Elementary Math

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 + b2

-

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 90o

Obtuse Angles: angles between 90o and 190o

Obtuse Angles: angles between 180o and 360o

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

Angle properties

No.

1

2

3

4

5

6

7

8

9

10

Property

Angles on a straight line

Angles at a point

Vertically opposite angles

Angles formed by parallel lines

Angles formed by parallel lines

Angles formed by parallel lines

Angle properties of triangles

Angle properties of triangles

Angle properties of polygons

Angle properties of polygons

Explanation

Example

    • 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

Angles at a point add up to 360o

Vertically opposite angles are equal

Alternate interior angles are equal

Alternate exterior angles are equal

Corresponding angles are equal

The sum of angles in a triangle adds up to 180o

The sum of 2 interior opposite angles is equal to the exterior angle

    • 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

    • 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

Circle

Sector

A=b x h

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

Cylinder

Sphere

Prism

Pyramid

Cone

V = l x b x h

SA = 2bl + 2hb + 2hl

V = base area x height

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 360o

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