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Important Notes and Formulas

Numbers

Test of Divisibility

Standard form

This is a convenient way to write very large or very small numbers, using the from a x 10ⁿ, where n is a positive or negative integer, and a s between 1 to 10 inclusive.

An example:

More examples:
123 400 written as standard form is 1.234 x 10⁵
0.0000987 written as standard form is 9.87 x 10^-5

Multiplying numbers in standard form

Dividing numbers in standard form

Adding and Subtracting numbers in standard form

- Make the index between the 2 numbers the same so that it is easier to factorise the numbers before adding
eg

Scales and Maps

Given that a map has a scale of 1:10 000, this means that 1cm on the map represents 10,000cm on the actual ground.

1cm : 200m = 1cm : 0.2km = 1cm² : 0.04km²

Proportion

A. Direct Proportion

This means that when y increases, x increases, and vice versa.

Use this equation: y = kx

B. Indirect Proportion

This means that when y increases, x  decreases, and vice versa.

Use this equation: y=k/x

Percentage Change

Percentage Profit and Loss

Simple Interest and Compound Interest

A. Simple Interest Formula

B. Compound Interest Formula

C. Compound interest compounded MONTHLY

Formula:
S = P(1 + r/k)ⁿ

S = final value 
P = principal
r = interest rate (expressed as decimal eg 4% = 0.04)
k = number of compounding periods 

Note: 

• if compounded monthly, number of periods = 12
• if compounded quarterly, number of periods = 4

Example:

If $4000 is invested at an annual rate of 6.0% compounded monthly, what will be the final value of the investment after 10 years?

Since the interest is compounded monthly, there are 12 periods per year, so, k = 12.
Since the investment is for 10 years, or 120 months, there are 120 investment periods, so, n = 120.

S = P(1 + r/k)ⁿ

S = 4000(1 + 0.06/12)¹²⁰
S = 4000(1.005)¹²⁰
S = 4000(1.819396734)
S = $7277.59

Coordinate Geometry Formulas

From: http://www.dummies.com/how-to/content/coordinate-geometry-formulas.html

Algebraic Manipulation

ax + bx = x(a+b)

ax + bx + kay + kby = x(a+b) + ky(a+b) = (a+b)(x+ky)

(a+b)² = a² + 2ab + b²

(a-b)² = a² - 2ab + b^2
-
a² - b² = (a + b)(a - b)

Solving algebraic fractional equations

Avoid these common mistakes!

Solution of Quadratic Equations

Completing the Square

Step 1: Take the number or coefficient before x and square it
Step 2: Divide the square of the number by 4

Eg. y = x² + 6x - 11

y = x² + 2x(6/2) + (6/2)² - 11 - (6/2)²

y = (x + 3)² - 20

Sketching Graphs of Quadratic Equations

A. eg. y= +/-(x - h)² + k

Steps
1. Identify shape of curve 

• look at sign in front of(x - h) to determine if it is "smiley face" or "sad face".

2. Find turning point 

• (h, -k)

3. Find y-intercept 

• sub x = 0 into the equation --> (0, y)

4. Line of symmetry reflect

• x = h, reflect to get (2x, y)

B. eg. y = +/-(x - a)(x - b)


Steps
1. Identify shape of curve 

• look at the formula ax2 + bx + c.
• if a>1, it is positive; otherwise, it is negative

2. Find turning point 

• (a + b)/2, sub answer into equation --> (a,b)

3. Find y-intercept 

• sub x = 0 into the equation --> (0, y)

4. Line of symmetry reflect

• x = a, reflect to get (2a, y)

Inequalities

Ways to solve equalities: 

1. Add or subtract numbers from each side of the inequality
eg 10 - 3 < x - 3

2. Multiply or divide numbers from each side of the inequality by a constant
eg 10/3 < x/3

3. Multiply or divide by a negative number AND REVERSE THE INEQUALITY SIGNS
eg. 10 < x  becomes 10/-3 > x/-3

Example

Geometrical terms and relationships

Parallel Lines

Perpendicular Lines

Right Angle

Acute Angles: angles less than 90^o

Obtuse Angles: angles between 90^o and 190^o

Obtuse Angles: angles between 180^o and 360^o

^Polygons

Polygon: a closed figure made by joining line segments, where each line segment intersects exactly 2 others

Irregular polygon: all its sides and all its angles are not the same
Regular Polygon: all its sides and all its angles are the same

The sum of angles in a polygon with n sides, where n is 3 or more, is

Name of Polygons

Triangles

Quadrilaterals

Similar Plane Figures

Figures are similar only if

• their corresponding sides are proportional
• their corresponding angles are equal

Similar Solid Figures

Solids are similar if their corresponding linear dimensions are proportional.

Congruent Figures

Congruent figures are exactly the same size and shape.

2 triangles are congruent if they satisfy any of the following:

a. SSS property: All 3 sides of one triangle are equal to the corresponding sides of the other triangle.

b. SAS property: 2 given sides and a given angle of one triangle are equal to the corresponding sides and angle of the other triangle.

c. AAS property: 2 given angles and a given side of one triangle are equal to the corresponding angles and side of the other triangle.

d. RHS property: The hypothenuse and a given side of a right-angled triangle are equal to the hypothenuse and the corresponding side of the other right-angled triangle.

Bearings

A bearing is an angle, measured clockwise from the north direction.

Symmetry

Shape	Number of lines of symmetry	Order of rotational symmetry	Centre of point symmetry
Equilateral triangle	3	3	Yes
Isosceles triangle	1	1	None
Square	4	4	Yes
Rectangle	2	2	Yes
Kite	1	1	None
Isosceles trapezium	1	1	None
Parallelogram	0	2	Yes
Rhombus	2	2	Yes
Regular pentagon	5	5	Yes
Regular hexagon	6	6	Yes

Angle properties

No.	Property	Explanation	Example
1	Angles on a straight line	Angles on a straight line add up to 180o 2 angles are complementary is they add up to 90o 2 angles are called supplementary if they add up to 180o
2	Angles at a point	Angles at a point add up to 360o
3	Vertically opposite angles	Vertically opposite angles are equal
4	Angles formed by parallel lines	Alternate interior angles are equal
5	Angles formed by parallel lines	Alternate exterior angles are equal
6	Angles formed by parallel lines	Corresponding angles are equal
7	Angle properties of triangles	The sum of angles in a triangle adds up to 180o
8	Angle properties of triangles	The sum of 2 interior opposite angles is equal to the exterior angle
9	Angle properties of polygons	sum of interior angles of an n-sided polygon = (n-2) x 180o each interior angle of a regular n-sided polygon = (n-2) x 180o / n
10	Angle properties of polygons	sum of exterior angles of an n-sided polygon is 360o each exterior angle of a regular n-sided polygon = 360o / n

Angle Properties of Circles

Mensuration

All the mensuration formulas you'll ever need can by found here...
http://oscience.info/math-formulas/mensuration-formulas/

But here's a quick reference for the important ones...

Area of Figures

Triangle
Trapezium
Parallelogram		A=b x h
Circle
Sector

Radian Measure

• Radian is another common unit to measure angles.
• A radian is a measure of the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
• To convert radians to degrees and vice versa, use these formulas:
• π rad = 180º
• 1 rad = 180º/π
• 1º = π/180 rad

Volume of Figures

Cube
Cuboid		V = l x b x h SA = 2bl + 2hb + 2hl
Cylinder
Sphere
Prism		V = base area x height
Pyramid
Cone

Trigonometry

Pythagora's theorem

Trigonometrical Ratio

SINE RULE

To find an angle, can write as follows:

COSINE RULE

Area of Triangle

Statistics

Mean

Mode

The mode is the most frequent value.

Median

The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude (in increasing order).

If you have n numbers in a group, the median in:

Types of Chart

1. Bar chart: the heights of the bars represent the frequency. The data is discrete.

2. Pie chart: the angles formed by each part adds up to 360^o

3. Histogram: it is a vertical bar graph with no gaps between the bars. The area of each bar is proportional to the frequency it represents.

4. Stem-and-leaf diagram: a diagram that summarises while maintaining the individual data point. The stem is a column of the unique elements of data after removing the last digit. The final digits (leaves) of each column are then placed in a row next to the appropriate column and sorted in numerical order.

5. Simple frequency distribution and frequency polygons: a plot of the cumulative frequency against the upper class boundary with the points joined by line segments.

6. Quartiles

Probability

Probability is the likelihood of an event happening

• The probability that a certain event happening is 1
• The probability that a certain event cannot happen is 0
• The probability that a certain event not happening is 1 minus he probability that it will happen

2 events are independent if the outcome of one of the events does not affect the outcome of another
2 events are dependent if the outcome of one of the events depends on the outcome of another

• If 2 events A and B are independent of each other, then the probability of both A and B occurring is found by P(A) x P(B)
• If it is impossible for both events A and B to occur, then the probability of A or B occurring is P(A) and P(B)

Set Notation