FRACTIONS - Basic Operations

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

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

${4\over 5}\times{5\over 6}={4\times 5\over 5\times 6}={20\over 30}={2\over 3}$
${4\over 5}\div{5\over 6}={4\over 5}\times{6\over 5}={4\times 6\over 5\times 5}={24\over 25}$

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

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

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: $1 m^2=1 kL, 1 cm^2=1 mL,1 L =1000 mL$

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 $m^2:n^2$. 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. }

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). }