYTHAGORAS' THEOREM applies to right angled triangles: $$a^2+b^2=c^2$$ where the longer side is $c$. This is useful when finding some surface areas and volumes for pyramids where right angled-triangles can occur. }